\section{Results used}
\label{rw:sec:inputs}

We collect the theorems on which the construction rests, in the
form in which they are used. Precise references to the individual
statements are repeated at each point of use.

\subsection{From the literature}

\begin{extinput}[Restricted geometric Langlands in characteristic zero]
\label{rw:in:gr}
Let \(C\) be a smooth proper curve over an algebraically closed
field of characteristic zero. Gaitsgory and Raskin construct a
functor \(\mathbb L^r_G\) from the category of sheaves with
nilpotent singular support on \(\operatorname{Bun}_G(C)\) to the
category of ind-coherent sheaves with nilpotent singular support on
the stack \(\operatorname{LS}^r_{\widehat G}(C)\) of local systems
with restricted variation, and prove that in characteristic zero it
is an equivalence \cite[Theorem~1.3.9(i)--(ii)]{GR2025}. (Over a
ground field of positive characteristic their theorem gives an
equivalence onto the sheaves supported on a union of connected
components; the hypotheses on the characteristic are those of
\cite[\S0.1.9]{GR2025}.) We use three of its properties:
it intertwines the Hecke action with the spectral action of
\cite[Theorems~8.1.4 and~14.3.2]{AGKRRV2022}; on the image of
quasi-coherent sheaves its inverse is given by the spectral action
on the nilpotent projection of the Poincar\'e object
\cite[\S\S1.1.3--1.1.8]{GR2025}; and the Whittaker coefficient of
the automorphic sheaf which corresponds to a quasi-coherent sheaf
is the complex of \(!\)-global sections of the latter on the
spectral side
\cite[Proposition~1.2.5]{GR2025}.
\end{extinput}

\begin{extinput}[Universal local acyclicity over a valuation ring]
\label{rw:in:hs}
Let \(S\) be the spectrum of an absolutely integrally closed
valuation ring with generic point \(\eta\), and let \(Y/S\) be
separated and of finite presentation. Restriction to the generic
fibre is an equivalence between complexes on \(Y\) that are
universally locally acyclic over \(S\) and constructible complexes
on \(Y_\eta\), with inverse \(Rj_*\)
\cite[Theorem~4.1]{HS}. Universal local acyclicity can be tested on
rank-one valuation rings \cite[Corollary~3.10]{HS}.
\end{extinput}

\begin{extinput}[Vanishing cycles over a two-dimensional base]
\label{rw:in:saito}
We use Saito's results on nearby cycles for a bounded
constructible complex and a morphism of finite type. If the
morphism is locally acyclic relative to the complex outside a
closed subscheme which is quasi-finite over the base, the nearby
cycles and the vanishing cycles are constructible and commute with
finite base change; if it is universally locally acyclic outside
this subscheme, they commute with every base change
\cite[Proposition~1.8 and Corollary~1.10]{SaitoCC}.
\end{extinput}

\begin{extinput}[The Whittaker functional as a microstalk]
\label{rw:in:microstalk}
For a smooth projective curve \(C\) over \(\mathbb C\) and sheaves
in the classical topology with nilpotent singular support on
\(\operatorname{Bun}_G(C)\), Nadler and Taylor define a Whittaker
functional as the vanishing cycles, at the canonical bundle, of the
sum of the simple root coordinates on the stack of twisted
\(N\)-bundles, applied to the \(!\)-pullback of a sheaf. They
prove that, after a shift, this functional computes the microstalk
at the regular nilpotent Higgs field on the canonical bundle
\cite[(3.8) and Theorem~3.3.4]{NT2026}. Two further steps are
needed here and are carried out in
Proposition~\ref{fl:lem:vacuum-microstalk-comparison}: the comparison of
this functional with the Whittaker coefficient of
Input~\ref{rw:in:gr}, and the transfer to \(\ell\)-adic sheaves
over an arbitrary algebraically closed field of characteristic
zero.
\end{extinput}

Some of these sources are recent preprints, among them
\cite{GR2025}, \cite{NT2026} and \cite{BBMY2026}. The first two
enter the proof of Theorem~\ref{fl:thm:weyl-prime-realization}
exactly as stated in Inputs~\ref{rw:in:gr}
and~\ref{rw:in:microstalk}; from the third we take a method and no
statement (Section~\ref{fl:sec:conservation}). The results of
L.~Lafforgue, V.~Lafforgue and Xue on which the arithmetic part
rests are listed in \cite[Section~\xnum{I}{rw:sec:inputs}]{HuI}.

\subsection{From the first paper}
\label{rw:sec:from-I}

The arithmetic part of the argument is contained in \cite{HuI}. We
use the following definitions and statements; a reader who accepts
them does not need the rest of that paper.
\begin{enumerate}
\item \emph{Excursion operators on functions.} For a level $N$ the
excursion operators $S_e$, for excursion data
$e=(I,f,\boldsymbol\gamma)$ with $\boldsymbol\gamma\in W_F^I$, act
on the functions with compact support on
$Y_N=G(F)\backslash G(\mathbb A_F)/K_N$, and
$S_e^{\mathrm{fun}}$ is defined on all functions on $Y_N$ by
transposition
(Definition~\xref{I}{fl:def:function-excursion-action}). On cusp
forms it is the operator of V.~Lafforgue. The operators $S_e$ are
defined through the cohomology of the stacks of shtukas with
integral coefficients
(Section~\xref{I}{fl:sec:compact-foundations}), and the partial
Frobenius maps on this cohomology have the properties stated in
Lemma~\xref{I}{fl:lem:unperiodized-specialization}.
\item \emph{Purity.} A semisimple elliptic parameter is strongly
pure (Theorem~\xref{I}{fl:thm:arithmetic-strong-purity}). We also
use Lemma~\xref{I}{fl:lem:arithmetic-finite-coefficients} on
compact subgroups of $\mathrm{GL}_n(k)$, and
Lemma~\xref{I}{fl:lem:arithmetic-monodromy-quotient}, by which the
quotient of the arithmetic monodromy group of such a parameter by
the geometric one is finite and cyclic.
\item \emph{From a functional to a cusp form.} Let $\varphi\ne0$ be
a function on $Y_N$ and $\sigma_{\mathrm{coeff}}$ a semisimple
elliptic parameter unramified outside $N$. If the linear form
$c\mapsto[c,\varphi]$ on functions with compact support has the
excursion eigenvalues of $\sigma_{\mathrm{coeff}}$ for all tuples
in the Weil group, then $\varphi$ is a cusp form with parameter
$\vartheta_G\sigma_{\mathrm{coeff}}$, for all tuples in $\Gamma_F$
(Proposition~\xref{I}{fl:prop:cuspidal-excursion-character}; see
also Lemma~\xref{I}{rw:lem:chevalley-dual-summand}).
\end{enumerate}
Those conventions of
\cite[Section~\xnum{I}{fl:sec:parameters-parameters}]{HuI} which are
used in this paper are repeated in
Section~\ref{fl:sec:parameters-parameters}.