% =========================================================================
% THE RESTRICTED GEOMETRIC LANGLANDS EQUIVALENCE IN POSITIVE CHARACTERISTIC
% =========================================================================
%
% This is a self-contained rewrite: same theorem as v1, tighter narrative,
% clearer routing of the three regimes and their inputs. Read the "Roadmap"
% at the end of Section 1 before diving into Sections 2-7.

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\DeclareMathOperator{\Spec}{Spec}
\DeclareMathOperator{\Rep}{Rep}
\DeclareMathOperator{\QCoh}{QCoh}
\DeclareMathOperator{\IndCoh}{IndCoh}
\DeclareMathOperator{\Shv}{Shv}
\DeclareMathOperator{\Nilp}{Nilp}
\DeclareMathOperator{\QLisse}{QLisse}
\DeclareMathOperator{\IndLisse}{IndLisse}
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\title{The Restricted Geometric Langlands Equivalence in Positive Characteristic}


\begin{document}
\maketitle

\begin{abstract}
We prove the $\overline{\mathbb Q}_\ell$-linear restricted geometric Langlands
equivalence for a smooth projective connected curve and a connected reductive
group over a field of characteristic $p>0$, with $\ell\ne p$, in two regimes.

Over $\overline{\mathbb F}_q$, we assume only the four characteristic
conditions of the restricted theory (Hypotheses~\ref{hyp:common}
and~\ref{hyp:A}). Over an arbitrary algebraically closed field, we assume
these conditions plus that $p$ is very good and does not divide the order
of the Weyl group (Hypothesis~\ref{hyp:B}). In both regimes we prove
Gaitsgory--Raskin's full-support conjecture
\cite[Conj.~1.3.10]{GR}: the primed equivalence is already an equivalence
onto the entire spectral stack.

We then record three consequences: the local category on each spectral
component is nonzero and corresponds to its own summand of automorphic
sheaves; in the finite-field regime the arithmetic trace formula holds
with the full derived Frobenius fixed-point stack; and every parameter
of the spectral stack admits a nonzero Hecke eigenobject. A final section
handles rational coefficients by coefficient comparison.
\end{abstract}

\tableofcontents

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

\subsection{The question}

Geometric Langlands predicts that automorphic objects on the moduli
$\Bun_G(X)$ of $G$-bundles on a smooth projective curve $X$ are matched
with spectral objects depending on local systems on $X$ for the dual
group $\check G$. The modern formulation compares whole sheaf categories,
and this comparison is organised by the stack of local systems itself:
each point of the spectral stack should determine a summand of the
automorphic category, and this pinning-down should be coherent in families.

Over characteristic zero this is a theorem, established in the five-part
project of Arinkin, Beraldo, Campbell, Chen, Faergeman, Gaitsgory, Lin,
Raskin, and Rozenblyum \cite{GLCI,GLCV}. Its input is a comparison between
$\ell$-adic sheaves on $\Bun_G$ and ind-coherent sheaves on the
\emph{stack of local systems with restricted variation} introduced by
Arinkin--Gaitsgory--Kazhdan--Raskin--Rozenblyum--Varshavsky \cite{AGKRRV}.
Positive characteristic is different: the theory has additional
arithmetic inputs, and the comparison one can construct naturally
only reaches an \emph{open-and-closed union of spectral components}. The
question left open by \cite{GR} is whether any component can be missing.
We prove that none is, in the two regimes stated in the abstract.

Our proof is geometric. A missing component $C$ would give a nonzero
constructible object $F$ on $\Bun_G$ over the geometric generic fibre
of a lift $\mathscr X/S$ of $X$ to the Witt trait, satisfying $\Psi(F)=0$
for geometric nearby cycles $\Psi$. We then construct a small, finite
slice transverse to the support of $F$ on which $\Psi(F)$ cannot vanish,
provided the slice suitably avoids the nilpotent cone. That contradiction
finishes the proof.

\subsection{The spectral stack}

Let $k$ be an algebraically closed field of characteristic $p>0$,
let $\ell\ne p$ be a prime, and set $E=\overline{\mathbb Q}_\ell$. Let
$X/k$ be a smooth projective connected curve and $G/k$ a connected
reductive group; such a group is split over $k$. Write
$\check G/E$ for the Langlands dual group and
$\mathfrak g=\Lie(G)$.

We work with the $E$-linear ind-constructible sheaf categories of
\cite{AGKRRV}. In particular $\Shv(X)$ is ind-constructible, and
$\QLisse_E(X)\subset\Shv(X)$ is the full subcategory whose ordinary
cohomology sheaves are filtered colimits of finite-rank lisse
$E$-sheaves \cite[Def.~1.2.6]{AGKRRV}. The restricted spectral prestack
$Y=LS^{\mathrm{restr}}_{\check G}(X)$ assigns to a derived affine
$E$-scheme $T$ the groupoid
\begin{equation}\label{eq:Y-def}
 Y(T)=\Fun^{\otimes,\mathrm{r.t.ex}}\!\bigl(\Rep_E(\check G),\,
   \QCoh(T)\otimes_E\QLisse_E(X)\bigr),
\end{equation}
of right-$t$-exact symmetric monoidal functors. An \emph{$E$-parameter}
$\sigma\in Y(E)$ is such a functor with $T=\Spec E$: concretely, an
assignment of a lisse sheaf $\sigma(V)$ to each finite-dimensional
representation of $\check G$, monoidal in $V$.

The reason for the restricted variation in \eqref{eq:Y-def} is the same
as in characteristic zero. Ind-coherent sheaves alone with no support
condition do not capture the Arthur-type data that automorphic sheaves
must carry. On the automorphic side one restricts to sheaves with
nilpotent singular support, i.e.\ those whose wavefront set is contained
in the global nilpotent cone $\mathcal N\subset T^*\Bun_G$; write this
subcategory $\Shv_{\Nilp}(\Bun_G(X))$ (see \S\ref{sec:intro-cat} below
for the precise definition). On the spectral side one imposes an
analogous condition and uses $\IndCoh_{\Nilp}(Y)$. These categories
already exist in \cite{AGKRRV}; our new assertion is about the
spectral components on which they are supported.

Both sides carry a coherent action of $\QCoh(Y)$, coming from the
universal family of local systems on $Y\times X$ and the construction
of Hecke operators. This makes the comparison functor
$\QCoh(Y)$-linear, and lets us localise to individual components.

\subsection{Two characteristic regimes and the four standing hypotheses}

All four hypotheses below are already required by the construction of
$\Shv_{\Nilp}$ in \cite{AGKRRV}. They hold for $G=\mathrm{GL}_n$ in
any characteristic and for many other groups; each captures a way in
which positive characteristic can complicate Lie theory.

\begin{hypothesis}[Common characteristic assumptions]\label{hyp:common}
The following hold for $G/k$.
\begin{enumerate}[label=\textup{(\roman*)}]
\item There is a nondegenerate $G$-invariant symmetric bilinear form
  $\kappa$ on $\mathfrak g$ whose restriction to $\Lie Z(M)$ is
  nondegenerate for every Levi subgroup $M$.
\item Chevalley restriction $k[\Lie M]^M\to k[\Lie T_M]^{W_M}$ is an
  isomorphism for every Levi $M$ and every maximal torus $T_M\subset M$.
\item The scheme-theoretic centraliser in $G$ of every semisimple
  element of $\mathfrak g$ is a Levi subgroup.
\item Every nilpotent element of $\mathfrak g$ lies in $\Lie R_u(P)$ for
  some parabolic $P\subset G$, after any field extension.
\end{enumerate}
\end{hypothesis}

Conditions (i)--(iii) are \cite[\S14.4.1]{AGKRRV}; condition (iv) is
\cite[\S D.1.1]{AGKRRV}. Both hypotheses below keep the notion of
semisimple and nilpotent as in \cite{AGKRRV}: semisimple elements of
$\mathfrak g$ are exactly the ones geometrically conjugate into a
Cartan subalgebra, nilpotent ones the ones whose adjoint orbit closure
contains $0$.

\begin{hypothesis}[Regime A: finite-field descent]\label{hyp:A}
There are $q=p^r$, a smooth projective geometrically connected curve
$X_0/\mathbb F_q$, and a split connected reductive group
$G_0/\mathbb F_q$ with $k=\overline{\mathbb F}_q$,
$X=X_0\otimes_{\mathbb F_q}k$, $G=G_0\otimes_{\mathbb F_q}k$.
Hypothesis~\ref{hyp:common} holds. No restriction on $|W_G|$ is imposed.
\end{hypothesis}

\begin{hypothesis}[Regime B: arbitrary algebraically closed base]
\label{hyp:B}
$k$ is any algebraically closed field of characteristic $p>0$.
Hypothesis~\ref{hyp:common} holds, $p$ is very good for $G$, and
$p\nmid|W_G|$. No finite-field descent is assumed.
\end{hypothesis}

``Very good'' is the usual root-system condition. For type $A_n$ it
means $p\nmid n+1$; for $B_n,C_n,D_n$ it means $p\ne2$; for
$G_2,F_4,E_6,E_7$ that $p\ne2,3$; for $E_8$ that $p\ne2,3,5$; applied on
each irreducible factor. A torus has no such restriction, and every
prime is very good for a torus.

The two regimes differ only in one Lie-theoretic input
(Section~\ref{sec:lie}): how to produce a central cocharacter with a
nonvanishing pairing with the constant term of a Higgs field. Regime A
solves this with a lattice argument that uses only $\kappa$ and
Hypothesis~\ref{hyp:common}(i)-(iii). Regime B solves it with the
separately assumed invertibility of $|W_G|$. Neither is used as a
substitute for the other; both produce the identical statement of
Proposition~\ref{lie:input}, and everything downstream is common to
the two regimes.

\subsection{Nilpotent supports and the relevant categories}
\label{sec:intro-cat}

The subcategories we use are the standard ones from \cite{AGKRRV}, but
it is worth pinning them down once, because several comparisons later
depend on the precise definition.

On the automorphic side, the \emph{global nilpotent cone}
$\mathcal N\subset T^*\Bun_G(X)$ consists of Higgs pairs $(P,A)$ with
$A\in H^0(X,\Ad P\otimes\omega_X)$ pointwise nilpotent. A bounded
constructible $E$-sheaf on $\Bun_G(X)$ has nilpotent singular support
if its wavefront set lies in $\mathcal N$; the full subcategory
$\Shv_{\Nilp}(\Bun_G(X))\subset\Shv(\Bun_G(X))$ consists of the
ind-objects whose every cohomology sheaf has this property. This
definition uses Beilinson's theory of singular support for constructible
sheaves in arbitrary characteristic \cite{BeilinsonSingularSupport};
it is not the coherent singular-support theory of \cite{ArinkinGaitsgory},
which lives on the spectral side.

On the spectral side, ind-coherent sheaves on $Y$ with nilpotent singular
support, denoted $\IndCoh_{\Nilp}(Y)$, are defined in \cite[\S\S21.2.1--21.2.5]{AGKRRV}.
The nilpotent condition there is a condition on a horizontal section of
the adjoint local system, matching Laumon's prediction. This condition,
its correspondence to Laumon's conjecture for Hecke eigensheaves, and the
compatibility with geometric Satake are all proved in \cite{AGKRRV}.

The two categories play different roles: the automorphic nilpotent cone is
geometric, the spectral nilpotent support is Arthur-theoretic. The
correspondence we prove matches them.

\subsection{The primed equivalence and full support}

By \cite[Main Thm.~1.3.9]{GR}, in either regime there is a
$\QCoh(Y)$-linear equivalence of the form
\begin{equation}\label{eq:primed}
 \mathbb L_G^{\mathrm{restr}}\colon
 \Shv_{\Nilp}(\Bun_G(X))
 \xrightarrow{\ \sim\ }\IndCoh_{\Nilp}(Y')
 \hookrightarrow\IndCoh_{\Nilp}(Y),
\end{equation}
where $Y'\subset Y$ is an open-and-closed union of connected components
of $Y$. Equivalently, it is a $\QCoh(Y)$-linear equivalence onto the
direct summand $\IndCoh_{\Nilp}(Y')$ of $\IndCoh_{\Nilp}(Y)$. The
inclusion is the natural fully faithful functor, and its image is a
summand.

What is missing from \eqref{eq:primed} is the assertion $Y'=Y$: \emph{all}
components, not just some, support a nonzero summand of automorphic
sheaves. This is the content of the following theorem.

\begin{theorem}[Full support]\label{thm:full-support}
Under either Hypothesis~\ref{hyp:A} or Hypothesis~\ref{hyp:B}, the
inclusion $Y'\subset Y$ in \eqref{eq:primed} is equality of spectral
prestacks, and the natural functor
\[
 \mathbb L_G^{\mathrm{restr}}\colon
 \Shv_{\Nilp}(\Bun_G(X))\xrightarrow{\ \sim\ }
 \IndCoh_{\Nilp}(Y)
\]
is a $\QCoh(Y)$-linear equivalence.
\end{theorem}

The primed equivalence used as input comes from \cite[Main Thm.~1.3.9(i)]{GR};
the full characteristic-zero equivalence and the case of $\mathrm{GL}_n$
are also established there. Theorem~\ref{thm:full-support} upgrades the
positive-characteristic case from a summand to the whole category, and
in so doing proves \cite[Conj.~1.3.10]{GR} in the two regimes.

\subsection{Consequences and rational coefficients}

Three consequences are developed in Section~\ref{sec:consequences}:
\begin{enumerate}[label=(\alph*)]
\item localising at any one component $C\subset Y$ gives a nonzero
  automorphic category $\mathcal C_{s,C}$ equivalent to
  $\IndCoh_{\Nilp}(C)$;
\item in regime A, the arithmetic trace formula holds with the full
  derived fixed-point stack $Y^{\Frob}$ rather than the primed
  $(Y')^{\Frob}$;
\item every $E$-parameter $\sigma\in Y(E)$ admits a nonzero Hecke
  eigenobject with coherent tensor and fusion structure.
\end{enumerate}
None of these requires any additional hypothesis beyond
Theorem~\ref{thm:full-support}.

Section~\ref{sec:coefficients} proves the same support statement for
rational coefficients $\mathbb Q_\ell$. The theorem there is conditional
on a single input: the existence of the primed trace formula over
$\mathbb Q_\ell$ itself, intertwining the excursion and spherical Hecke
actions with the Frobenius and Satake conventions. We do not construct
that map. What we prove is that once it is given, the primed formula
unprimes without any further hypothesis, at the level of the whole
spectral prestack.

\subsection{Roadmap to the proof}

The rest of the paper is organised so that each section proves one point.

\smallskip\noindent
\emph{Section~\ref{sec:specialization}} (specialisation). We lift
$(X,G)$ to $(\mathscr X/S,G/S)$ over the Witt trait, write
$\mathcal B=\Bun_G(\mathscr X/S)$ and set up geometric specialisation
$\Psi$ from the geometric generic fibre to the closed fibre. We show
that a missing component $C$ produces a nonzero compact $F$ on
$\mathcal B_K$ with $\Psi(F)=0$, and that this vanishing persists
after pulling back along smooth maps $a\colon T\to\mathcal B$ and after
applying any Hecke operator $\mathrm H_V$. Crucially, the leg direction
of the Hecke map is arbitrary; only the projection to $\mathcal B$ must
be smooth. This is the freedom later needed to cut Hecke correspondences
by a hypersurface.

\smallskip\noindent
\emph{Section~\ref{sec:lie}} (Lie-theoretic reduction). Starting from a
non-nilpotent Higgs field $A$ at some point $x$ on the curve, we find a
Levi subgroup $M\subset G$, a cocharacter $\lambda$ of its centre, and a
change of formal frame, such that $A$ lies in $\mathfrak m[[t]]$,
$\kappa(A,d\lambda)\ne0$, and $\ad_{A}$ is invertible on
$\mathfrak g/\mathfrak m$. This is the input for constructing a Hecke
modification whose moving-point covector does not vanish.

\smallskip\noindent
\emph{Section~\ref{sec:hecke}} (Hecke modification). Applying the above
modification to a fixed bundle, we get a nonzero leg covector $\xi$ and
two relative cotangent sections (fixed-input and fixed-output) whose
derivatives are isomorphisms on the two Hecke fibres. This is the
tranversality input.

\smallskip\noindent
\emph{Section~\ref{sec:twohecke}} (two Hecke correspondences). Composing
two Hecke correspondences with common output and common leg, and cutting
by a hypersurface dual to $\xi$, we isolate one stalk of $\Psi(F)$ and
identify its local neighbourhood with a split quadratic equation
$\sum_{i=1}^m u_iv_i+f=0$.

\smallskip\noindent
\emph{Section~\ref{sec:slices}} (from quadric to slices). A proper
projective compactification of that quadric shows that the derived
restriction to the hypersurface is a constant Tate line; vanishing on
the quadric therefore forces vanishing on the hypersurface. A projective
incidence construction then handles higher-codimension transverse slices.

\smallskip\noindent
\emph{Section~\ref{sec:support}} (contradiction). The support of
$\Psi(F)$ can be cut down to a finite scheme over the trait, on which
nearby cycles are a finite direct sum of stalks. Since the summands
cannot all vanish (the object is nonzero), we get a slice on which
$\Psi$ does not vanish, contradicting the transverse test. Since the
nilpotent cone has dimension at most $\dim\mathcal B_s$, a slice of
sufficiently high codimension always exists. This finishes
Theorem~\ref{thm:full-support}.

\smallskip\noindent
\emph{Sections~\ref{sec:consequences}--\ref{sec:coefficients}} apply the
theorem to the previously listed consequences and prove the rational
coefficient statement by coefficient comparison.

% =========================================================================
\section{Specialisation and a missing spectral component}
\label{sec:specialization}
% =========================================================================

Throughout this section we assume Hypothesis~\ref{hyp:A} or
Hypothesis~\ref{hyp:B}; we never assume Theorem~\ref{thm:full-support}
itself. The goal is a single statement: if $Y'\ne Y$, then there exists
a nonzero sheaf $F$ on the geometric generic fibre for which all the
nearby-cycle vanishings we need later do actually hold. The
construction has three parts. First we place everything over the Witt
trait. Second we extract such an $F$ from the missing component.
Third we propagate its vanishing along smooth pullbacks and Hecke
transforms.

\subsection{The Witt trait and geometric specialisation}

Let
\[
 R=W(k),\qquad S=\Spec(R),\qquad K=\overline{\operatorname{Frac}(R)},
\]
where $W(k)$ is the ring of Witt vectors of $k$. Since $k$ is perfect,
$R$ is a complete Noetherian discrete valuation ring with uniformizer
$p$ and residue field $k$. It is strictly henselian, and it is
excellent because it is a complete Noetherian local ring
\cite[Tag~07QS]{Stacks}.

The curve $X/k$ admits a smooth projective lift $\mathscr X/S$.
Concretely, smooth-lifting theory gives a smooth formal lift over
$\operatorname{Spf}R$, since the obstruction to lifting a smooth curve
lies in $H^2(X,T_X)=0$; the ample line bundle $O_X(1)$ lifts for the
same reason ($H^2(X,\mathcal O_X)=0$); Grothendieck's existence theorem
algebraises the resulting formal data to a projective lift. The lift
is smooth near the closed fibre, and the nonsmooth locus is a closed
subset of a proper $S$-scheme; it cannot meet the closed fibre without
meeting it as a horizontal divisor, so it is empty. See
\cite[Tags~0DZQ,~0E7R]{Stacks}; the same lift is used in
\cite[\S3.1.1]{GR}. This construction does not require that $X$ descends
to a finite field.

The root datum of the split $G$ gives a split reductive group over
$\mathbb Z$, and base change to $S$ gives a lift, which we also denote
$G$ \cite[Thms.~6.1.16(2) and~6.1.17]{Conrad}. Write
\[
 \mathcal B=\Bun_G(\mathscr X/S),\qquad
 s=\Spec(k),\qquad \eta=\Spec(K).
\]
Then $\mathcal B=\Hom_S(\mathscr X,B_SG)$ is an algebraic stack locally
of finite type over $S$ with affine diagonal
\cite[Thm.~1.2(i)--(iii)]{HallRydh}: $\mathscr X/S$ is proper, flat and
of finite presentation, and $B_SG$ is locally of finite presentation
and quasi-separated, with affine stabilisers and affine diagonal. The
obstruction to deforming a bundle on a curve lies in $H^2(X,\Ad P)=0$,
so $\mathcal B$ is smooth over $S$.

Later we will replace $R$ by its integral closure in a finite extension
of $\operatorname{Frac}(R)$ inside $K$. Such an integral closure is a
finite complete discrete valuation ring with residue field $k$, still
excellent and strictly henselian \cite[Tags~0EXQ,~0323]{Stacks}. All
subsequent models are understood to have been replaced in this way,
and we continue to use $R,S,\mathscr X,\mathcal B$. The point is that
every dimension argument below takes place over a Noetherian trait
which is not the integral closure in the whole of $K$.

For a finite-type $S$-scheme or algebraic space $T$, geometric
specialisation with monodromy forgotten is a functor
\[
 \Psi_T\colon \Shv(T_K)\longrightarrow\Shv(T_s).
\]
Here $\Shv$ always means the ind-completion of bounded constructible
$E$-adic complexes, using ordinary $*$-pullback, ordinary tensor and
ordinary stalks. It extends to locally finite-type algebraic stacks by
smooth descent, and $\Psi_T$ is defined on the whole sheaf category.
The construction is as in \cite[\S\S4.1.1--4.1.5]{GR}: at finite
coefficients and finite extensions of the base field one takes
geometric nearby cycles; one passes through the compatible coefficient
limits, inverts $\ell$, extends coefficients to $E$, and extends
continuously from constructibles. The result is exact, continuous,
preserves constructibility, and commutes with smooth pullback and
representable proper pushforward \cite[\S4.1.6]{GR}. These properties
are unchanged by the finite trait replacements above; the coefficient
functors and the models at successive stages agree on the identified
geometric-fibre coefficients.

We will use one tensor identity repeatedly. If $a\colon T\to\mathcal B$
is smooth and $\mathcal L$ is a finite-rank lisse sheaf on $T$, then
for every $D\in\Shv(\mathcal B_K)$
\begin{equation}\label{eq:spec-lisse-tensor}
 \Psi_T\bigl(a_K^*D\otimes\mathcal L_K\bigr)
 \simeq a_s^*\Psi_{\mathcal B}(D)\otimes\mathcal L_s.
\end{equation}
A lisse sheaf on the source of a smooth morphism is universally locally
acyclic over the target; \eqref{eq:spec-lisse-tensor} is exactly the
universally-locally-acyclic tensor-pullback formula
\cite[Lemma~4.1.9]{GR}. Continuity makes it applicable to arbitrary
$D$ in the indicated ind-category.

\subsection{The spectral summand attached to a missing component}
\label{sec:spec-summand}

Set
\[
 \mathcal C_s=\Shv_{\Nilp}(\mathcal B_s),\qquad
 \mathcal C_K=\Shv_{\Nilp}(\mathcal B_K),\qquad
 Y_K=LS^{\mathrm{restr}}_{\check G}(\mathscr X_K).
\]
Restriction to the closed fibre gives an identification
$\QLisse(\mathscr X)\simeq\QLisse(X)$ \cite[\S3.1.3]{GR}. Its inverse,
composed with restriction to the geometric generic fibre, gives a
symmetric monoidal functor
\[
 \rho\colon\QLisse(X)\to\QLisse(\mathscr X_K).
\]
The corresponding map
\[
 i\colon Y\hookrightarrow Y_K
\]
identifies $Y$ with an open-and-closed union of components of $Y_K$
\cite[\S3.1.3]{GR}. In particular the tensor product
\begin{equation}\label{eq:spec-DY}
 \mathcal D_Y:=\QCoh(Y)\underset{\QCoh(Y_K)}{\otimes}\mathcal C_K
\end{equation}
is a direct summand of $\mathcal C_K$.

We need the actual specialisation functor to commute with this summand,
not merely a functor obtained by transporting $\mathbb L_G$ across the
specialisation. Corollary~4.3.6 and \S4.3.9 of \cite{GR} show that the
chartwise geometric nearby-cycle functor we built above restricts to
\[
 \Psi_{\mathcal B}\colon \mathcal D_Y\longrightarrow \mathcal C_s,
\]
and it is $\QCoh(Y)$-linear by \cite[Thm.~3.1.6(B) and \S4.3.10]{GR}.

The characteristic-zero equivalence of \cite[Main Thm.~1.3.9(ii)]{GR},
together with its spectral linearity, identifies
\begin{equation}\label{eq:spec-generic-equivalence}
 \mathcal D_Y\simeq\IndCoh_{\Nilp}(Y).
\end{equation}
In positive characteristic we use only the \emph{primed} version:
$\mathcal C_s\simeq\IndCoh_{\Nilp}(Y')$ for some open-and-closed
$Y'\subset Y$.

\begin{proposition}\label{spec:missing}
If a connected component $C\subset Y$ is disjoint from $Y'$, then the
summand
\[
 \mathcal D_C:=\QCoh(C)\underset{\QCoh(Y)}{\otimes}\mathcal D_Y
\]
of $\mathcal D_Y$ is nonzero; $\Psi_{\mathcal B}(D)=0$ for every
$D\in\mathcal D_C$; and there exists a nonzero $F\in\mathcal D_C$
whose image under $\mathcal D_Y\subset\mathcal C_K\subset
\Shv(\mathcal B_K)$ is compact, and which restricts to a bounded
constructible complex on every finite-type smooth chart $U\to\mathcal B$.
\end{proposition}

\begin{proof}
Let $e_C\in\QCoh(Y)$ be the structure sheaf of $C$ extended by zero.
Its action is the projection onto $\mathcal D_C$. On $\mathcal C_s$
the same object acts by zero, because $C\cap Y'=\varnothing$ and the
primed equivalence is spectral-linear. Therefore for every
$D\in\mathcal D_C$,
\[
 \Psi_{\mathcal B}(D)
 \simeq\Psi_{\mathcal B}(e_C\cdot D)
 \simeq e_C\cdot\Psi_{\mathcal B}(D)=0.
\]
Under \eqref{eq:spec-generic-equivalence}, $\mathcal D_C$ is
$\IndCoh_{\Nilp}(C)$, which is nonzero: the fully faithful
$\QCoh(C)\hookrightarrow\IndCoh_{\Nilp}(C)$ sends $\mathcal O_C\ne0$ to
a nonzero object \cite[\S\S1.3.1--1.3.4]{GR}.

The category $\mathcal D_Y$ is compactly generated
\cite[\S3.1.7]{GR}. In the binary decomposition
$\mathcal D_Y\simeq\mathcal D_C\times\mathcal D_{Y\setminus C}$ both
projections preserve compactness, because both their right adjoints
(the inclusions of the summands) are continuous. Project a set of
compact generators of $\mathcal D_Y$ under $e_C$; at least one
projection is nonzero, since otherwise the colimit closure would give
$\mathcal D_C=0$. Choose such an $F$.

Compactness in $\mathcal C_K$ is the same argument: $F$ is compact in
$\mathcal D_Y$, which is a summand of $\mathcal C_K$, and the inclusion
of a summand preserves compactness. The inclusion
$\mathcal C_K\hookrightarrow\Shv(\mathcal B_K)$ preserves compactness
by \cite[Thm.~1.1.7]{GR}. Finally, a compact sheaf on an algebraic
stack restricts to a bounded constructible complex on every smooth
finite-type chart \cite[\S F.2.2]{AGKRRV}.
\end{proof}

\begin{remark}
The argument uses only one missing component and binary decomposition.
It does not require finitely many components of $Y$ or compactness of
$e_C$. If $Y'\ne Y$ then such a $C$ exists because $Y'$ is a union of
open-and-closed components of $Y$.
\end{remark}

For the remainder of the geometric argument we fix a missing component
$C$ and the associated $F$ as in Proposition~\ref{spec:missing}. If
$Y'=Y$ there is nothing to prove.

\subsection{Hecke transforms of a specialising object}
\label{sec:spec-hecke}

Let $V$ be a finite-dimensional representation of $\check G$ and let
\[
 \mathrm H_V\colon\mathcal C_K\longrightarrow
   \Shv(\mathcal B_K\times_K\mathscr X_K)
\]
be the one-leg Hecke functor, normalised so that it realises the
spectral action on the spectral side. The correspondence and kernel
are pinned down in Section~\ref{sec:hecke}; here we only need two
facts about the universal family.

The universal objects are
\[
 \mathcal E_{V,s}\in\QCoh(Y)\otimes\QLisse(X),\qquad
 \mathcal E_{V,K}\in\QCoh(Y_K)\otimes\QLisse(\mathscr X_K).
\]
On an affine test $T\to Y$ the restriction of $\mathcal E_V$ is the
family of local systems obtained by applying the corresponding tensor
functor to $V$. Acting by its first tensor factor and tensoring with
the exterior product gives $\mathrm H_V$ on the spectral side
\cite[\S14.3.3]{AGKRRV}.

By definition of $i$,
\begin{equation}\label{eq:spec-universal-family}
 (i^*\otimes\id)(\mathcal E_{V,K})
 \simeq(\id\otimes\rho)(\mathcal E_{V,s}).
\end{equation}
On every affine test the two sides are the same local system; the
passage from these identities to \eqref{eq:spec-universal-family} uses
dualizability of $\QLisse(X)$. Concretely, $\QLisse(X)$ for a smooth
projective curve is compactly generated
\cite[\S1.2.11,~\S\S E.2.6--E.2.7]{AGKRRV}, hence dualizable as a
presentable dg category, and tensoring with a dualizable category
commutes with the descent limit defining $\QCoh(Y)$.

The second, distinct input is \cite[Rem.~14.3.9]{AGKRRV}:
\begin{equation}\label{eq:spec-IndLisse}
 \mathcal E_{V,s}\in\QCoh(Y)\otimes\IndLisse(X),
\end{equation}
where $\IndLisse(X)=\operatorname{Ind}(\operatorname{Lisse}(X))$ and
$\operatorname{Lisse}(X)$ consists of bounded constructible complexes
with finite-rank lisse cohomology sheaves. We do \emph{not} identify
$\IndLisse(X)$ with $\QLisse(X)$; that identification fails for
$X=\mathbb P^1$. The two properties used above (dualizability for
descent, membership in $\IndLisse$ for propagation) are separate.

Now apply the $\QCoh(Y)$-action in \eqref{eq:spec-IndLisse} to $F$. The
result is an object of $\mathcal D_C\otimes\IndLisse(X)$. After applying
$\id\otimes\rho$ and the exterior-product functor, it is precisely
$\mathrm H_V(F)$ by \eqref{eq:spec-universal-family}. Elementary tensors
generate $\mathcal D_C\otimes\IndLisse(X)$ under colimits, and bounded
truncations build every object of $\operatorname{Lisse}(X)$ from
finite-rank lisse sheaves. Therefore $\mathrm H_V(F)$ lies in the stable
colimit closure of objects
\begin{equation}\label{eq:spec-exterior-generators}
 D\boxtimes\mathcal L_K,\qquad D\in\mathcal D_C,
\end{equation}
where $\mathcal L_K$ is the restriction of a finite-rank lisse sheaf
$\mathcal L_S$ on $\mathscr X$.

\begin{lemma}\label{spec:vanishing}
Let $a\colon T\to\mathcal B$ be smooth, with $T$ a finite-type $S$-scheme
or algebraic space, and let $\ell_T\colon T\to\mathscr X$ be any
$S$-morphism. Then for $F$ as in Proposition~\ref{spec:missing} and
every finite-dimensional $V$,
\[
 \Psi_T(a_K^*F)=0,
 \qquad
 \Psi_T\bigl((a,\ell_T)_K^*\mathrm H_V(F)\bigr)=0.
\]
The same statements hold on the smooth locus of $a$, and after any of
the finite trait replacements described above.
\end{lemma}

\begin{proof}
The first equality follows from $\Psi_{\mathcal B}(F)=0$ and smooth base
change along $a$. For an exterior generator in
\eqref{eq:spec-exterior-generators}, ordinary pullback gives
\[
 (a,\ell_T)_K^*(D\boxtimes\mathcal L_K)
 =a_K^*D\otimes\ell_{T,K}^*\mathcal L_K.
\]
The sheaf $\ell_T^*\mathcal L_S$ is lisse on $T$, regardless of whether
$\ell_T$ is smooth. Formula~\eqref{eq:spec-lisse-tensor} and
$\Psi_{\mathcal B}(D)=0$ therefore give
\[
 \Psi_T\bigl(a_K^*D\otimes\ell_{T,K}^*\mathcal L_K\bigr)
 \simeq a_s^*\Psi_{\mathcal B}(D)\otimes\ell_{T,s}^*\mathcal L_s=0.
\]
Both pullback and specialisation are continuous, so the same vanishing
holds for the stable colimit closure of these exterior generators. That
closure contains $\mathrm H_V(F)$. Finite trait replacement preserves
geometric specialisation, so no new hypothesis is needed there.
\end{proof}

\begin{remark}[What is and is not used]
The proof uses smoothness only of $a\colon T\to\mathcal B$; the leg
map $\ell_T$ may be arbitrary. It makes no smooth-base-change assertion
for the combined map $(a,\ell_T)$. This is exactly the freedom needed
in Section~\ref{sec:twohecke}, where we cut the Hecke correspondence
by a hypersurface in $T$: after the cut, the leg coordinate floats
freely over the legacy of the modification, and we do not control
whether the cut is smooth in the leg direction.
\end{remark}

Sections~\ref{sec:lie} and~\ref{sec:hecke} convert a non-nilpotent
cotangent direction into a modification that exhibits this vanishing,
nonzero incoming covector and both transverse derivatives.
% =========================================================================
\section{A Levi reduction and a central cocharacter}
\label{sec:lie}
% =========================================================================

Section~\ref{sec:specialization} reduced the main theorem to the following
local problem. Suppose $P\in\mathcal B_s(k)$ is a bundle and
$A\in T^*_P\mathcal B_s$ is a \emph{non-nilpotent} Higgs field: a section
of $\Ad P\otimes\omega_X$ whose values are not all nilpotent. We want to
find a Hecke modification of $P$ whose moving-point covector is nonzero,
and whose fixed-input / fixed-output cotangent sections have invertible
derivatives. Both properties will come from a central cocharacter of a
Levi subgroup.

The Lie-theoretic input is the following. Choose a point $x\in X(k)$
where $A$ is non-nilpotent. In a formal coordinate $t$ and bundle frame
at $x$, write $A=a(t)\,dt$ with $a(t)\in\mathfrak g[[t]]$. Chevalley
restriction identifies the nilpotent cone in $\mathfrak g$ with
$\chi^{-1}(\chi(0))$, where $\chi\colon\mathfrak g\to\mathfrak g/\!/G$ is
the adjoint quotient; so the requirement that $A$ be non-nilpotent at $x$
becomes $\chi(a(0))\ne\chi(0)$.

\begin{proposition}[Lie-theoretic reduction]\label{lie:input}
Let $k$ and $G$ satisfy Hypothesis~\ref{hyp:common} and either
Hypothesis~\ref{hyp:A} or Hypothesis~\ref{hyp:B}, and let $\kappa$ be the
invariant symmetric form of Hypothesis~\ref{hyp:common}(i). Let
$a(t)\in\mathfrak g[[t]]$ satisfy $\chi(a(0))\ne\chi(0)$. Then after a
change of frame by an element of $G(k[[t]])$, there are a Levi subgroup
$M\subset G$ and a cocharacter $\lambda\colon\mathbb G_m\to Z(M)$ such
that
\begin{equation}\label{eq:lie-output}
 a(t)\in\mathfrak m[[t]],\qquad \kappa(a(0),d\lambda)\ne0,\qquad
 \operatorname{ad}_{a(0)}\colon\mathfrak g/\mathfrak m
  \xrightarrow{\ \sim\ }\mathfrak g/\mathfrak m,
\end{equation}
where $\mathfrak m=\Lie M$ and $d\lambda=d\lambda(1)\in\mathfrak g$ is the
image of $1\in\Lie(\mathbb G_m)=k$. The subgroup $M$ may equal $G$; the
invertibility includes the zero vector space.
\end{proposition}

The proof has three layers. First, root-pairing and Jordan facts
(\S\ref{sec:lie-basic}): $\kappa$ pairs root vectors with their opposite
partners, $\mathfrak t$ is nondegenerate, and every element is conjugate
into the Lie algebra of a Borel subgroup. Second, one of two
regime-specific arguments produces a central cocharacter
(\S\ref{sec:lie-A} and~\S\ref{sec:lie-B}). Third, a formal smoothing
argument lifts the constant-term reduction to the full formal arc
(\S\ref{sec:lie-formal}).

\subsection{Root pairings and constant terms}
\label{sec:lie-basic}

Fix a maximal torus $T\subset G$. Set
$\Lambda=X_*(T)$, $\Lambda^\vee=X^*(T)$, $\mathfrak t=\Lie T=\Lambda\otimes k$.
For a root $\alpha$ write $d\alpha\colon\mathfrak t\to k$ for the
differential of the character $\alpha$ and $d\alpha^\vee\in\mathfrak t$
for the differential of the coroot.

\begin{lemma}\label{lie:root-pairing}
The form $\kappa$ is nondegenerate on $\mathfrak t$. For each root $\alpha$
there is a scalar $c_\alpha\in k^\times$ such that
\begin{equation}\label{eq:root-coroot-pairing}
 \kappa(d\alpha^\vee,h)=c_\alpha\,d\alpha(h)\qquad(h\in\mathfrak t).
\end{equation}
In particular neither $d\alpha$ nor $d\alpha^\vee$ vanishes. Every element
of $\mathfrak g$ is conjugate into the Lie algebra of a Borel subgroup
containing $T$, and every semisimple element is conjugate into
$\mathfrak t$.
\end{lemma}

\begin{proof}
Under the $T$-weight decomposition
$\mathfrak g=\mathfrak t\oplus\bigoplus_\alpha\mathfrak g_\alpha$,
invariance of $\kappa$ makes $\mathfrak t$ orthogonal to each root space
and makes $\mathfrak g_\alpha$ pair only with $\mathfrak g_{-\alpha}$.
Nondegeneracy of $\kappa$ on $\mathfrak g$ then gives nondegeneracy on
$\mathfrak t$ and on each root pairing. Choose compatible root vectors
$e_\alpha,e_{-\alpha}$ with $[e_\alpha,e_{-\alpha}]=d\alpha^\vee$; this
follows from the $\mathrm{SL}_2$-root homomorphism and holds for every
isogeny type \cite[\S\S8.1.1--8.1.4]{Springer}, \cite[Cor.~5.1.12, Eq.~(5.1.4)]{Conrad}.
Invariance of $\kappa$ gives \eqref{eq:root-coroot-pairing} with
$c_\alpha=\kappa(e_\alpha,e_{-\alpha})\ne0$. Hence one of
$d\alpha,d\alpha^\vee$ vanishes iff the other does. Simultaneous
vanishing would put $\alpha\in p\Lambda^\vee$ and $\alpha^\vee\in p\Lambda$,
so $\langle\alpha,\alpha^\vee\rangle=2$ would be divisible by $p^2$,
impossible. This argument uses characters, not just their differentials,
and works in characteristic two.

For the Borel-conjugation statement: consider the incidence morphism
\[
 G\times^B\mathfrak b\longrightarrow\mathfrak g,\qquad
 [g,x]\mapsto\Ad(g)x,
\]
where $B\supset T$ is a Borel subgroup and $\mathfrak b=\Lie B$. Its
source is closed in $(G/B)\times\mathfrak g$, so the morphism is proper.
Choose $c\in\mathfrak t$ outside the root differential hyperplanes.
At $[1,c]$, the differential of the incidence morphism is onto (the
negative-root tangent directions map isomorphically to the negative root
spaces, and $\mathfrak b$ supplies the rest), so the morphism is dominant,
hence surjective.

For the semisimple-conjugation statement: write a semisimple element of
$\mathfrak b$ as $c+u$ with $c\in\mathfrak t$ and $u$ in the positive
root Lie algebra. Conjugate by positive-root groups in increasing height.
If $d\alpha(c)\ne0$, conjugation by the $\alpha$-root group changes the
$\alpha$ coefficient by $-z\,d\alpha(c)$ and only touch higher-height
coefficients; it therefore kills that coefficient. After all steps we
are at $c+u_0$ with $[c,u_0]=0$. In a faithful representation $c$ is
semisimple, $u_0$ is nilpotent, and their sum is semisimple, so the
$(\text{semisimple}+\text{nilpotent})$ uniqueness forces $u_0=0$.
\end{proof}

Choose a closed faithful representation $G\hookrightarrow\mathrm{GL}(V)$.
Its differential realises $\mathfrak g$ as a restricted Lie subalgebra of
$\End(V)$, with $x\mapsto x^{[p]}$ equal to matrix $p$th power. We use
the ordinary matrix Jordan decomposition; the two lemmas below prove
its two parts lie in $\mathfrak g$ in each regime; compare
\cite[\S4.4, Thm.~4.4.20]{Springer}.

\begin{lemma}\label{lie:constant-parts}
Suppose $a_0=s+n\in\mathfrak g$ where $s,n\in\mathfrak g$, $s$ is
semisimple, $n$ is nilpotent in a faithful representation, and
$[s,n]=0$. Suppose also $\chi(a_0)\ne\chi(0)$. After conjugation, $s$ is
a nonzero element of $\mathfrak t$, $M=Z_G(s)$ is a Levi subgroup,
$a_0\in\mathfrak m$, and $\ad_{a_0}$ is invertible on
$\mathfrak g/\mathfrak m$. Moreover for every $\lambda\colon\mathbb G_m
\to Z(M)$,
\begin{equation}\label{eq:nilpotent-orthogonality}
 \kappa(a_0,d\lambda)=\kappa(s,d\lambda).
\end{equation}
\end{lemma}

\begin{proof}
The root-pairing lemma conjugates $s$ into $\mathfrak t$. If $s=0$ then
$n=a_0$ is nilpotent. The projection $\mathfrak b\to\mathfrak t$ preserves
restricted $p$-operations; since $n$ is killed by some iterated
$p$-operation and that operation is injective on $\mathfrak t$, its
toral projection vanishes. A strictly dominant cocharacter contracts
the positive-root Lie algebra to zero. Therefore every invariant
polynomial has the same value on $n$ as on $0$, contradiction with
$\chi(a_0)\ne\chi(0)$. Thus $s\ne0$.

By Hypothesis~\ref{hyp:common}(iii), $M=Z_G(s)$ is a Levi subgroup. Its
Lie algebra is $\ker\ad_s$, which contains $a_0$. On
$\mathfrak g/\mathfrak m$ the operator $\ad_s$ is diagonalisable with
nonzero eigenvalues, while $\ad_n$ is nilpotent (indeed
$\ad_n^{p^e}=\ad_{n^{[p^e]}}=0$ for large $e$) and commutes with
$\ad_s$. Therefore on each $\ad_s$-eigenspace the sum
$\ad_{a_0}=\ad_s+\ad_n$ is invertible.

For \eqref{eq:nilpotent-orthogonality}: conjugate $n$ inside $M$ into
the Lie algebra of a Borel of $M$ containing $T$; the incidence
argument of the previous lemma applies to $M$ too, and its roots have
nonzero differentials and $\kappa$ is nondegenerate on $\mathfrak m$ by
the same weight decomposition. Restricted nilpotence again kills the
toral projection, so the conjugated $n$ lies in the positive-root Lie
algebra of $M$. That Lie algebra is orthogonal to $\mathfrak t$, so
$\kappa(n,d\lambda)=0$ for $d\lambda\in\mathfrak t$ fixed by $M$; by
invariance the same holds before conjugation.
\end{proof}

The remaining work is choosing the central cocharacter with
nonvanishing pairing against $s$. This is where the two regimes
diverge; everything else is common.

\subsection{Regime A: a lattice argument for actual central cocharacters}
\label{sec:lie-A}

Assume Hypothesis~\ref{hyp:A}, so $k=\overline{\mathbb F}_q$. Two facts
are needed. First, the \emph{Jordan parts} $s,n$ of $a_0$ constructed in
Lemma~\ref{lie:constant-parts} actually lie in $\mathfrak g$; they start
life only in an ambient $\End(V)$. Second, some $\lambda\in
X_*(Z(M))$ has $\kappa(s,d\lambda)\ne0$. Under Hypothesis~\ref{hyp:A},
both are integral.

The Jordan parts first. In a faithful representation write $a_0=s+n$ for
the matrix Jordan decomposition. The eigenvalues of $s$ lie in a finite
field $\mathbb F_{p^h}$; choose $e$ divisible by $h$ and large enough
to have $n^{p^e}=0$. Then $a_0^{[p^e]}=a_0^{p^e}=s^{p^e}+n^{p^e}=s^{p^e}$
so $s=a_0^{[p^e]}$, hence $s\in\mathfrak g$ as the $p^e$th restricted
power of an element of $\mathfrak g$. Consequently $n\in\mathfrak g$ too.
Lemma~\ref{lie:constant-parts} then supplies a nonzero $s\in\mathfrak t$,
$M=Z_G(s)$ and the quotient invertibility.

For the cocharacter, we use the following lattice statement. It uses
only Hypothesis~\ref{hyp:common}(i) and~(iii); no Weyl-order assumption.

\begin{lemma}\label{lie:lattice}
Assume $\kappa$ and Hypothesis~\ref{hyp:common}(i),~(iii). Then every Levi
Cartan determinant is a $p$-adic unit. Moreover, for every Levi
$M\supset T$,
\begin{equation}\label{eq:central-span}
 \operatorname{span}_k\{d\lambda:\ \lambda\in X_*(Z(M))\}
 =\mathfrak z(\mathfrak m),
\end{equation}
where the $d\lambda\in\mathfrak t$ are differentials of \emph{actual}
central cocharacters of $Z(M)$.
\end{lemma}

\begin{proof}
Fix a positive system and let $I=\{\alpha_1,\ldots,\alpha_r\}$ be the
simple roots of $M$. Set
$J_I=\operatorname{span}_k\{d\alpha_i^\vee\}\subset\mathfrak t$. A
central element of $\mathfrak m$ has no root-space component because all
root differentials are nonzero (Lemma~\ref{lie:root-pairing}), so
\eqref{eq:root-coroot-pairing} gives
\[
 \mathfrak z(\mathfrak m)=\bigcap_i\ker d\alpha_i=J_I^\perp.
\]
The form on this center is nondegenerate by hypothesis, and on
$\mathfrak t$ by Lemma~\ref{lie:root-pairing}; nondegeneracy on $J_I$
follows.

Define integral maps $U\colon\Lambda\to\mathbb Z^r$,
$y\mapsto(\langle\alpha_i,y\rangle)_i$, and $V\colon\mathbb Z^r\to
\Lambda$, $e_j\mapsto\alpha_j^\vee$. Their composite is the Cartan matrix
$C_I=UV$. Reduction to $k$ is denoted by a subscript; by
\eqref{eq:root-coroot-pairing} we have $V_k^*\kappa=D U_k$ for $D$ a
diagonal invertible matrix. The radical of $V_k^*\kappa V_k$ is exactly
$\ker V_k$, again by nondegeneracy on $J_I=\im V_k$. Hence
\begin{equation}\label{eq:cartan-ranks}
 \operatorname{rank}(C_{I,k})=\operatorname{rank}(V_k)
 =\operatorname{rank}(U_k).
\end{equation}

If some Levi Cartan determinant were divisible by $p$, the standard
determinant list
\cite[Thm.~C.56,~\S24(c)]{Milne}
\[
 \begin{array}{c|ccccccccc}
 \text{type}&A_r&B_r&C_r&D_r&E_6&E_7&E_8&F_4&G_2\\ \hline
 \det&r+1&2&2&4&3&2&1&1&1
 \end{array}
\]
would give a further Levi of type $A_{p-1}$: in $A_r$ take a consecutive
string of $p-1$ vertices; in determinant-$2$ and determinant-$4$ cases
take one vertex (forcing $p=2$); in $E_6$ take two adjacent vertices
(forcing $p=3$). This new Levi is a simple-root subdiagram of $G$, so
\eqref{eq:cartan-ranks} applies to it.

For this $A_{p-1}$ Levi, $r=p-1$ and $\det C_I=p$. Choose dual bases of
the full lattices (rank $N_T$); the matrices of $U,V$ are $r\times N_T$
and $N_T\times r$. By \eqref{eq:cartan-ranks} both reductions have rank
$<r$, so every maximal minor of either matrix is divisible by $p$.
Cauchy--Binet gives
\[
 \det C_I
 =\sum_{\substack{J\subset\{1,\ldots,N_T\}\\|J|=r}}
   \det(U_{:,J})\,\det(V_{J,:}),
\]
each summand divisible by $p^2$, contradicting $\det C_I=p$. This uses
the full lattices, with no isogeny-type assumption.

For an arbitrary Levi set $d_I=\det C_I$ and define
$\Pi_I=d_I\id_\Lambda-V\operatorname{adj}(C_I)U\in\End(\Lambda)$. Then
$U\Pi_I=0$; the lattice $\ker U$ is exactly the set of integral
cocharacters into $Z(M)$. For $z\in\ker U_k$, $\Pi_{I,k}(z)=d_I z$; since
$d_I$ is invertible in $k$, the differentials of $\ker U$ span
$\ker U_k=\mathfrak z(\mathfrak m)$, giving
\eqref{eq:central-span}. For $r=0$ take $d_I=1$ and $\Pi_I=\id$; the
same proof covers tori.
\end{proof}

Now return to the regime-A story. Under Hypothesis~\ref{hyp:A}, the
Jordan parts $s,n\in\mathfrak g$ have been constructed. Lemma~\ref{lie:constant-parts}
gives $s\in\mathfrak z(\mathfrak m)\setminus\{0\}$ and the quotient
invertibility. Since $\kappa$ is nondegenerate on $\mathfrak z(\mathfrak
m)$ by Hypothesis~\ref{hyp:common}(i),
Lemma~\ref{lie:lattice} produces an \emph{actual} $\lambda\in X_*(Z(M))$
with $\kappa(s,d\lambda)\ne0$. Orthogonality
\eqref{eq:nilpotent-orthogonality} gives the required
$\kappa(a_0,d\lambda)\ne0$. This used no Weyl-order assumption.

\subsection{Regime B: Weyl-order averaging}
\label{sec:lie-B}

Now assume Hypothesis~\ref{hyp:B}. Eigenvalues of a matrix over $k$ need
not be fixed by any positive power of Frobenius, so the restricted-power
trick of regime A does not apply. Instead we interpolate the inverse of
Frobenius using an $\mathbb F_p$-linear additive polynomial.

In the faithful representation write $a_0=s+n$ and choose $e$ with
$n^{p^e}=0$, so $a_0^{p^e}=s^{p^e}$. Let $V_0\subset k$ be the
finite-dimensional $\mathbb F_p$-span of the eigenvalues of $s$, and let
$V_e=\{v^{p^e}:v\in V_0\}$. The inverse of Frobenius $V_e\to V_0$ is
$\mathbb F_p$-linear. If $V_0=0$ then $s=0$ and $a_0=n\in\mathfrak g$,
contradicting $\chi(a_0)\ne\chi(0)$
(Lemma~\ref{lie:constant-parts}). Hence $r=\dim_{\mathbb F_p}V_e\ge1$ and
we may assume $V_0\ne0$.

Take a basis $\beta_1,\ldots,\beta_r$ of $V_e$ over $\mathbb F_p$. The
Moore matrix $(\beta_i^{p^j})_{1\le i\le r,\ 0\le j<r}$ is invertible:
if not, some nonzero polynomial $\sum_{j=0}^{r-1}c_j z^{p^j}$ of degree
$\le p^{r-1}$ would vanish on all $p^r$ elements of $V_e$, contradicting
$|V_e|=p^r$. Therefore there is an additive polynomial
$Q(z)=\sum_{j=0}^{r-1}c_j z^{p^j}$ agreeing with inverse-Frobenius on
$V_e$. Applying it to the diagonalisable matrix $s^{p^e}$ gives
\[
 s=Q(a_0^{p^e})=\sum_{j=0}^{r-1}c_j a_0^{[p^{e+j}]}\in\mathfrak g.
\]
Hence $n\in\mathfrak g$ as well. Lemma~\ref{lie:constant-parts} now
supplies $s\in\mathfrak t\setminus\{0\}$, $M=Z_G(s)$ and the quotient
invertibility.

To find $\lambda$: let $W_M=N_M(T)/T$ be the Weyl group of $M$ and set
$h_M=|W_M|$. Since $W_M\subset W_G$, Hypothesis~\ref{hyp:B} implies
$h_M$ is invertible in $k$. The averaging operator
\[
 e_M=\frac1{h_M}\sum_{w\in W_M}w\colon\mathfrak t\to\mathfrak t
\]
is a self-adjoint projection for $\kappa$, so $\kappa$ is nondegenerate
on $\mathfrak t^{W_M}=\im e_M$. That subspace contains $s$ because $M$
centralises $s$. In fact $\mathfrak t^{W_M}=\mathfrak z(\mathfrak m)$:
the reflection formula is $s_\alpha(h)=h-d\alpha(h)d\alpha^\vee$
and $d\alpha^\vee\ne0$ by Lemma~\ref{lie:root-pairing}. Thus nondegeneracy
on $\mathfrak z(\mathfrak m)$ again follows from $\kappa$ alone, without
needing Hypothesis~\ref{hyp:common}(i)'s second clause.

For an integral cocharacter, choose a basis $\mu_1,\ldots,\mu_{N_T}$ of
$\Lambda$ and set $\nu_i=\sum_{w\in W_M}w\mu_i\in\Lambda^{W_M}$. Their
differentials span $\mathfrak t^{W_M}$ because $d\nu_i=h_M e_M(d\mu_i)$.
The integral reflection formula $s_\alpha(\lambda)=\lambda-
\langle\alpha,\lambda\rangle\alpha^\vee$ and torsion-freeness of
$\Lambda$ show that $\Lambda^{W_M}=X_*(Z(M))$: a $W_M$-invariant
cocharacter has zero pairing with every root of $M$, and conversely. No
division of a cocharacter by $h_M$ is made. Nondegeneracy of $\kappa$ on
$\mathfrak t^{W_M}$ now gives an integral $\lambda\in X_*(Z(M))$ with
$\kappa(s,d\lambda)\ne0$, and \eqref{eq:nilpotent-orthogonality} gives
$\kappa(a_0,d\lambda)\ne0$.

This argument applies to any algebraically closed $k$. Its use of
$p\nmid|W_G|$ is specific to Hypothesis~\ref{hyp:B}.

\subsection{Formal smoothing: from constant term to whole arc}
\label{sec:lie-formal}

Both regimes have now produced the same constant-term data: a Levi $M$,
a central cocharacter $\lambda$, and $a_0=a(0)\in\mathfrak m$ with
$\kappa(a_0,d\lambda)\ne0$ and $\ad_{a_0}$ invertible on
$\mathfrak g/\mathfrak m$. The remaining step moves the entire formal
arc $a(t)$ into $\mathfrak m[[t]]$.

\begin{lemma}\label{lie:formal}
Let $G$ be a connected reductive group over an algebraically closed
field $k$, $M\subset G$ a Levi subgroup, $a(t)\in\mathfrak g[[t]]$ with
$a_0=a(0)\in\mathfrak m$. If $\ad_{a_0}$ is invertible on
$\mathfrak g/\mathfrak m$, then there is $g(t)\in G(k[[t]])$ with
$g(0)=1$ and $\Ad(g(t))a(t)\in\mathfrak m[[t]]$.
\end{lemma}

\begin{proof}
Proceed by successive approximation. Suppose the arc already lies in
$\mathfrak m[[t]]$ modulo $t^r$, $r\ge1$, and let $c_r\in
\mathfrak g/\mathfrak m$ be the class of its $t^r$-coefficient. Choose
$D\in\mathfrak g$ with $c_r+[D,a_0]=0$ in the quotient (possible by
invertibility). Smoothness of $G$ gives $g_r(t)\in G(k[[t]])$ with
$g_r(t)\equiv1+t^rD\pmod{t^{r+1}}$; formally, this is a lift of a point
in the tangent kernel of the square-zero ideal $(t^r)/(t^{r+1})$ to
$G(k[t]/(t^{r+1}))$, obtained by iterating formal smoothness at $G$. Then
conjugation by $g_r(t)$ changes the $t^r$-coefficient by $[D,a_0]$ and
leaves all lower coefficients fixed, so it places the arc in
$\mathfrak m[[t]]$ modulo $t^{r+1}$.

Iterating, the products $\prod g_r(t)$ converge in the $t$-adic
topology because $g_r\equiv1\pmod{t^r}$. The affine group $G$ then
produces a single $g(t)\in G(k[[t]])$ with $g(0)=1$ and
$\Ad(g(t))a(t)\in\mathfrak m[[t]]$. No exponential series is used.
\end{proof}

\begin{proof}[Proof of Proposition~\ref{lie:input}]
Under Hypothesis~\ref{hyp:A}, use the restricted-power construction of
Jordan parts together with Lemma~\ref{lie:lattice}. Under
Hypothesis~\ref{hyp:B}, use the Moore-matrix construction together with
integral Weyl averaging. In both cases Lemma~\ref{lie:constant-parts}
gives the Levi $M$, the quotient invertibility and
$\kappa(a_0,d\lambda)\ne0$. Apply Lemma~\ref{lie:formal} after the
initial constant conjugation; the formal change of frame fixes $a_0$ by
construction, so all of \eqref{eq:lie-output} holds.
\end{proof}

\begin{remark}\label{lie:zero-orbit}
Centrality of $\lambda$ in $M$ gives $\Ad(\lambda(t))a(t)=a(t)$, and
every nonzero $\lambda$-weight space of $\mathfrak g$ injects into
$\mathfrak g/\mathfrak m$, so the adjoint action is invertible on the
truncated nonzero-weight modules used below. We do \emph{not} need
$Z_G(\lambda)=M$. If $\lambda$ is central in $G$ the Hecke orbit has
dimension zero, but the nonzero pairing in \eqref{eq:lie-output} still
supplies the moving-point covector. Tori are therefore included in every
allowed characteristic.
\end{remark}

% =========================================================================
\section{Hecke modifications and transverse cotangent sections}
\label{sec:hecke}
% =========================================================================

Section~\ref{sec:lie} produced Levi data. Now we turn it into a
modification of a bundle. Let $P\in\mathcal B_s(k)$ and let
$A\in T^*_P\mathcal B_s$ be non-nilpotent. We will choose a point
$x\in X(k)$, a cocharacter $\lambda$, and a modification
$(P,P',x)\in H^\lambda_s(k)$ such that:
\begin{enumerate}[label=(\roman*)]
\item the moving-point displacement $(d\lambda/t)|_x$ pairs nonzero
  against $A$ to give a leg covector $\xi\ne0$;
\item the fixed-input cotangent section $S_A$ and the fixed-output one
  $S_{A'}$ both vanish at the modification and both have invertible
  derivatives on their respective tangent spaces.
\end{enumerate}
These are the "transversality" and "isolation" conditions.

\subsection{Correspondence and kernels}

We keep the trait $S$, the curve $\mathscr X/S$ and the bundle stack
$\mathcal B$ from Section~\ref{sec:specialization}. A one-point Hecke
modification is a quadruple $(P,P',x,\alpha)$ where $x\in\mathscr X$
and $\alpha$ identifies $P'$ and $P$ away from the graph of $x$. Write
\[
 p(P,P',x,\alpha)=P,\qquad o(P,P',x,\alpha)=P',\qquad
 \ell_H(P,P',x,\alpha)=x.
\]
Convention: in a formal coordinate $t$ at $x$, modification by
$\lambda$ means the output frame is $\lambda(t)$ times the input frame.
Relative position depends only on the Weyl orbit of $\lambda$. Let
$\mu=\lambda^+$ be its dominant conjugate, $m=\langle 2\rho,\mu\rangle$
where $2\rho$ is the sum of positive roots, $\mu^\vee=-w_0\mu$, and
$\mathcal Z=\mathcal B\times_S\mathscr X$. Denote the exact-position
correspondence by $H=H^\lambda$, and set
\[
 \widetilde p=(p,\ell_H)\colon H\to\mathcal Z,\qquad
 q=(o,\ell_H)\colon H\to\mathcal Z.
\]
Both $\widetilde p$ and $q$ forget some of the modification data.

\begin{lemma}[Hecke models and kernel]\label{hecke:models}
Both $\widetilde p$ and $q$ are representable, separated and smooth of
relative dimension $m$. The correspondence $H$ is open in a bounded
correspondence $\overline H$ for which both projections to $\mathcal Z$
are representable and proper. On the output side the bound on
$\overline H$ has dominant label $\mu^\vee$.

Let $V$ be the irreducible representation of $\check G$ corresponding
to this bound, with the indicated input/output convention. Then the
generic Hecke functor has the form
\begin{equation}\label{hecke:kernel}
 \mathrm H_V(F)=\overline q_{K,*}\mathcal I_K,
\end{equation}
with $\mathcal I_K$ on $H_K$ equal to $p_K^*F$ tensored with a fixed
invertible constant complex (the perverse intersection-cohomology
normalisation $E[m]$, plus a Tate line if weight-normalised).
\end{lemma}

\begin{proof}
Consider the split integral affine Grassmannian with loops $G(A((t)))$
and positive loops $G(A[[t]])$. Its integral Schubert bound is
projective and the positive-loop orbit is smooth, open, fibre-dense of
relative dimension $m$ \cite[\S\S1.2,~2.1]{RicharzScholbach}. We use these
models by base change from $\mathbb Z$.

A graph coordinate and a formal-disc frame identify modifications of
$P$ with the corresponding Grassmannian spaces; these descriptions glue
by Beauville--Laszlo descent \cite{BeauvilleLaszlo}. More explicitly,
the graph is an effective Cartier divisor. Gluing applies to the
coordinate algebra, multiplication and coaction of an affine $G$-torsor
(as it does to modules); the torsor identity is checked on the two
pieces. A disc frame exists locally by lifting a frame on the graph
successively through its infinitesimal neighbourhoods using smoothness
of $G$. Changes of frame and of coordinate preserve the orbit and
its bound, so properness, smoothness and dimension all descend to the
input-and-leg projection.

Swapping the two bundles and inverting $\alpha$ is an involution of the
Hecke correspondence. On double cosets it sends $\mu$ to $-\mu$, whose
dominant representative is $\mu^\vee$, and it carries the bound to the
oppositely-labelled bound. This gives the assertions for $q$, and
$\langle 2\rho,\mu^\vee\rangle=\langle 2\rho,\mu\rangle=m$. Separatedness
of the open correspondence follows from that of the proper bound. This
involution is an involution of the correspondence itself, not of the
right-quotient affine Grassmannian.

The fibres here are relative-position spaces: an automorphism of the
moving bundle compatible with its off-graph identification is the
identity; no quotient by automorphisms of the fixed bundle is taken.

For the last assertion, geometric Satake constructs the intersection
complex of the bound. Its restriction to the smooth open orbit is the
shifted constant sheaf
\cite[Prop.~4.1]{Richarz}. The Hecke construction tensors this kernel
with $p_K^*F$ and pushes forward along the proper output-and-leg
projection \cite[\S\S4.2.1--4.2.4]{GR}. Relabelling by $\mu^\vee$ if
needed matches these projections with the chosen $V$. A fixed overall
shift or Tate line changes neither the description on the open orbit nor
any vanishing assertion below. Only the open-orbit restriction of the
kernel is identified here; on the boundary the kernel remains its
intersection complex.
\end{proof}

\subsection{The cotangent identity and the transverse section}

On the special fibre a cotangent vector to $\mathcal B_s$ at $P$ is an
element of $H^0$ of the cotangent-complex fibre. Deformation theory and
Serre duality \cite[Tag~0BS4]{Stacks}, together with $\kappa$, identify
this space as
\[
 T_P^*\mathcal B_s=H^1(X,\mathfrak g_P)^*=H^0(X,\mathfrak g_P\otimes\omega_X),
\]
where $\mathfrak g_P$ is the adjoint bundle. We call such a vector $A$ a
\emph{Higgs field}. For a representable smooth map, the cotangent
pullback is injective and its image is the kernel of passage to the
relative cotangent; we use this in the form supplied by the cotangent
triangle.

\begin{proposition}[Transverse modification]\label{hecke:transverse}
Let $P\in\mathcal B_s(k)$ and let $A\in T^*_P\mathcal B_s$ be
non-nilpotent. There exist a point $x\in X(k)$, a cocharacter $\lambda$,
a modification $h\in H^\lambda_s(k)$ from $P$ to a bundle $P'$, and a
covector
\[
 \theta=(A',\xi)\in T^*_{(P',x)}(\mathcal B_s\times X),\qquad \xi\ne0,
\]
with $A'=A$ away from $x$, satisfying
\begin{equation}\label{hecke:cotangent-identity}
 dp_h^*A=dq_h^*\theta.
\end{equation}
Moreover, on the fixed-side fibres
\[
 H^{\mathrm{in}}_{P,x}=\widetilde p^{-1}(P,x),\qquad
 H^{\mathrm{out}}_{P',x}=q^{-1}(P',x),
\]
with fixed-bundle identifications retained, the following hold. The
restriction of $p^*A$ to the $q$-relative tangent bundle defines a
section $S_A$ of $\Omega_{H_s/\mathcal Z_s,q}$; the restriction of
$o^*A'$ to the $\widetilde p$-relative tangent bundle defines a section
$S_{A'}$. Both sections vanish at $h$, and both derivatives
\begin{align*}
 (dS_A)_h&\colon T_hH^{\mathrm{in}}_{P,x}
              \longrightarrow (T_hH^{\mathrm{out}}_{P',x})^*,\\
 (dS_{A'})_h&\colon T_hH^{\mathrm{out}}_{P',x}
              \longrightarrow (T_hH^{\mathrm{in}}_{P,x})^*
\end{align*}
are isomorphisms. In particular each section has an isolated reduced
zero at $h$.
\end{proposition}

\begin{proof}
\emph{Choice of $\lambda$.} Since $A$ is non-nilpotent, choose
$x\in X(k)$ where its value is non-nilpotent. Choose a formal coordinate
$t$, a bundle frame at $x$, and write $A=a(t)\,dt$. Chevalley restriction
identifies the nilpotent fibre with $\chi^{-1}(\chi(0))$, so
$\chi(a(0))\ne\chi(0)$. Proposition~\ref{lie:input} then supplies a
formal change of frame, a Levi $M$, and a cocharacter
$\lambda\colon\mathbb G_m\to Z(M)$ with
\begin{equation}\label{hecke:lie-data}
 a(t)\in\mathfrak m[[t]],\quad
 \ad_{a(0)}\text{ invertible on }\mathfrak g/\mathfrak m,\quad
 \kappa(a(0),d\lambda)\ne0.
\end{equation}
After constant conjugation we regard $\lambda$ as an element of the
fixed cocharacter lattice, so it labels an integral model of
Lemma~\ref{hecke:models}. Modify by $\lambda(t)$. Since $\lambda$
centralises $M$, the transported field in the output frame is again
$a(t)\,dt$; combined with $A$ away from $x$, this defines the global
Higgs field $A'$ on $P'$. This central-in-a-Levi modification and the
resulting leg covector originate in
\cite[\S\S20.4--20.6]{AGKRRV}. What follows makes both derivative
assertions explicit.

\emph{Tangent spaces.} In the input frame, put
$L=\mathfrak g[[t]]$, $L'=\Ad(\lambda(t))L$, $I=L\cap L'$. Let
$\mathfrak g_j$ be the weight-$j$ space of $\lambda$; the integers $j$
are genuine characters of $\mathbb G_m$, not reduced mod $p$. Then
\begin{equation}\label{hecke:weight-quotients}
 L'=\bigoplus_j t^j\mathfrak g_j[[t]],\quad
 P_1:=L/I=\bigoplus_{j>0}\mathfrak g_j\otimes k[[t]]/(t^j),\quad
 P_2:=L'/I=\bigoplus_{j<0}\mathfrak g_j\otimes (t^jk[[t]]/k[[t]]).
\end{equation}
An infinitesimal disc automorphism $1+\epsilon D$, $D\in L$, acts on
the output lattice while fixing $(P,x)$, and its stabiliser tangent is
$I$. Its orbit is smooth of dimension $\dim_k(L/I)=m$ by the root
formula, so this computes the full tangent space:
\begin{equation}\label{hecke:tangents}
 T_hH^{\mathrm{in}}_{P,x}=P_1,\qquad
 T_hH^{\mathrm{out}}_{P',x}=P_2.
\end{equation}
The same argument with output fixed gives the second identity. The
kernel over dual numbers is identified with the Lie algebra via the
canonical isomorphism, no exponential map needed.

There is a perfect pairing
\begin{equation}\label{hecke:residue-pairing}
 \beta\colon P_1\times P_2\to k,\qquad
 \beta(D,C)=\operatorname{Res}_{t=0}\kappa(D,C)\,dt.
\end{equation}
Indeed, $L$ is its own residue-annihilator on $\mathfrak g((t))$, and
invariance of $\kappa$ gives the same statement for $L'$. So $\beta$
descends to the quotients. Concretely, $\kappa$ pairs $\mathfrak g_j$
perfectly with $\mathfrak g_{-j}$ and the coefficient of $t^a$ in the
first space, for $0\le a<j$, pairs with the coefficient of $t^{-a-1}$
in the second. This is perfect, including when $j$ is divisible by $p$.

\emph{Cotangent identity and moving leg.} Take the gluing sign
convention where an infinitesimal transition $1+\epsilon C$ gives the
boundary class of the principal part $C$. Serre duality pairs it with
$A$ by $\operatorname{Res}\kappa(a,C)\,dt$. For output-fixed tangent
$C\in L'$ the residue is zero because $a\in L'$ and $L'$ is self-annihilating.
Therefore $dp_h^*A$ vanishes in the $q$-relative cotangent, so it is
$dq_h^*\theta$ for a unique covector $\theta$.

The bundle component of $\theta$ is $A'$. Indeed, choose $x'\ne x$ and
vary both bundle transitions only at $x'$, leaving the modification at
$x$ unchanged. Principal parts at $x'$ surject to
$H^1(X,\mathfrak g_{P'})$ (Serre vanishing for $N\gg0$), and the
pairings agree on these directions since $A=A'$ near $x'$. Write
$\theta=(A',\xi)$.

Now keep the input fixed and move the leg from $t=0$ to $t=\epsilon$.
The output transition is $g_\epsilon=\lambda(t-\epsilon)$ and to first
order in $\epsilon$,
$(\delta g)g^{-1}=-d\lambda/t$. Evaluating the cotangent identity on
this motion gives
\[
 0=-\kappa(a(0),d\lambda)+\xi(\partial_t),
 \qquad \xi=\kappa(a(0),d\lambda)\,dt\big|_x\ne0,
\]
using $\kappa(a(0),d\lambda)\ne0$ from \eqref{hecke:lie-data}. This is
also the residue formula of \cite[Eq.~(20.20)]{AGKRRV}; reversing the
gluing boundary convention reverses the common sign and leaves
everything below unchanged.

\emph{The first derivative.} The section $S_A$ is defined along the
whole fixed-input fibre, using the specified identification with $P$.
Its value at $h$ is zero from the previous paragraph. Deform the output
lattice to $L'_\epsilon=\Ad(1+\epsilon D)L'$ for $D\in L$; a test
vector $C\in L'$ extends as $C_\epsilon=\Ad(1+\epsilon D)C$. The input
field remains $a(t)\,dt$. Differentiating the residue pairing gives
\begin{equation}\label{hecke:derivative}
 \bigl\langle(dS_A)_h(D\bmod I),C\bmod I\bigr\rangle
 =\operatorname{Res}\kappa(a,[D,C])\,dt
 =\operatorname{Res}\kappa([a,D],C)\,dt
 =\beta(\ad(a)D,C).
\end{equation}
Since $S_A(h)=0$ the derivative is well-defined, and since $\ad(a)$
preserves $L,L',I$, the formula is independent of representatives.

For invertibility: $\ad(a(t))$ preserves every weight summand in
\eqref{hecke:weight-quotients}. Because $\lambda$ is central in $M$,
each nonzero weight space injects as an invariant summand into
$\mathfrak g/\mathfrak m$; its endomorphism from $\ad(a(0))$ is therefore
invertible by \eqref{hecke:lie-data}. The corresponding $k[[t]]$-linear
block has unit determinant, so $\ad(a(t))$ is invertible on every
truncated summand of $P_1$ and $P_2$. Combined with
\eqref{hecke:derivative} and \eqref{hecke:residue-pairing}, this shows
$(dS_A)_h\colon P_1\to P_2^*$ is an isomorphism. Zero-weight directions
do not appear in these quotients, so no extra regularity of $\lambda$
outside $M$ is needed.

\emph{The reverse derivative.} The value of $S_{A'}$ at $h$ is zero
because $a\in L$ and $L$ is self-annihilating. Keeping the output fixed,
vary the input lattice by $C\in L'$ and extend $D\in L$ to
$\Ad(1+\epsilon C)D$. The same calculation, with the same gluing
convention, gives
\[
 \bigl\langle(dS_{A'})_h(C\bmod I),D\bmod I\bigr\rangle
 =\operatorname{Res}\kappa(a,[C,D])\,dt
 =\operatorname{Res}\kappa([a,C],D)\,dt,
\]
and invertibility on $P_2$ with residue duality proves the second
assertion. Both sections are sections of rank-$m$ vector bundles on
smooth $m$-dimensional fibres; invertible derivatives make their zeros
reduced and isolated.

When $m=0$, both tangent spaces vanish and both derivatives are
isomorphisms between zero spaces. The smooth zero-dimensional fibres
already isolate $h$, and the moving-leg computation still applies.
Modifications of this form, and tori, are thus included.
\end{proof}

\begin{remark}
The proposition gives both properties we need. The nonvanishing
$\xi\ne0$ allows a suitable hypersurface cut of the Hecke correspondence
to be smooth over $\mathcal B$, while the two relative derivatives
isolate $h$ and set up the quadratic coordinates around it. In
Section~\ref{sec:twohecke}, these two facts are used in reverse order:
the isolating one first, and the nonvanishing one later.
\end{remark}

% =========================================================================
\section{Two Hecke correspondences and a quadratic test}
\label{sec:twohecke}
% =========================================================================

We now convert Lemma~\ref{spec:vanishing} into a local test for a
non-nilpotent cotangent direction, in one big step. The first Hecke
correspondence supplies a proper pushforward whose fibre is a smooth
curve; the second makes the pushed-forward sheaf vanish everywhere
except at a single point; the two transverse sections of
Proposition~\ref{hecke:transverse} identify the local neighbourhood of
that point with a split quadratic equation. The next section removes
the extra variables in that equation.

Throughout this section, $F\in\Shv(\mathcal B_K)$ is the specialising
object of Proposition~\ref{spec:missing}, and we use the vanishings of
Lemma~\ref{spec:vanishing} freely.

\begin{proposition}[Vanishing at the quadratic vertex]
\label{two:vertex}
Let $b\colon U\to\mathcal B$ be a smooth chart with $U$ finite type over
$S$. Let $f\in\mathcal O(U)$ and $u\in U_s(k)$ satisfy $f(u)=0$, and
suppose $df_u=b^*A$ for some non-nilpotent $A\in T^*_{b(u)}\mathcal B_s$.
Then there exist $x\in\mathscr X_s(k)$ and an integer $m\ge0$ with the
following property. Put $T=U\times_S\mathscr X$ and let $F_T$ be the
pullback of $F_U=b_K^*F$. Consider the affine quadric
\[
 Q^a=\left\{\sum_{i=1}^m u_iv_i+f=0\right\}\subset T\times_S\mathbb A_S^{2m},
 \qquad \tau\colon Q^a\to T,
\]
where $u_i,v_i$ are affine coordinates (distinct symbols from the point
$u$) and $f$ is pulled back from $U$. Then at the point
$\nu=(u,x,0,0)$,
\begin{equation}\label{eq:two-vertex}
 \bigl(\Psi_{Q^a}(\tau_K^*F_T)\bigr)_\nu=0.
\end{equation}
When $m=0$, this is the vanishing on $Q^a=V(f)\times_S\mathscr X$.
\end{proposition}

\begin{proof}
Apply Proposition~\ref{hecke:transverse} to $A$ to obtain a
modification $h$, its leg $x$, its orbit dimension $m$, and a covector
$\theta=(A',\xi)$ with $\xi\ne0$, satisfying
\begin{equation}\label{eq:two-cotangent}
 p^*A=q^*\theta\quad\text{at }h.
\end{equation}
Use the exact-position correspondence $H=H^\lambda$, its proper bound
$\overline H$, and its generic Satake integrand $\mathcal I_K$ from
Lemma~\ref{hecke:models}. On $H_K$ this integrand is $p_K^*F$ tensored
with a fixed invertible shift and Tate line; we suppress that factor
after restricting to the open orbit.

\paragraph{Step 1: a proper fibre with vanishing cohomology.}
Write $D=V(f)$. Take two copies $H_1,H_2$ of $H$ and form
\[
 W_0=H_2\times_{p_2,\mathcal B,b}U,\qquad
 W=H_2\times_{\mathcal B}D\subset W_0.
\]
Both map to $\mathcal Z=\mathcal B\times_S\mathscr X$ by
$q_2=(o_2,\ell_{H_2})$. The map $W_0\to\mathcal Z$ is smooth: its
projection to $H_2$ is a base change of the smooth chart $b$, and $q_2$
is smooth. At $w=(h,u)$, the differential of the equation cutting out
$W$ from $W_0$ is the pullback of $\theta$ by \eqref{eq:two-cotangent}.
Its relative differential over the output bundle stack is nonzero,
because the leg component is $\xi\ne0$. The smooth-hypersurface
criterion therefore makes $W\to\mathcal B$ (via the output bundle)
smooth near $w$. The arbitrary-leg assertion of
Lemma~\ref{spec:vanishing} gives
\begin{equation}\label{eq:two-hecke-zero}
 \bigl(\Psi_W(q_{2,K}^*\mathrm H_V(F))\bigr)_w=0.
\end{equation}
Note: smoothness is used only for the projection to $\mathcal B$, matching
the hypothesis of Lemma~\ref{spec:vanishing}.

Now form fibre products
\[
 P=H_1\times_{\mathcal Z}W_0,\qquad
 P_D=H_1\times_{\mathcal Z}W\subset\overline P_D=\overline H_1\times_{\mathcal Z}W.
\]
The following squares are cartesian; in the second the vertical map
$\pi$ is proper. In the first, $\Delta$ repeats the modification in the
two factors.
\[
\begin{tikzcd}[column sep=large,row sep=large]
 P \arrow[r] \arrow[d] & H_1 \arrow[d,"q_1"] \\
 W_0 \arrow[r,"q_2"'] \arrow[u,bend left=38,"\Delta"]
       & \mathcal Z
\end{tikzcd}
\qquad
\begin{tikzcd}[column sep=large,row sep=large]
 \overline P_D \arrow[r] \arrow[d,"\pi"']
       & \overline H_1 \arrow[d,"\overline q_1"] \\
 W \arrow[r,"q_2"'] & \mathcal Z.
\end{tikzcd}
\]
Write $c\colon P\to U$ for the map through $W_0$, and
$a\colon P\to\mathcal B$ for the input map $p_1$. So $a$ records the
input from which the sheaf is pulled back, and $c$ records the second
input in $U$ on which we impose $f=0$. The distinguished point is
$z=(h,h,u)\in P_D(k)$.

Let $\overline{\mathcal I}_K$ be the pullback of the Satake integrand
to $(\overline P_D)_K$. Proper base change on the generic fibre gives
$R\pi_{K,*}\overline{\mathcal I}_K\simeq q_{2,K}^*\mathrm H_V(F)$.
Apply proper compatibility of specialisation, then proper base change
at $w$, to \eqref{eq:two-hecke-zero}. With
\[
 Z_w=\pi^{-1}(w),\qquad
 \mathcal K=(\Psi_{\overline P_D}\overline{\mathcal I}_K)|_{Z_w},
\]
we get
\begin{equation}\label{eq:two-fiber-zero}
 R\Gamma(Z_w,\mathcal K)=0.
\end{equation}
Here $Z_w$ is the proper bounded Hecke fibre with output bundle and leg
fixed, identifications retained.

\paragraph{Step 2: isolating the distinguished stalk.}
The map $a\colon P\to\mathcal B$ is smooth: $P\to H_1$ is a base change
of $W_0\to\mathcal Z$, and $p_1$ is smooth. At a point $(h_1,w)$ in the
exact-position part of $Z_w$, the differential of $f(c)$ is $q_1^*\theta$
pulled back. If the relative differential over $\mathcal B$ of $f(c)$
were zero, cotangent injectivity for the smooth map $P\to H_1$ would
imply that $q_1^*\theta$ comes from the input cotangent under $p_1$.
Then passing to the relative cotangent of $\widetilde p_1=(p_1,\ell_{H_1})$,
the leg term drops out, and the necessary condition becomes
$S_{A'}(h_1)=0$. The second half of
Proposition~\ref{hecke:transverse} says this section has an isolated
zero at $h$ on the fixed-output-and-leg fibre. Hence $P_D\to\mathcal B$
via $a$ is smooth at every point of a punctured neighbourhood of $z$
in $Z_w$. On the exact-position open, the generic integrand equals
$a_K^*F$. Lemma~\ref{spec:vanishing} therefore gives a neighbourhood
$O$ of $z$ in $Z_w$ with
\[
 \mathcal K|_{O\setminus\{z\}}=0.
\]

This local vanishing prevents any cancellation with the boundary of the
Hecke bound. Put $C_w=Z_w\setminus O$ (reduced structure). The closed
subsets $\{z\}$ and $C_w$ are disjoint, so open-closed localization
identifies
\[
 \mathcal K\simeq i_{z,*}\mathcal K_z\oplus i_{C_w,*}(i_{C_w}^*\mathcal K),
 \qquad
 R\Gamma(Z_w,\mathcal K)\simeq
 \mathcal K_z\oplus R\Gamma(C_w,i_{C_w}^*\mathcal K).
\]
The residue field of $z$ is algebraically closed, so the first summand
is its stalk complex. Equation~\eqref{eq:two-fiber-zero} forces it to
vanish. Removing the invertible Satake normalization gives
\begin{equation}\label{eq:two-isolated-zero}
 \bigl(\Psi_{P_D}(a_K^*F)\bigr)_z=0.
\end{equation}
No boundary behaviour of the Satake kernel is used.

\paragraph{Step 3: chart lift along the whole diagonal.}
We must now translate \eqref{eq:two-isolated-zero} into coordinates.
First, the distinguished diagonal $\Delta\simeq W_0\subset P$ is a
section of the smooth separated map $P\to W_0$ of relative dimension
$m$, hence a closed regular immersion near $z$. On $\Delta$ there is a
specified isomorphism $bc\simeq a$, coming from the two identical
modifications.

We claim that after an \emph{\'etale} replacement of $P$ near $z$, there
is a map $r\colon P\to U$ with an isomorphism
\begin{equation}\label{eq:two-chart-lift}
 br\simeq a,\qquad r|_\Delta=c|_\Delta,
\end{equation}
whose restriction to $\Delta$ is the specified isomorphism. For this,
form the smooth space $E_P=P\times_{a,\mathcal B,b}U\to P$. The
diagonal data give a section over $\Delta$. First take an \'etale
scheme neighbourhood in $P$. Since $\mathcal B$ has affine diagonal
(Section~\ref{sec:specialization}), $E_P\to P\times_S U$ is affine, so
$E_P$ is a scheme as well. Near the section point, smooth coordinates
give an \'etale map $E_P\to\mathbb A_P^{e}$
\cite[Tag~039P]{Stacks}. After shrinking, the section over $\Delta$
lands in this coordinate neighbourhood. Extend its coordinate functions
$\Delta\to P$, then pull back the \'etale map along the extended tuple.
This gives an \'etale neighbourhood of $P$ with a map to $E_P$. Its
inverse image of $\Delta$ contains the open branch specified by the
original section; remove the closed complement of this branch inside
that inverse image. The retained diagonal is the prescribed one, and
the map to $E_P$ proves \eqref{eq:two-chart-lift} including its
isomorphism of bundles. We keep the notation $P,\Delta,z$.

\paragraph{Step 4: exact quadratic coordinates.}
After shrinking $U$, suppose $U/S$ has constant relative dimension $N$.
The map
\[
 \Delta\to T=U\times_S\mathscr X,\qquad(c,\text{leg}),
\]
is smooth of relative dimension $m$. So $T,\Delta,P$ are smooth over
$S$ of relative dimensions
\[
 N+1,\qquad N+1+m,\qquad N+1+2m.
\]
Choose generators $v_1,\ldots,v_m$ of the diagonal ideal whose classes
form a conormal frame. Under the identification with
$\Omega_{P/W_0}|_\Delta$, the classes $dv_i$ form a basis. Because
$r=c$ on $\Delta$, there exist regular functions $u_1,\ldots,u_m$ with
the \emph{exact} identity
\begin{equation}\label{eq:two-exact-quadratic}
 f(c)-f(r)=\sum_{i=1}^m u_iv_i.
\end{equation}
The restrictions $u_i|_\Delta$ are determined by the conormal class of
the left side.

Consider the whole local diagonal fibre $\Delta_{(u,x)}$ on the special
fibre. This parametrises modifications with input bundle $b(u)$, its
identification, and leg $x$ fixed. The relative differential of $f(c)$
for $P\to W_0$ is zero. On this diagonal fibre, the relative
differential of $f(r)$ is induced by $df_u=b^*A$: the restriction
$r|_{\Delta_{(u,x)}}$ is constant at value $u$, and the bundle
isomorphism in \eqref{eq:two-chart-lift} is the canonical one along the
whole diagonal. The conormal coefficient in
\eqref{eq:two-exact-quadratic} is therefore
\begin{equation}\label{eq:two-coefficients}
 \sum_i(u_i|_\Delta)[dv_i]=-S_A
 \quad\text{on }\Delta_{(u,x)}.
\end{equation}
In particular $u_i(z)=0$, and the derivative of $(u_i|_\Delta)$ is an
isomorphism on $T_z\Delta_{(u,x)}$ by
Proposition~\ref{hecke:transverse}. This uses the equality on the whole
fibre in \eqref{eq:two-coefficients}, not just at $z$.

Now set
\[
 \Phi=(r,\text{leg},u_1,\ldots,u_m,v_1,\ldots,v_m)\colon
 P\to T\times_S\mathbb A_S^{2m}.
\]
Its relative differential at $z$ is an isomorphism: a tangent vector in
its kernel is first killed by every $dv_i$, so lies in $T_z\Delta$; its
$(dr,d\text{leg})$ then places it in $T_z\Delta_{(u,x)}$; then the
derivative isomorphism in \eqref{eq:two-coefficients} forces the vector
to vanish. Source and target are both smooth over $S$ with the same
relative dimension $N+1+2m$, so this injective differential is an
isomorphism, and the relative Jacobian criterion makes $\Phi$ \'etale
at $z$.

By \eqref{eq:two-exact-quadratic}, the cut $P_D=V(f(c))$ is exactly
$\Phi^{-1}(Q^a)$. So the induced map $\Phi_D\colon P_D\to Q^a$ is
\'etale near $z$, sending $z$ to $\nu$. Moreover \eqref{eq:two-chart-lift}
identifies
\[
 a_K^*F\simeq r_K^*F_U\simeq\Phi_{D,K}^*\tau_K^*F_T
 \quad\text{on }(P_D)_K,
\]
so $\Phi_D$ identifies the actual generic sheaf and the equation
simultaneously. \'Etale base change transforms \eqref{eq:two-isolated-zero}
into \eqref{eq:two-vertex}.

If $m=0$, the diagonal is open near $z$, the sum in
\eqref{eq:two-exact-quadratic} is empty, and the same argument gives
$Q^a=D\times_S\mathscr X$. Then smooth base change along $D\times_S
\mathscr X\to D$ yields $(\Psi_D(F_U|_{D_K}))_u=0$ directly.
\end{proof}

\begin{remark}
The argument used no division by 2 and no singular-support claim in
mixed characteristic. Its output for $m>0$ is a vanishing on the quadric
$Q^a$; the next section recovers the corresponding vanishing on the
hypersurface $V(f)$ itself.
\end{remark}

% =========================================================================
\section{From quadratic models to transverse slices}
\label{sec:slices}
% =========================================================================

Section~\ref{sec:twohecke} gave a vanishing on the affine quadric $Q^a$
at its vertex. To get a test on the hypersurface $D=V(f)$ itself, we
compactify the quadric to a projective family and use the relation
between hyperplane powers and the closed support $D$. Concretely, the
cone of the hyperplane-class map is a shifted constant Tate line
supported on $f=0$; after tensoring with the ambient sheaf, the
hyperplane terms have vanishing nearby cycles from the smooth-chart
result of Lemma~\ref{spec:vanishing}, and the whole proper quadric
vanishing then forces the hypersurface vanishing. A second proper
comparison, using projective incidence, handles an arbitrary collection
of equations simultaneously.

\subsection{The projective quadric comparison}

\begin{lemma}[Quadric comparison]\label{slice:quadric}
Let $K_0$ be an algebraically closed field of characteristic $\ne\ell$,
$E=\overline{\mathbb Q}_\ell$, and $B$ a finite-type $K_0$-scheme. For
$f\in\Gamma(B,\mathcal O_B)$ put $i\colon D=V(f)\hookrightarrow B$. If
$m\ge1$, form
\[
 g\colon Q_f=\Bigl\{\sum_{a=1}^m u_av_a+fw^2=0\Bigr\}
       \subset\mathbb P_B^{2m}\longrightarrow B.
\]
Then the hyperplane-class map defines a distinguished triangle
\begin{equation}\label{eq:slice-quadric-triangle}
 \bigoplus_{j=0}^{2m-1}E_B(-j)[-2j]
 \longrightarrow Rg_*E_{Q_f}
 \longrightarrow i_*E_D(-m)[-2m]
 \longrightarrow \Bigl(\bigoplus_{j=0}^{2m-1}E_B(-j)[-2j]\Bigr)[1].
\end{equation}
The identification of the third term with the Tate line requires only a
fixed one-dimensional generator. Neither $B$ nor $D$ needs to be smooth
or reduced.
\end{lemma}

\begin{proof}
Write $h=c_1(\mathcal O_{Q_f}(1))\in H^2(Q_f,E(1))$. Adjunction on each
$h^j$ gives the morphism $E_B(-j)[-2j]\to Rg_*E_{Q_f}$. Let
$\mathcal K$ be the cone of their direct sum, and let
$j_B\colon B\setminus D\hookrightarrow B$.

We recall the cohomology of a split smooth projective quadric of
dimension $n$: no odd cohomology, one $E(-r)$ in degree $2r$ for
$0\le r\le n$, and one additional copy when $n$ is even and $r=n/2$
\cite[Example~16.3,~Thm.~23.2]{MilneEtale}. This follows by induction on
the equation $x_0x_1+q'(x_2,\ldots)=0$; the chart $x_0\ne0$ is affine
$n$-space, whose complement is the projective cone over the quadric
of dimension $n-2$ with the vertex removed (a line bundle over that
smaller quadric). The resulting filtration by affine cells has one cell
in every dimension $0,\ldots,n$, and a second middle-dimensional cell
when $n$ is even. Initial cases are a conic ($\cong\mathbb P^1$) and
the split zero-dimensional quadric (two points). Localization plus
$R\Gamma_c(\mathbb A^r,E)=E(-r)[-2r]$ gives the cohomology. On a smooth
quadric of positive dimension the top hyperplane power has degree $2$,
hence is nonzero in $E$; all lower powers are nonzero as well.

Over $B\setminus D$, the geometric fibres of $g$ are smooth quadrics of
odd dimension $2m-1$, so the calculation shows that the hyperplane-power
map is an isomorphism on their cohomology. Proper base change therefore
gives $j_B^*\mathcal K=0$.

Over $D$ the family, with its hyperplane line bundle, is the constant
family with fibre
\[
 V_m=\Bigl\{\sum_{a=1}^m u_av_a=0\Bigr\}\subset\mathbb P_{K_0}^{2m}.
\]
Let $e=[0:\cdots:0:1]$ be its vertex. Projection away from $e$ identifies
$V_m\setminus\{e\}$ with the total space of a line bundle over the split
smooth quadric
\[
 C_m=\Bigl\{\sum_{a=1}^m u_av_a=0\Bigr\}\subset\mathbb P_{K_0}^{2m-1}.
\]
Compactly supported cohomology of this line bundle, plus localization at
the vertex, gives
\begin{equation}\label{eq:slice-cone-cohomology}
 H^0(V_m,E)=E,\qquad H^1(V_m,E)=0,\qquad
 H^r(V_m,E)=H^{r-2}(C_m,E)(-1)\quad(r\ge2).
\end{equation}
So $V_m$ has the cohomology of $\mathbb P^{2m-1}$ plus one extra
$E(-m)$ in degree $2m$.

The hyperplane powers on $V_m$ remain nonzero. To check the top one
without requiring smoothness of $V_m$, resolve the vertex by the
projective bundle of lines
\[
 \pi\colon\widetilde V_m=\mathbb P_{\mathrm{lines},C_m}
   (\mathcal O_{C_m}(-1)\oplus\mathcal O_{C_m})
   \longrightarrow C_m.
\]
The inclusion $\mathcal O_{C_m}(-1)\oplus\mathcal O_{C_m}\hookrightarrow
(\text{trivial rank-}(2m+1))$ defines $\widetilde V_m\to V_m$; its
fibre over a point of $C_m$ is the projective line joining that point
to $e$. If $\zeta$ is the pullback of the hyperplane class and
$h_C=c_1(\mathcal O_{C_m}(1))$, the projective-bundle relation is
$\zeta^2=\pi^*h_C\,\zeta$ \cite[\S23]{MilneEtale}, hence
\[
 \int_{\widetilde V_m}\zeta^{2m-1}=\int_{C_m}h_C^{2m-2}=2.
\]
So the top hyperplane power on $V_m$ is nonzero; by multiplication, so
are the lower powers. This includes $m=1$, where $C_1$ is two points
and $V_1$ is two lines meeting at $e$.

Therefore the cone of the vector-space morphism
$\bigoplus_{j=0}^{2m-1}E(-j)[-2j]\to R\Gamma(V_m,E)$ has exactly one
nonzero cohomology group, $E(-m)$ in degree $2m$: injectivity in every
degree (from nonvanishing of the hyperplane powers) rules out any others.
The cone is $E(-m)[-2m]$.

Proper base change identifies $i^*\mathcal K$ with the constant
pullback of this vector-space cone (both the family over $D$ and the
hyperplane-power morphism are pulled back from $V_m$). Open-closed
localization gives
\[
 (j_B)_!j_B^*\mathcal K\to\mathcal K\to i_*i^*\mathcal K\to\cdot,
\]
whose first term vanishes, so $\mathcal K\simeq i_*E_D(-m)[-2m]$. In
particular the third term in \eqref{eq:slice-quadric-triangle} is
constant along $D$; no middle-cohomology local system is left. Localization
and proper base change hold with finite coefficients
\cite[Tags 095L, 0DDE]{Stacks} and in their adic forms
\cite[Tags 09AH, 09C9]{Stacks}; we use them after inverting $\ell$ and
extending to $E$.
\end{proof}

\begin{remark}\label{rem:slice-rational-coefficients}
Lemma~\ref{slice:quadric} uses rational coefficients, including
$\ell=2$. Its hyperplane-power comparison is not an isomorphism off $D$
with $\mathbb Z/2^a$ coefficients: on a conic the hyperplane class has
degree $2$. We use finite coefficients to build the standard
compatibilities, then use \eqref{eq:slice-quadric-triangle} after
rationalizing. This imposes no restriction on residue characteristic.
The case $m=0$ is handled below directly; formally $Q_f=D\subset
\mathbb P_B^0$ and the hyperplane sum is empty.
\end{remark}

\subsection{The hypersurface test}

Work over the trait $S$, with $\mathcal B$ and $F\in\mathcal D_C$ as in
Proposition~\ref{spec:missing}. For a smooth finite-type chart
$b\colon U\to\mathcal B$, write $F_U=b_K^*F$, and let
$\mathcal N_U\subset T^*U_s$ be the image of the global nilpotent cone
under smooth cotangent pullback.

\begin{proposition}[Hypersurface test]\label{slice:hypersurface}
Let $b\colon U\to\mathcal B$ be smooth with $U$ finite type over $S$.
Let $f\in\mathcal O(U)$, $D=V(f)$, and $u\in D_s(k)$. If
\[
 d(f_s)_u\notin(\mathcal N_U)_u,
\]
then
\[
 \bigl(\Psi_D(F_U|_{D_K})\bigr)_u=0.
\]
\end{proposition}

\begin{proof}
If $d(f_s)_u$ does not lie in the image of $T^*_{b(u)}\mathcal B_s$,
its image in the relative cotangent of $b$ is nonzero, so $D\to\mathcal B$
is smooth near $u$; Lemma~\ref{spec:vanishing} gives the claim.

Otherwise $d(f_s)_u=b^*A$ for some non-nilpotent
$A\in T^*_{b(u)}\mathcal B_s$. Proposition~\ref{two:vertex} then provides
$x\in\mathscr X_s(k)$ and $m\ge0$ such that nearby cycles of the
pullback of $F_U$ to
$Q^a=\{\sum_{a=1}^m u_av_a+f=0\}$ vanish at $(t_0,0,0)$, where
$T=U\times_S\mathscr X$ and $t_0=(u,x)$. If $m=0$ this affine model is
$D_T$ and smooth compatibility for $D_T\to D$ gives the result.

Assume $m\ge1$. Compactify to
\[
 g\colon Q_f=\Bigl\{\sum_{a=1}^m u_av_a+fw^2=0\Bigr\}
   \subset\mathbb P_T^{2m}\to T.
\]
The fibre over $t_0$ is the split cone of Lemma~\ref{slice:quadric}. At
every point of this fibre except its vertex, some $u_a$ or $v_a$ is
nonzero: in the chart $u_a=1$ its equation solves uniquely for $v_a$
and $g$ is smooth there (works also in characteristic 2). Lemma~\ref{spec:vanishing}
gives $\Psi_TF_T=0$, and smooth compatibility gives vanishing of nearby
cycles of $g_K^*F_T$ at all those nonvertex points. The vertex is in
the chart $w=1$, where Proposition~\ref{two:vertex} gives the vanishing.

So the nearby-cycle complex restricts to zero on the entire proper
fibre. Proper compatibility plus the projection formula give
\begin{equation}\label{eq:slice-quadric-pushforward-zero}
 \bigl(\Psi_T(F_T\otimes Rg_{K*}E)\bigr)_{t_0}=0.
\end{equation}
Apply Lemma~\ref{slice:quadric} on $T_K$ and tensor with $F_T$: the
projection formula identifies the third term with
$i_{K,*}(F_T|_{(D_T)_K})(-m)[-2m]$ for
$i\colon D_T\hookrightarrow T$. After $\Psi_T$ and stalk at $t_0$,
the first term vanishes ($\Psi_TF_T=0$) and the second by
\eqref{eq:slice-quadric-pushforward-zero}; proper compatibility for $i$
identifies the third with
$(\Psi_{D_T}(F_T|_{(D_T)_K}))_{t_0}(-m)[-2m]$, which therefore also
vanishes. Smooth compatibility for $D_T\to D$ finishes the proof.
\end{proof}

The passage through the projective quadric uses no purity theorem for
the possibly singular hypersurface $D$; it uses the actual closed-support
term of the comparison triangle. The same principle handles several
equations simultaneously.

\subsection{Projective incidence and higher codimension}

\begin{lemma}[Incidence comparison]\label{slice:incidence}
Let $B$ be a finite-type scheme over an algebraically closed field of
characteristic $\ne\ell$, let $f_1,\ldots,f_d\in\Gamma(B,\mathcal O_B)$
with $d\ge1$, and set $i\colon J=V(f_1,\ldots,f_d)\hookrightarrow B$.
For the incidence family
\[
 r\colon I=\Bigl\{\sum_{a=1}^d c_af_a=0\Bigr\}
   \subset B\times\mathbb P^{d-1}\longrightarrow B,
\]
the hyperplane powers give a distinguished triangle
\begin{equation}\label{eq:slice-incidence-triangle}
 \bigoplus_{j=0}^{d-2}E_B(-j)[-2j]
 \longrightarrow Rr_*E_I
 \longrightarrow i_*E_J(-(d-1))[-2(d-1)]
 \longrightarrow \Bigl(\bigoplus_{j=0}^{d-2}E_B(-j)[-2j]\Bigr)[1].
\end{equation}
For $d=1$ the first term is zero and $I=J$.
\end{lemma}

\begin{proof}
If $d=1$, the equation in $B\times\mathbb P^0$ is $f_1=0$ and the
statement is immediate. Suppose $d\ge2$. Off $J$, the map
$\mathcal O_B^d\to\mathcal O_B$ with coefficients $(f_1,\ldots,f_d)$ is
surjective, so its kernel is a vector bundle of rank $d-1$ and $I$ is
its projective bundle of lines. The restricted hyperplane powers identify
its direct image with the first term of
\eqref{eq:slice-incidence-triangle} (check on geometric fibres, using
proper base change and the cohomology of $\mathbb P^{d-2}$).

On $J$, the family is exactly $J\times\mathbb P^{d-1}$ and its hyperplane
class comes from the second factor. The restricted cone is therefore the
constant top class $E_J(-(d-1))[-2(d-1)]$. The global cone vanishes off
$J$, so open-closed localization identifies it with the pushforward of
this restricted cone.
\end{proof}

\begin{proposition}[Higher-codimension test]\label{slice:higher}
Let $b\colon U\to\mathcal B$ be smooth with $U$ finite type over $S$,
and let $F_U=b_K^*F$. Suppose $u\in U_s(k)$ and $f_1,\ldots,f_d$ are
regular on $U$ vanishing at $u$. Suppose their special-fibre
differentials at $u$ are linearly independent and
\[
 \operatorname{span}_k\{d(f_{1,s})_u,\ldots,d(f_{d,s})_u\}
 \cap(\mathcal N_U)_u=\{0\}.
\]
Then for $J=V(f_1,\ldots,f_d)$,
\[
 (\Psi_J(F_U|_{J_K}))_u=0.
\]
The case $d=0$ is allowed, with $J=U$.
\end{proposition}

\begin{proof}
For $d=0$ this is Lemma~\ref{spec:vanishing}. For $d\ge1$ form the
proper incidence morphism
\[
 r\colon I=\Bigl\{\sum_{a=1}^d c_af_a=0\Bigr\}
 \subset U\times_S\mathbb P_S^{d-1}\longrightarrow U.
\]
Each affine projective-space chart in $U\times_S\mathbb P_S^{d-1}$ is
still smooth over $\mathcal B$. On the chart $c_a=1$, the hypersurface
equation is $f_a+\sum_{b\ne a}c_bf_b=0$, with special-fibre differential
at $(u,[c])$ equal to $\sum_{b=1}^dc_b\,d(f_{b,s})_u$ (the projective
parameter term is zero since every $f_b(u)=0$). This combination is
nonzero by linear independence and, by hypothesis, lies outside
$(\mathcal N_U)_u$. The nilpotent cone on the product chart is the
pullback of $\mathcal N_U$ with zero parameter component, so
Proposition~\ref{slice:hypersurface} gives vanishing of nearby cycles of
$r_K^*F_U$ at every closed point over $u$.

The restriction of this nearby-cycle complex to the projective fibre is
constructible; a nonzero constructible complex on a finite-type scheme
over $k$ has a nonzero stalk at some closed $k$-point, so it must
vanish on the entire fibre. Proper compatibility gives
\[
 (\Psi_U Rr_{K*}r_K^*F_U)_u=0.
\]
Tensor the triangle of Lemma~\ref{slice:incidence} on $U_K$ with $F_U$
and use the projection formula: the first term is a finite sum of shifts
and twists of $F_U$, the second is $Rr_{K*}r_K^*F_U$, and the third is
$i_{K*}(F_U|_{J_K})(-(d-1))[-2(d-1)]$ for $i\colon J\hookrightarrow U$.
After $\Psi_U$, the first two terms vanish at $u$, and proper
compatibility for $i$ shows that the third is a shift and twist of
$(\Psi_J(F_U|_{J_K}))_u$. Thus the third vanishes. For $d=1$ the first
sum is empty and the same argument applies.
\end{proof}

We have thus obtained all finite transverse-slice tests, while applying
the hypersurface argument only to schemes smooth over $\mathcal B$. It
remains to find one such slice that meets $\Psi$ on the support of $F$
in a nonzero way.

% =========================================================================
\section{A finite slice of the support}
\label{sec:support}
% =========================================================================

We finish the proof of Theorem~\ref{thm:full-support} by applying the
slicing theorem to the actual support of $F$. The structure of the
argument: we first arrange that a nonzero generic stalk specializes to
the closed fibre; then we cut the support down to a finite scheme over
the trait; on a finite scheme nearby cycles are a finite direct sum of
stalks, so they cannot vanish if any single one is nonzero.

\subsection{Extending a generic bundle}

\begin{lemma}\label{support:extension}
Let $\mathscr X\to S$ be the smooth projective curve from
Section~\ref{sec:specialization}, and let $K$ be the fixed algebraic
closure of $\operatorname{Frac}(R)$. Every $G$-bundle on $\mathscr X_K$
extends to a $G$-bundle on $\mathscr X_{R'}$ for some finite extension
$\operatorname{Frac}(R')/\operatorname{Frac}(R)$ inside $K$, where
$R'$ is the integral closure of $R$ in that extension.
\end{lemma}

\begin{proof}
Let $\mathcal P_K$ be the bundle. By
\cite[Thm.~2,~Rem.~2(b)]{DrinfeldSimpson} it is trivial on some
nonempty open subset of $\mathscr X_K$ (this theorem applies to connected
reductive groups over any field). Choose a trivialization away from
distinct points $x_1,\ldots,x_r\in\mathscr X(K)$. Gluing at those points
gives a chain of one-point modifications
\[
 \mathcal P_0\dashrightarrow\mathcal P_1\dashrightarrow\cdots
 \dashrightarrow\mathcal P_r=\mathcal P_K,\qquad
 \mathcal P_0\text{ trivial},
\]
whose $i$th modification has leg $x_i$. Choose a Schubert bound for each
of the finitely many relative positions.

Start with the trivial bundle over $S=\Spec R$ and form the space of
bounded chains (legs varying on $\mathscr X$). At each stage the bounded
Hecke space is representable and proper over the preceding bundle and
leg. Since $\mathscr X/S$ is proper, induction shows that the space of
chains is a proper algebraic space of finite type over $S$. It carries
the universal final bundle and contains the chosen chain as a
$K$-point. This point descends to a finite extension of
$\operatorname{Frac}(R)$ (the space is finite type and $K$ is algebraic).
The valuative criterion extends the descended point over $R'$, and the
universal final bundle gives the extension. The construction allows
legs to coincide on the closed fibre, since it uses successive one-point
modifications.
\end{proof}

Now suppose a spectral component is missing and let $F$ be the nonzero
compact object from Proposition~\ref{spec:missing}. It is constructible
on finite-type smooth charts, so there is a $K$-valued bundle at which
its stalk is nonzero (a nonzero constructible complex on a finite-type
scheme has a nonzero stalk at some closed $K$-point). Apply
Lemma~\ref{support:extension} to that bundle and replace $R$ by the
resulting $R'$; all specialization statements from
Section~\ref{sec:specialization} remain valid with the same geometric
generic field $K$.

Choose a finite-type smooth affine chart $b\colon U\to\mathcal B$
through the special value of the extended bundle. The fibre product of
this chart with its trait section is smooth over $S$ and has a
$k$-point. Formal smoothness lifts that $k$-point to compatible points
modulo every power of the uniformizer, and completeness of $R$ then
gives an honest $R$-point of an affine \'etale neighbourhood; thus the
trait section lifts to $U$. After shrinking around its closed point, $U/S$
has constant relative dimension $N$. The generic stalk of $F_U=b_K^*F$
on the lifted section is nonzero, so the support closure of $F_U$ meets
$U_s$.

\subsection{Finding a slice with nonzero nearby cycles}

The next lemma is purely about the dimension of a closed cone in the
special-fibre cotangent bundle. We state it separately to make clear
that only the \emph{support strata}, not the sheaf itself, are descended
to a finite extension.

\begin{lemma}\label{support:finite-slice}
Let $R$ be a complete discrete valuation ring with algebraically closed
residue field $k$ of characteristic $\ne\ell$, $S=\Spec R$, and $K$ an
algebraic closure of its fraction field. Let $U/S$ be a smooth affine
scheme of constant relative dimension $N$, $\mathcal F$ a bounded
constructible $E$-complex on $U_K$, and
$\Lambda\subset T^*U_s$ a closed conical subset with $\dim\Lambda\le N$.
Suppose the closure in $U$ of the nonzero-stalk locus of $\mathcal F$
meets $U_s$. Then after a finite extension of the trait and shrinking
$U$, there exist $u\in U_s(k)$, $0\le d\le N$ and functions
$f_1,\ldots,f_d$ vanishing at $u$ whose special-fibre differentials are
independent, such that
\[
 \operatorname{span}\{d(f_{1,s})_u,\ldots,d(f_{d,s})_u\}
 \cap\Lambda_u\subseteq\{0\},
\]
and for $J=V(f_1,\ldots,f_d)$,
\begin{equation}\label{eq:support-nonzero}
 \bigl(\Psi_J(\mathcal F|_{J_K})\bigr)_u\ne0.
\end{equation}
For $d=0$ the list is empty and $J=U$.
\end{lemma}

\begin{proof}
\emph{Step 1: descend the finite support data.}
Let $A_1,\ldots,A_t$ be the irreducible components of the closure of the
nonzero-stalk locus in $U_K$. Bounded constructibility gives a proper
closed subset $B_i\subsetneq A_i$ such that every stalk on
$A_i\setminus B_i$ is nonzero. Give these closed sets reduced
structures. Their finitely generated ideals descend to a common finite
extension of $\operatorname{Frac}(R)$; enlarge that extension to descend
all containments too. Replace the trait accordingly. The new support
closure still meets the special fibre: the finite map from the new
model to the old is closed and agrees with the geometric generic
support, so it maps the new closure onto the old. This step does not
require $\mathcal F$ to descend.

Write $\overline A_i,\overline B_i$ for the horizontal closures in $U$,
and $e_i=\dim A_i$. Each $\overline A_i$ is integral and dominates $S$.
Every nonempty special fibre $(\overline A_i)_s$ is pure of dimension
$e_i$ \cite[Tag~0B2J]{Stacks}. Applying this to the irreducible
components of $B_i$ shows that whenever
$(\overline B_i)_s\ne\varnothing$,
\[
 \dim(\overline B_i)_s<e_i.
\]
The trait is excellent, hence universally catenary (a complete Noetherian
local ring is excellent \cite[Tag~07QW]{Stacks}), so at any closed
$v\in(\overline A_i)_s$,
\begin{equation}\label{eq:support-local-dimension}
 \dim\mathcal O_{\overline A_i,v}=e_i+1
 \qquad\text{\cite[Tag~02JT]{Stacks}}.
\end{equation}

\emph{Step 2: choose the special point.}
Let $d$ be the maximum of the $e_i$ such that $(\overline A_i)_s\ne
\varnothing$; such exists by the support assumption. Choose a reduced
irreducible component $T\subset(\overline A_i)_s$ of dimension $d$ for
some such $i$. At a general closed point of $T$ the reduced structure is
smooth, no other distinct reduced component of $\bigcup_i(\overline A_i)_s$
passes through the point, and no $\overline B_i$ passes through it: an
$i$ with empty special fibre contributes nothing, while every remaining
bad special locus has dimension $<d$. Different horizontal components
may share the same reduced $T$; we keep them all.

We may also arrange
\begin{equation}\label{eq:support-cone-fiber}
 \dim\Lambda_u\le N-d.
\end{equation}
Indeed, restrict $\Lambda$ to $T$. Its components that do not dominate
$T$ can be excluded over a proper closed subset of $T$. The generic
fibre of each remaining component has dimension $\le N-d$ because
$\dim\Lambda\le N$; the same bound holds over a nonempty open subset of
$T$. Choose $u\in T(k)$ satisfying all requirements.

\emph{Step 3: choose transverse equations.}
Let $V=T_u^*U_s$, so $\dim V=N$. If $d>0$, a general $d$-plane
$L\subset V$ satisfies $L\cap\Lambda_u\subseteq\{0\}$: from
\eqref{eq:support-cone-fiber},
$\dim\mathbb P(\Lambda_u)\le N-d-1$, and the incidence of $d$-planes
meeting this projective set is a proper closed subset of $\Gr(d,V)$. The
condition that $L\to T_u^*T$ be an isomorphism is another nonempty open
condition; two nonempty opens meet because the Grassmannian is
irreducible. Pick such an $L$ and a basis of it; for $d=0$ take $L=0$.

Lift the basis to functions $f_1,\ldots,f_d$ vanishing at $u$, on an
affine neighbourhood in $U$. Their restrictions to the smooth
$d$-dimensional $T$ form a regular system of parameters at $u$. Hence
for $J=V(f_1,\ldots,f_d)$ and $Z=J\cap\bigcup_i\overline A_i$, the point
$u$ is isolated in $Z_s$ after shrinking; the reduced special fibre of
the union of all support closures near $u$ is precisely $T$, even if
several horizontal components specialize to $T$.

\emph{Step 4: a generic generization in the slice.}
Choose $i$ with $e_i=d$ and $T$ a component of $(\overline A_i)_s$. By
\eqref{eq:support-local-dimension}, $\dim\mathcal O_{\overline A_i,u}
=d+1$. Quotienting by $f_1,\ldots,f_d$ leaves dimension $\ge1$; then
quotienting by a uniformizer $\pi$ gives dimension zero (isolation). So
some prime of the cut local ring does not contain $\pi$ -- otherwise
$\pi$ would be nilpotent and the ring would be dimension zero. This gives
a generic-fibre point $z$ specializing to $u$.

The morphism $Z\to S$ is quasi-finite at $u$ (closed point isolated in
its fibre \cite[Tag~01TH]{Stacks}); its quasi-finite locus is open, so
it contains $z$. Thus $z$ has residue field finite over
$\operatorname{Frac}(R)$. Its geometric lifts to $K$ avoid $B_i$: a
point in the closed set $\overline B_i$ cannot specialize to
$u\notin\overline B_i$. Each such lift therefore has a nonzero stalk of
$\mathcal F|_{J_K}$.

\emph{Step 5: nearby cycles on a finite neighbourhood.}
Take an affine neighbourhood of $u$ inside the quasi-finite locus of
$Z\to S$. The henselian decomposition of a finite-type algebra
\cite[Tag~04GJ]{Stacks} gives a finite local factor containing $u$: an
open neighbourhood $Q$ of $u$ in $Z$, finite over $S$, whose special
fibre has only the point $u$ (equivalently, Zariski's main theorem
embeds a quasi-finite neighbourhood into a finite $S$-scheme; the
henselian local factor at $u$ lies in the original open because an open
subset of a local spectrum containing the closed point is the whole
spectrum). Every generization of $u$, including $z$, lies in $Q$.

The restriction $\mathcal F|_{J_K}$ is supported on $Z_K$. To use the
finite neighbourhood, remove $Z\setminus Q$ from $J$: this complement is
closed in $J$ hence in $U$, so remove it from $U$ and pass to an affine
neighbourhood of $u$. That neighbourhood contains $Q$ because every
point of the finite local $Q$ specializes to $u$. The new $J$ is still
cut out by the same functions in a smooth affine chart, and the change
leaves its nearby-cycle stalk at $u$ unchanged while making $Q$ closed
in $J$. If $\iota\colon Q\hookrightarrow J$ is this closed immersion
and $\mathcal H=\mathcal F|_{Q_K}$, localization gives
$\mathcal F|_{J_K}\simeq\iota_{K,*}\mathcal H$. Proper compatibility for
$\iota$ reduces \eqref{eq:support-nonzero} to the stalk of $\Psi_Q
\mathcal H$ at $u$.

Let $a\colon Q\to S$ be the finite structural map. Its special fibre
has only $u$, so proper compatibility gives
\[
 (\Psi_Q\mathcal H)_u\simeq\Psi_S(\RGamma(Q_K,\mathcal H))
 \simeq\RGamma(Q_K,\mathcal H).
\]
The second identification uses geometric specialization on the trait,
which is the identity on coefficient complexes. The finite $K$-scheme
$Q_K$ has the \'etale topos of a finite discrete set, so
$\RGamma(Q_K,\mathcal H)\simeq\bigoplus_{x\in|Q_K|}\mathcal H_x$. At
least one summand is nonzero (Step 4), so this complex is nonzero,
proving \eqref{eq:support-nonzero}.

These are geometric nearby cycles with monodromy forgotten; no inertia
invariants. The finite direct-sum calculation is compatible with finite
coefficients, their $\ell$-adic limit, inversion of $\ell$, and
extension to $E$, so it applies to the specialization functor of
Section~\ref{sec:specialization} without requiring a field of descent
for $\mathcal F$.
\end{proof}

\subsection{Completion of the full-support proof}

\begin{proof}[Proof of Theorem~\ref{thm:full-support}]
If $Y'\ne Y$, Proposition~\ref{spec:missing} supplies the nonzero
compact $F$ used above. Lemma~\ref{support:extension} and the
subsequent chart construction give a smooth affine $U\to\mathcal B$ of
constant relative dimension $N$ for which the closure of the nonzero-stalk
locus of $F_U$ meets $U_s$.

Let $\mathcal N_U\subset T^*U_s$ be the pullback of the global nilpotent
cone. Under either hypothesis regime, the nilpotent-parabolic
assumption and invariant form give the half-dimensionality theorem of
\cite[App.~D.1.1--D.1.8]{AGKRRV}. In its smooth-chart formulation
\cite[App.~D.1.2]{AGKRRV}, this says $\dim\mathcal N_U\le\dim U_s=N$.

Apply Lemma~\ref{support:finite-slice} with $\mathcal F=F_U$ and
$\Lambda=\mathcal N_U$. It gives a slice $J=V(f_1,\ldots,f_d)$ and
$u\in J_s(k)$ such that
\[
 (\Psi_J(F_U|_{J_K}))_u\ne0
\]
while the independent differentials of its equations span a plane
meeting $(\mathcal N_U)_u$ only at zero. The higher-codimension vanishing
of Proposition~\ref{slice:higher} applies to the same chart and slice
and says that this stalk is zero. Contradiction. This also handles $d=0$,
where the test is the original smooth-chart vanishing.

Thus the complementary union is empty. The primed equivalence recalled
in Section~\ref{sec:specialization} is consequently the full restricted
equivalence asserted in Theorem~\ref{thm:full-support}.
\end{proof}

% =========================================================================
\section{Component categories and consequences}
\label{sec:consequences}
% =========================================================================

Throughout this section assume Hypothesis~\ref{hyp:A} or
Hypothesis~\ref{hyp:B}, so Theorem~\ref{thm:full-support} holds.
Put
\[
 \mathcal C_s=\Shv_{\Nilp}(\Bun_G),\qquad
 \mathcal A=\QCoh(Y),\qquad
 \mathcal D=\IndCoh_{\Nilp}(Y),
\]
and write $\mathbb L_G^{\mathrm{restr}}\colon\mathcal C_s\to\mathcal D$
for the restricted Langlands equivalence. Its $\mathcal A$-linear
structure is the one furnished by \cite[Lemma~1.3.7]{GR}.

\subsection{The category attached to one component}

For an open-and-closed component $i\colon Z\hookrightarrow Y$, define
\[
 \mathcal C_{s,Z}:=\QCoh(Z)\underset{\QCoh(Y)}\otimes\mathcal C_s,
\]
the tensor product being that of presentable dg categories. The
idempotent quasi-coherent $\mathcal O_Z$ (extended by zero) acts as the
projection to this summand.

\begin{corollary}\label{cor:components}
Under either hypothesis regime, every connected component $Z\subset Y$
has nonzero localized automorphic category, and the restriction of
$\mathbb L_G^{\mathrm{restr}}$ induces an equivalence
$\mathcal C_{s,Z}\xrightarrow{\sim}\IndCoh_{\Nilp}(Z)$.
\end{corollary}

\begin{proof}
The decomposition $Y=Z\sqcup(Y\setminus Z)$ decomposes quasi-coherent
and nilpotent ind-coherent sheaves into two summands, so
$\QCoh(Z)\otimes_{\mathcal A}\mathcal D\simeq\IndCoh_{\Nilp}(Z)$. Tensor
the $\mathcal A$-linear equivalence of Theorem~\ref{thm:full-support}
with $\QCoh(Z)$ over $\mathcal A$, and the assertion follows.

Every component contains a semisimple $E$-point
\cite[Prop.~3.7.2,~Cor.~3.7.4]{AGKRRV}. Pullback to this point shows
$\mathcal O_Z\ne0$, and the fully faithful
$\QCoh(Z)\hookrightarrow\IndCoh_{\Nilp}(Z)$ then gives nonvanishing.
\end{proof}

\begin{remark}
The component index and the parameter are not the same object. Two
geometric parameters lie in the same component iff their semisimp-
lifications are isomorphic \cite[Prop.~3.7.2]{AGKRRV}. The construction
of eigenobjects below uses the actual parameter, including its
extension data, not just its semisimple base point.
\end{remark}

\subsection{The arithmetic trace formula}

Assume Hypothesis~\ref{hyp:A}. The finite-field models $X_0,G_0$
determine geometric $q$-Frobenius, whose pullback action on $\QLisse(X)$
induces an automorphism $\Frob$ of $Y$. Define the derived fixed-point
stack
\[
 Y^{\mathrm{arithm}}:=Y^{\Frob}=Y\underset{Y\times_EY}{\overset{\mathrm R}\times}Y,
\]
where the two maps to $Y\times_EY$ are the diagonal and the graph of
$\Frob$. An $E$-point of $Y^{\mathrm{arithm}}$ is a geometric local
system together with a Weil structure. See \cite[\S24.1]{AGKRRV}
and \cite[\S1.5.1]{GR}.

\begin{corollary}\label{cor:arithmetic}
For the finite-field datum of Hypothesis~\ref{hyp:A} there is a
canonical isomorphism
\[
 \Funct_c(\Bun_{G_0}(\mathbb F_q),E)
 \simeq\Gamma^{\IndCoh}(Y^{\mathrm{arithm}},\omega_{Y^{\mathrm{arithm}}}).
\]
The left side is $H^0$ of compactly supported $E$-valued functions on
isomorphism classes of rational bundles; the right side uses the
ind-coherent dualizing object and its ind-coherent global sections on the
derived fixed-point stack.
\end{corollary}

\begin{proof}
The primed union $Y'$ is Frobenius-stable. The spectral trace
calculation of Beraldo--Lin--Reeves
\cite[\S\S6.4.12--6.4.13]{BeraldoLinReeves} enters the known primed
formula \cite[\S\S1.5.3--1.5.5,~Cor.~1.5.6]{GR}:
\[
 \Funct_c(\Bun_{G_0}(\mathbb F_q),E)
 \simeq\Gamma^{\IndCoh}((Y')^{\Frob},\omega_{(Y')^{\Frob}}).
\]
Its automorphic trace input is \cite[Thm.~0.2.6]{AGKRRVTrace} with
endofunctor $q$-Frobenius pushforward on $\Bun_G$. By
Theorem~\ref{thm:full-support}, the inclusion $Y'\hookrightarrow Y$ is
equality. Taking derived fixed points identifies $(Y')^{\Frob}$ with
$Y^{\mathrm{arithm}}$ and transports the dualizing object. Substitution
gives the stated formula.
\end{proof}

\begin{remark}
Keeping $\Gamma^{\IndCoh}$ explicit avoids a convention in
\cite[\S1.5.3]{GR}, whose "ordinary global sections" notation uses the
image of $\omega$ under the dual of $\Upsilon\colon\QCoh\to\IndCoh$. We
neither truncate $Y^{\mathrm{arithm}}$ classically nor replace its
dualizing object by its structure sheaf.
\end{remark}

\subsection{Constructing Hecke eigenobjects}

We now prove that every parameter has a nonzero eigenobject. The
categorical input is simple.

\begin{lemma}\label{lem:parameter-adjoint}
Let $\mathcal A_0$ be a compactly generated, presentable, stable,
$E$-linear symmetric monoidal category whose compact objects are
dualizable, and suppose its tensor product preserves colimits
separately. Let $a\colon\mathcal A_0\to\Vect_E$ be a continuous
$E$-linear symmetric monoidal functor, and give $\Vect_E$ the
$\mathcal A_0$-action through $a$. Then the right adjoint $R_a$ of $a$
is continuous, carries a coherent $\mathcal A_0$-linear structure, and
$R_a(E)\ne0$.
\end{lemma}

\begin{proof}
Presentability gives $R_a$. Every compact object of $\mathcal A_0$ is
dualizable, so its image under $a$ is a perfect complex; thus $a$
preserves compact objects. Testing against compact generators and using
adjunction shows $R_a$ preserves filtered colimits. Both categories are
stable, so $R_a$ is exact, and exact + colimit-preserving implies
colimit-preserving.

The canonical projection $b\otimes R_a(V)\to R_a(a(b)\otimes V)$ is
invertible for dualizable $b$: for any $c$,
\begin{align*}
 \RHom_{\mathcal A_0}(c,b\otimes R_a(V))
 &\simeq\RHom_{\mathcal A_0}(b^\vee\otimes c,R_a(V))\\
 &\simeq\RHom_E(a(b)^\vee\otimes a(c),V)\\
 &\simeq\RHom_E(a(c),a(b)\otimes V).
\end{align*}
Both sides are colimit-preserving in $b$, so compact generation proves
invertibility for all $b$. These projections are adjunction mates of
the module structure on $a$, hence satisfy the unit and associativity
coherences before invertibility is imposed; invertibility then gives a
coherent $\mathcal A_0$-linear structure on $R_a$.

Finally $R_a(E)\ne0$ because
$\RHom_{\mathcal A_0}(\mathbf 1,R_a(E))
\simeq\RHom_E(E,E)\simeq E$; this last argument does not require a
compact unit.
\end{proof}

For our spectral stack, $\mathcal A=\QCoh(Y)$ satisfies the
hypotheses of Lemma~\ref{lem:parameter-adjoint} by
\cite[\S7.9,~Cor.~C.4.4]{AGKRRV}. On a disjoint union of components the
compact generators have finite component support. We use the continuous,
fully faithful $\mathcal A$-linear functor
\[
 \jmath_Y\colon\QCoh(Y)\xrightarrow[\sim]{\Upsilon_Y}
 \IndCoh_{\{0\}}(Y)\hookrightarrow\IndCoh_{\Nilp}(Y),
\]
where $\Upsilon_Y(Q)=Q\otimes\omega_Y$
\cite[\S1.3.1,~Eq.~(1.2)]{GR}. Its essential image has zero singular
support, hence lies in the nilpotent category.

For a finite set $I$, put $\mathcal Q_I=\QLisse(X)^{\otimes I}$. Given
$V\in\Rep(\check G)^{\otimes I}$, let $H_I(V,-)$ denote the multi-leg
Hecke functor and let
$\mathcal E_V^I\in\mathcal A\otimes\mathcal Q_I$ be the universal
evaluation object. For a parameter $\sigma\colon\Spec E\to Y$ define
$\sigma^I(V):=(\sigma^*\otimes\id)(\mathcal E_V^I)\in\mathcal Q_I$.

\begin{definition}\label{def:eigenstructure}
A Hecke eigenobject of eigenvalue $\sigma$ with coherent structure is an
$M\in\mathcal C_s$ equipped, for all finite sets $I$, with isomorphisms
$H_I(V,M)\simeq M\boxtimes\sigma^I(V)$, natural in $V\in
\Rep(\check G)^{\otimes I}$ and compatible with units, tensor products,
permutations of legs, collisions of legs, and all homotopy-coherent
compatibilities \cite[\S15.2]{AGKRRV}.
\end{definition}

\begin{corollary}\label{cor:eigenobjects}
Under either hypothesis regime, every parameter $\sigma\colon\Spec E\to
Y$ admits a nonzero object $M_\sigma\in\Shv_{\Nilp}(\Bun_G)$ with a
coherent Hecke eigenstructure of eigenvalue $\sigma$. If $\sigma$ lies
in a component $Z$, then $M_\sigma\in\mathcal C_{s,Z}$. The assertion
is in the ind-constructible dg category; no boundedness, perversity
or Weil structure is included.
\end{corollary}

\begin{proof}
Apply Lemma~\ref{lem:parameter-adjoint} to $a=\sigma^*\colon\mathcal A
\to\Vect_E$, and write $\sigma_*$ for its continuous right adjoint.
The composite
\[
 \Phi_\sigma\colon\Vect_\sigma\xrightarrow{\sigma_*}\mathcal A
 \xrightarrow{\jmath_Y}\mathcal D
 \xrightarrow{(\mathbb L_G^{\mathrm{restr}})^{-1}}\mathcal C_s
\]
is continuous $\mathcal A$-linear, where $\Vect_\sigma$ denotes
$\Vect_E$ with the action through $\sigma^*$. Set
$M_\sigma=\Phi_\sigma(E)$. It is nonzero by the lemma, full faithfulness
of $\jmath_Y$, and the Langlands equivalence. If $\sigma$ lies in $Z$,
the idempotent $e_Z$ pulls back to $E$, while its complementary
idempotent pulls back to zero; linearity puts $M_\sigma$ in
$\mathcal C_{s,Z}$.

For the eigenstructure, fix a finite set $I$. The action of
$\mathcal E_V^I$ on $\mathcal C_s$ is the multi-leg Hecke functor
$H_I(V,-)$ \cite[\S\S14.2--14.3]{AGKRRV}. Extend $\Phi_\sigma$ by the
identity on $\mathcal Q_I$. Its module structure gives
\[
 H_I(V,M_\sigma)\simeq(\Phi_\sigma\otimes\id)(\sigma^I(V))
 \simeq M_\sigma\boxtimes\sigma^I(V),
\]
the second equivalence using that a continuous $E$-linear functor from
$\Vect_E$ is just tensoring with its value on $E$. Naturality, tensor
and unit compatibility, and collision coherence follow from the
universal symmetric monoidal evaluation system; equivalently, from
restricting $\Phi_\sigma$ along the symmetric monoidal Ran-to-spectral
action \cite[\S15.1]{AGKRRV}. The eigenvalue is the actual $\sigma$,
semisimple or not.
\end{proof}

% =========================================================================
\section{Rational coefficients and spectral support}
\label{sec:coefficients}
% =========================================================================

We now prove the rational-coefficient form of full support. Put
$E_0=\mathbb Q_\ell$ and $E=\overline{\mathbb Q}_\ell$, keep $k=\overline{\mathbb F}_q$,
and assume the curve and group satisfy the rational hypotheses in the
Introduction (Hypothesis~\ref{hyp:A} plus a split dual group over
$E_0$). Put $\mathcal C_E=\Shv_{\Nilp,E}(\Bun_G)$, retain
$\mathcal C_0=\Shv_{\Nilp,E_0}(\Bun_G)$ and $Y_0$ from the Introduction,
and let $Y$ denote the spectral prestack over $E$. All categorical tensor
products below are presentable dg categorical tensor products, so
"extension of coefficients" means $\Vect_E\otimes_{E_0}(-)$, not
extension of the geometric field.

Recall the standing datum: a continuous $\QCoh(Y_0)$-action on
$\mathcal C_0$ whose restriction along universal evaluation is the usual
multi-leg Hecke system with its tensor, permutation and collision
compatibilities. We call this the rational spectral-action datum.

\begin{theorem}[Rational spectral support]\label{thm:rational-support}
For the rational datum above, every nonempty open-and-closed substack
$U\subset Y_0$ satisfies
\[
 \QCoh(U)\underset{\QCoh(Y_0)}\otimes\mathcal C_0\ne0.
\]
Consequently if $Y'_0\subset Y_0$ is an open-and-closed support union
whose complementary structure idempotent acts by zero on $\mathcal C_0$,
then $Y'_0=Y_0$ as spectral prestacks. In particular this applies to the
support of any $\QCoh(Y_0)$-linear primed equivalence.
\end{theorem}

The plan is: for $X\ne\mathbb P^1$ we compare the sheaf categories,
prestacks and actions so that Theorem~\ref{thm:full-support} in regime A
rules out the vanishing of any nonempty idempotent; faithful flatness of
$E/E_0$ then descends the absence of a missing summand. For
$X=\mathbb P^1$ connectedness of the spectral prestack gives the result
directly.

\subsection{Coefficient categories, including unbounded objects}

On a finite-type $T/k$, $\Shv_L(T)$ is the ind-completion of bounded
constructible $L$-adic complexes, $L\in\{E_0,E\}$. For a smooth $T$ and a
closed conical $N\subset T^*T$, let $\mathcal P_{N,L}(T)$ be the category
of constructible perverse sheaves with singular support in $N$; the
category $\Shv_{N,L}(T)$ consists of objects whose perverse cohomology
belongs in $\operatorname{Ind}\mathcal P_{N,L}(T)$ in every degree. This
is the cohomological support definition
\cite[\S\S E.5.1--E.5.8]{AGKRRV}; it is not necessarily the ind-completion
of support-bounded complexes. On a smooth scheme the zero-cone case is
$\QLisse_L$, with its equivalent ordinary-cohomology definition
\cite[\S\S1.2.5--1.2.11]{AGKRRV}.

\begin{lemma}[Coefficient comparison]\label{lem:coeff-sheaves}
Extension of coefficients induces equivalences
\begin{align}
 \Vect_E\otimes_{E_0}\Shv_{E_0}(T)&\simeq\Shv_E(T),
 \label{eq:coeff-ambient}\\
 \Vect_E\otimes_{E_0}\QLisse_{E_0}(X)&\simeq\QLisse_E(X),
 \label{eq:coeff-qlisse}\\
 \Vect_E\otimes_{E_0}\mathcal C_0&\simeq\mathcal C_E.
 \label{eq:coeff-nilp}
\end{align}
The first extends to locally finite-type algebraic stacks by smooth
descent. The second is symmetric monoidal and respects the t-structures
used in the tensor-functor definition of $Y_0,Y$. On smooth charts the
third respects cohomological nilpotent support. Free and forgetful
functors here are $t$-exact for the ordinary and perverse t-structures
whenever these are used.
\end{lemma}

\begin{proof}
\emph{Ambient statement.} For a finite extension $L/E_0$, coefficient
extension and restriction preserve bounded constructibles, and on
objects extended from $E_0$ the mapping complexes are obtained by
tensoring the originals with $L$. Every $L$-constructible $M$ is a
retract of $L\otimes_{E_0}\operatorname{Res}_{L/E_0}M$, the retraction
being given by the separability idempotent of $L/E_0$. So the induced
compact objects have the required mapping complexes and generate after
retracts. Constructible $E$-complexes and their morphisms come from
finite coefficient fields. These coefficient conventions and the
compact-object description are in
\cite[Prop.~5.2,~Thm.~7.7,~Lemma~7.9,~Prop.~8.2,~Cor.~8.3]{HemoRicharzScholbach};
see also \cite[\S2]{HansenScholze}. Cohomological finiteness
\cite[Lemma~8.6(1)]{HemoRicharzScholbach} holds on every finite-type
chart over the separably closed $k$ (either algebraic
$\mathbb Q_\ell$-coefficient field), so its compactness statements apply.
Passing to ind-completions proves \eqref{eq:coeff-ambient}.

\emph{Support condition.} View \eqref{eq:coeff-ambient} as internal
$E$-modules in $\Shv_{E_0}(T)$. An internal $E$-module has the $E$-support
condition iff its underlying $E_0$-object has the $E_0$-condition. Write
$F$ for the free and $U$ for the forgetful functor; both are t-exact for
ordinary and perverse cohomology. For $U$, using that a constructible
$E$-object over a finite extension $L/E_0$ has underlying
\[
 U(M_L\otimes_L E)=
 \mathop{\operatorname{colim}}_{\substack{L\subset L'\subset E\\ [L':E_0]<\infty}}
 \operatorname{Res}_{L'/E_0}(M_L\otimes_L L'),
\]
and that finite extension/restriction are t-exact and commute with the
ind-t-structure, one gets the claim first on constructibles, then on the
ambient ind-categories.

For bounded constructibles, coefficient extension preserves and
reflects singular-support bounds. The defining test pairs express the
bound by universal local acyclicity; use the dualizability criterion in
the algebraic $\mathbb Q_\ell$-setting \cite[Thm.~1.6]{HansenScholze}.
The correspondence category has compositions formed from pullback,
tensor, and $f_!$, all of which commute with coefficient extension by
the coefficient formalism and projection formula of \S2 there; extending
duality data preserves universal local acyclicity. For reflection,
suppose a bounded constructible complex over a finite extension
$L/E_0$ becomes universally locally acyclic over $E$. The relative dual
of its $E$-extension is constructible by
\cite[Prop.~3.4(ii)]{HansenScholze}; the dual, evaluation and
coevaluation maps plus both triangle identities are finite data on
fixed finite-type test schemes and their products, so by
\cite[Lemma~7.9]{HemoRicharzScholbach} they descend to a common finite
coefficient extension. Thus the complex is universally locally acyclic
over that finite extension already. Finite extension reflects local
acyclicity, giving reflection for each pair, hence for the bound.
Finite restriction preserves the same bound. The same arguments apply
to lissity.

It follows that $F$ and $U$ preserve the relevant ind-perverse hearts;
for $U$ use the colimit formula and filtered colimits, never treating a
finite $E$-rank as a finite $E_0$-rank. For the converse, let $P$ be a
perverse internal $E$-module with $U(P)$ in the $E_0$-ind-heart. The
counit $F(U(P))\to P$ is an epimorphism in the module heart because its
underlying map splits as $u\mapsto1\otimes u$. Its source is in the
$E_0$-ind-heart, which is closed under quotients by the Serre property
of the support heart; hence $P$ is as well. So the cohomological support
condition for an $E$-module coincides with the $E_0$-condition on its
underlying $E_0$-object, degree by degree.

\emph{Scalar extension and smooth descent.} We now realise these
subcategories as scalar extensions. The perverse Serre condition gives
closure under finite cones; since perverse cohomology commutes with
filtered colimits, these subcategories are closed under filtered
colimits and arbitrary direct sums, hence under all colimits including
bar realisations, and closed under scalar tensors. Internal modules in
each such subcategory are exactly ambient modules whose underlying
object belongs to it, and the free-forgetful bar resolution
constructs every such module from free modules inside the subcategory.
This proves \eqref{eq:coeff-qlisse} and the corresponding support
comparison on smooth charts. Ordinary tensor products and structural
maps commute with coefficient extension, giving the monoidal assertion.
Connective free modules generate the connective module category under
connective colimits, so the induced tensor t-structures agree.

Finally, take the smooth-chart limit defining sheaves on a stack.
Internal modules commute with these limits. Equivalently, $\Vect_E$ is
the module category of the continuous monad $E\otimes_{E_0}(-)$ on
$\Vect_{E_0}$ and is dualizable as a presentable $E_0$-linear category
\cite[Lemma~1.6.3]{GKRV}. Tensoring with a dualizable category preserves
limits, and the smooth pullbacks defining these limits are continuous;
chartwise support comparison therefore proves \eqref{eq:coeff-nilp}.
\end{proof}

\begin{remark}
The lemma exchanges scalar extension with the actual cohomological
categories, not with any unspecified left completion. It applies to
$\QLisse(\mathbb P^1)$ without identifying that category with
$\IndLisse(\mathbb P^1)$.
\end{remark}

\subsection{Affine families and coherent Hecke action}

The sheaf comparison controls the affine families defining the spectral
prestacks. We retain the right-t-exactness in their tensor functors.

\begin{lemma}[Spectral coefficient comparison]\label{lem:coeff-spectral}
There are natural identifications
\begin{equation}\label{eq:coeff-spectral}
 Y_0\times_{\Spec E_0}\Spec E\simeq Y,\qquad
 \Vect_E\otimes_{E_0}\QCoh(Y_0)\simeq\QCoh(Y),
\end{equation}
with the second symmetric monoidal. These identify the universal
evaluation systems for every finite set of legs.
\end{lemma}

\begin{proof}
Let $A$ be a connective commutative $E$-algebra, $\operatorname{Mod}_A
=\QCoh(\Spec A)$. Lemma~\ref{lem:coeff-sheaves} and associativity give
\[
 \operatorname{Mod}_A\otimes_{E_0}\QLisse_{E_0}(X)
 \simeq\operatorname{Mod}_A\otimes_E\QLisse_E(X).
\]
The tensor t-structures are generated by connective objects of the
factors; equivalently they are module t-structures with t-exact
forgetful functor \cite[\S1.4.1]{AGKRRV}, and they agree under this
identification by the connective assertion in
Lemma~\ref{lem:coeff-sheaves}.

For the split dual group, scalar extension identifies
$\Vect_E\otimes_{E_0}\Rep_{E_0}(\check G_0)$ with
$\Rep_E(\check G)$ as a symmetric monoidal category with t-structure: in
characteristic zero, finite-dimensional representations in the heart
generate connective objects, and irreducible highest-weight
representations over a split reductive group are defined over $E_0$ and
remain irreducible after extension
\cite[Thm.~22.2,~\S\S22.3--22.5,~Thm.~22.42]{Milne}. The universal
property of scalar extension therefore identifies tensor functors into
the displayed target, and right t-exactness is equivalent before and
after extension.

Present $Y_0$ as a colimit of affine test schemes. Base change presents
the left side of \eqref{eq:coeff-spectral} by base-changed affines, and
quasi-coherent categories form the corresponding limit. On an affine,
coefficient comparison is
$\Vect_E\otimes_{E_0}\operatorname{Mod}_B\simeq
\operatorname{Mod}_{B\otimes_{E_0}E}$. Affine pullbacks in this diagram
are continuous, and $\Vect_E$ is categorically dualizable, so tensoring
commutes with this limit and gives the second identification, monoidal
included. No finite-dimensionality of $E$ over $E_0$ is required.

Evaluation at the tensor functor parametrized by $\Spec A$ is unchanged
by these identifications; the assertion is natural in representations,
affine tests and maps of finite sets. It therefore identifies the whole
universal evaluation system, including tensor products and collisions
of legs.
\end{proof}

For $X\ne\mathbb P^1$ we can go further and compare the given rational
spectral action with the geometric one. This is the only case needed
below; for $\mathbb P^1$ we deduce support from connectedness of the
spectral prestack.

\begin{lemma}[Comparison of spectral actions]\label{lem:coeff-action}
Suppose $X\ne\mathbb P^1$ and equip $\mathcal C_0$ with the rational
spectral-action datum of Theorem~\ref{thm:rational-support}. Under
Lemmas~\ref{lem:coeff-sheaves} and~\ref{lem:coeff-spectral}, its scalar
extension is the established $\QCoh(Y)$-action on $\mathcal C_E$.
\end{lemma}

\begin{proof}
First, $\QLisse_E(X)=\IndLisse_E(X)$ for this curve
\cite[\S1.3.3,~\S E.2.8]{AGKRRV}. The same equality holds over $E_0$:
extend an object of $\QLisse_{E_0}(X)$ and use the constructible lisse
generators over $E$; forgetting coefficients takes those generators
into the colimit closure of $E_0$-constructible lisse objects, and the
original object is a retract of its extension-then-forgetting, using an
$E_0$-linear retraction $E\to E_0$ of the unit. Constructible lisse
objects are compact in the ambient category, so these
$\QLisse$-categories are compactly generated and dualizable; their
exterior-product and affine-limit evaluation constructions therefore
commute with coefficient comparison.

Fix the geometric Satake identification over $E_0$
\cite[Thm.~14.1]{MirkovicVilonen}, and take scalar extension over $E$,
with the same dual-group identification, cohomological shifts and
geometric Tate-line conventions. We compare the \emph{whole} finite-set
monoidal diagram giving the Hecke functors, not just objects indexed by
highest weights. On each bounded Schubert support, coefficient extension
commutes with exterior products, pullbacks, descent and proper
convolution pushforwards. It is perverse t-exact, so it also commutes
with intermediate extension (written as the image of the map from
perverse extension by zero to perverse direct image). So it takes the
Satake IC objects, their convolution maps, and the fusion diagrams with
unit and symmetry constraints, to the corresponding diagrams over $E$.
The fiber and weight functors in geometric Satake identify the same
split dual group under this operation
\cite[\S\S4--6,~Thm.~14.1]{MirkovicVilonen}.

Derived functors here are derived extensions of naive geometric Satake.
In \cite[\S\S B.3.5--B.3.8]{GKRV} the compatible monoidal diagram in
shifted perverse hearts is extended using the bounded-derived universal
property; the induced right-lax monoidal structure is strict, and one
restricts to compact representations and ind-extends. Applying this to
the whole coefficient-comparison diagram, naturality gives the
comparison of derived functors and all finite-set structural maps; the
monoidal equivalences persist under the exact coefficient functor. The
correspondence action of \cite[\S\S B.2.5--B.2.10]{GKRV} then compares
the continuous Hecke systems including units, composition maps and
base-change compatibilities.

The factorisation into nilpotent sheaves with $\QLisse$-valued legs goes
through the fully faithful exterior-product functors
\cite[\S14.2.5]{AGKRRV}; Lemma~\ref{lem:coeff-sheaves} compares these
subcategories, so the comparison holds after factorisation.
Lemma~\ref{lem:coeff-spectral} identifies universal evaluation. By
assumption, restricting the extended spectral action along evaluation
gives exactly the established coherent Hecke system over $E$. The
equivalence of spaces of actions in \cite[Thm.~8.1.4]{AGKRRV}, applied
over the algebraically closed $E$, identifies it with the established
spectral action of \cite[Thm.~14.3.2]{AGKRRV}.
\end{proof}

No Frobenius structure or normalised rational Langlands functor was
chosen. What this comparison establishes is the identification of the
given rational spectral action with the geometric action used in
Theorem~\ref{thm:full-support}.

\subsection{Descent of support and the arithmetic qualification}

For $X\ne\mathbb P^1$, the sheaf, prestack and coherent-action
comparisons are now in place.

\begin{lemma}[Detection of a clopen support]\label{lem:coeff-idempotent}
Suppose the coefficient comparisons \eqref{eq:coeff-nilp}
and~\eqref{eq:coeff-spectral} identify the two spectral actions. If an
open-and-closed substack $U\subset Y_0$ acts by zero on $\mathcal C_0$,
then $U=\varnothing$.
\end{lemma}

\begin{proof}
Let $e_U=\mathcal O_U$ extended by zero in $\QCoh(Y_0)$. Its extension
is the analogous idempotent $e_{U_E}$, where $U_E=U\times_{E_0}E$.
Compatibility of the actions and generation by scalar-extended objects
show $e_{U_E}$ acts by zero on $\mathcal C_E$.

The curve and split group descend to a finite field, so
Theorem~\ref{thm:full-support} in regime A identifies this action with
the action on $\IndCoh_{\Nilp}(Y)$. Apply $e_{U_E}$ to
$\jmath_Y(\mathcal O_Y)$; its value is $\jmath_Y(e_{U_E})$. Full
faithfulness of $\jmath_Y$ implies $e_{U_E}=0$, hence $U_E=\varnothing$.

To descend, take any affine test $T=\Spec A\to Y_0$. The inverse image
of $U$ is an open-and-closed affine summand of $T$, cut out by an
idempotent of $\pi_0(A)$. Its base change to $A\otimes_{E_0}E$ is
empty, and faithful flatness of $E/E_0$ forces the idempotent to
vanish. This holds on every affine test, so $U=\varnothing$. No
rational-point description of components is used.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:rational-support}]
For $X\ne\mathbb P^1$, Lemma~\ref{lem:coeff-action} supplies the
hypothesis of Lemma~\ref{lem:coeff-idempotent}. An open-and-closed
localization is the image of its idempotent projection, so vanishing of
the localized category is exactly zero action of that idempotent. The
lemma gives the claim.

If $X=\mathbb P^1$, the geometric spectral prestack $Y$ has one
connected component: components are indexed by semisimple geometric
local systems \cite[Prop.~3.7.2]{AGKRRV}, and the geometric \'etale
fundamental group of $\mathbb P^1$ is trivial (a connected finite
\'etale cover $C\to\mathbb P^1$ of degree $n$ satisfies
$2g(C)-2=-2n$ by Riemann--Hurwitz \cite[Tag~0C1B]{Stacks}, so $n=1$).
By Lemma~\ref{lem:coeff-spectral} and the affine test of
Lemma~\ref{lem:coeff-idempotent}, $Y_0$ has no nonempty proper
open-and-closed substack. The category $\mathcal C_0$ is nonzero (its
dualizing object has shifted constant restrictions on smooth charts and
zero singular support), so its only nonempty open-and-closed
localization is nonzero.

Finally, the stipulated complement of $Y'_0$ acts by zero, so it is
empty. In particular a given spectral-linear primed equivalence has
this property by linearity, and its specified inclusion is an equality
of prestacks.
\end{proof}

\begin{corollary}[Conditional rational unpriming]
\label{cor:rational-arithmetic}
Let $Y'_0\subset Y_0$ be a Frobenius-stable open-and-closed support union
as in Theorem~\ref{thm:rational-support}. In addition to the rational
hypotheses, assume a canonical primed trace isomorphism over $E_0$ is
given,
\[
 \Funct_c(\Bun_{G_0}(\mathbb F_q),E_0)
 \xrightarrow{\ \sim\ }
 \Gamma^{\IndCoh}((Y'_0)^{\Frob},\omega_{(Y'_0)^{\Frob}}),
\]
intertwining the excursion and spherical Hecke actions in the same
Frobenius and Satake conventions. Then the same canonical formula and
compatibilities hold with $Y_0^{\Frob}$ in place of $(Y'_0)^{\Frob}$.
\end{corollary}

\begin{proof}
Theorem~\ref{thm:rational-support} identifies the actual inclusion
$Y'_0\subset Y_0$ with equality, so the derived fixed-point stacks,
dualizing objects and ind-coherent global-sections functors agree.
Substitute.
\end{proof}

\begin{remark}
The extra comparison map in Corollary~\ref{cor:rational-arithmetic}
is not constructed here. The primed arithmetic theorem cited in
Corollary~\ref{cor:arithmetic} has coefficients $E$; its existence does
not by itself produce a canonical map over $E_0$. An unconditional
rational arithmetic formula still requires that map with its Frobenius,
dualizing-object, ind-coherent global-sections and excursion
compatibilities. Equality of derived fixed-point prestacks alone
supplies none of these additional data.
\end{remark}


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