==== TABLE OF CONTENTS ====
Donovans-Conjecture-over-Algebraic-Closures-of-Prime-Fields//00_introduction.tex
Donovans-Conjecture-over-Algebraic-Closures-of-Prime-Fields//01_preliminaries.tex
Donovans-Conjecture-over-Algebraic-Closures-of-Prime-Fields//02_geometry.tex
Donovans-Conjecture-over-Algebraic-Closures-of-Prime-Fields//03_families.tex
Donovans-Conjecture-over-Algebraic-Closures-of-Prime-Fields//04_labels.tex
Donovans-Conjecture-over-Algebraic-Closures-of-Prime-Fields//05_runners.tex
Donovans-Conjecture-over-Algebraic-Closures-of-Prime-Fields//06_endpoint.tex
Donovans-Conjecture-over-Algebraic-Closures-of-Prime-Fields//07_extensions.tex
Donovans-Conjecture-over-Algebraic-Closures-of-Prime-Fields//08_periodicity.tex
Donovans-Conjecture-over-Algebraic-Closures-of-Prime-Fields//common.tex
Donovans-Conjecture-over-Algebraic-Closures-of-Prime-Fields//main.tex
Donovans-Conjecture-over-Algebraic-Closures-of-Prime-Fields//proof-interfaces.tex
Donovans-Conjecture-over-Algebraic-Closures-of-Prime-Fields//references.tex
===========================

==== BEGIN Donovans-Conjecture-over-Algebraic-Closures-of-Prime-Fields//00_introduction.tex ====
\begin{abstract}
We prove Donovan's conjecture over every algebraically closed field
\(K\) of characteristic \(p>0\). For each fixed \(K\) and bound on
defect-group order, blocks of finite groups represent only finitely
many \(K\)-linear Morita equivalence classes. The defect groups need
not be abelian, and the result includes \(p=2\).
\end{abstract}

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

Let \(p\) be a prime and let \(K\) be an algebraically closed field
of characteristic \(p\).
A block of a finite group algebra \(KG\) is a summand \(KGb\)
defined by a primitive central idempotent \(b\).  Its defect group
is a \(p\)-subgroup of \(G\), determined up to conjugacy, that
measures how far the block is from being semisimple.  Two blocks
are \(K\)-linearly Morita equivalent when their module categories
are equivalent by a \(K\)-linear functor.  Donovan's conjecture
asks whether a fixed finite \(p\)-group can occur as the defect
group of only finitely many blocks up to this equivalence.
The question constrains an entire representation category by
local group data: neither the order of \(G\) nor the dimensions
of its simple modules are fixed, and the block need not be principal.
We prove the conjecture in the following bounded-order form.

\begin{theorem}[Donovan's conjecture]
\label{thm:donovan}
Fix a prime \(p\), an algebraically closed field \(K\) of
characteristic \(p\), and a positive integer \(M\).
As \(G\) ranges over finite groups and \(b\) over block idempotents of
\(KG\) with defect groups of order at most \(M\), only finitely many
\(K\)-linear Morita equivalence classes of \(KGb\) occur.
In particular, over this field \(K\), Donovan's conjecture holds for every finite
\(p\)-group, including nonabelian groups.
\end{theorem}

The bounded-order formulation is equivalent to the formulation with
one fixed defect group: there are only finitely many isomorphism
types of groups of bounded order.  The field \(K\) is fixed while
the finite group and block vary; the theorem concerns equivalences
of module categories over that field.  The resulting finite list also
bounds Cartan entries, Loewy lengths of basic algebras, and, in each
fixed degree, dimensions of extension groups between simple modules.

The proof first treats \(k=\overline{\mathbb F}_p\), which is the
coefficient field used in the main argument below.  After choosing an
embedding \(k\hookrightarrow K\), every block over \(K\) descends to a
unique block over \(k\) with the same defect groups, by the coefficient
comparison of \cite[Section~2.1]{BlockwiseAlperinWeight}.  Extending the
Morita bimodules then transfers the finite list from \(k\) to \(K\);
this final step is given in Section~\ref{sec:coefficient-extension}.

\subsection{Previous work}

The conjecture is attributed to Peter Donovan and appears as
Conjecture~M in Alperin's account of problems in group representation
theory \cite{Alperin1980}.
Its Cartan-boundedness component grows out of a question of Brauer;
Hiss emphasized the additional rationality obstruction
\cite{HissBrauer}.  Kessar proved that field-valued Donovan
finiteness separates into Cartan boundedness and bounded
Morita--Frobenius numbers \cite{KessarRemark}.

Scopes proved finiteness for symmetric-group blocks, and Kessar
treated their double-cover families \cite{Scopes,KessarSymmetric}.
For unipotent blocks of general linear groups, Jost's Morita
reductions bound the rank in terms of the defect \cite{Jost}.
Combined with rationality, this gives
finiteness across both ranks and field sizes; see
\cite[Example~5.3(2)--(3)]{HissBrauer} for these results and their
relationship.  Thus uniformity in these two parameters already has
an important type-A precedent.  Hiss--Kessar developed Scopes
reductions for unipotent blocks of classical groups, including
orthogonal blocks with degenerate-symbol characters in the sequel
\cite{HissKessarScopes,HissKessarScopesII}.

Eaton--Livesey proved the field-valued conjecture for abelian
\(2\)-groups \cite{EatonLivesey}.  Eaton--Eisele--Livesey established
an integral finiteness criterion and an abelian-defect reduction to
quasisimple blocks; together with Farrell--Kessar's rationality
bounds, this leaves Cartan boundedness as the remaining condition
in that reduction \cite{EEL,FarrellKessar}.
An--Eaton treated extraspecial groups of order \(p^3\) and exponent
\(p\) for \(p\ge5\), and reduced the corresponding case at \(p=3\)
to a quasisimple Cartan bound \cite{AnEaton}.
Eaton treated blocks with Suzuki \(2\)-group defects
\cite{EatonSuzuki}.  The August 2026 preprint
\cite[Theorem~1.1 and Corollary~1.2]{Quaternion2026} proves
integral finiteness for quaternion defect groups and deduces
finiteness for all tame blocks over \(k\).

For extensions, K\"ulshammer introduced crossed products and
algebraic-group actions into the reduction of Donovan's conjecture
\cite{Kulshammer1995}.  Eisele developed integral crossed-product
reductions and proved finiteness of integral block Picard groups
\cite{EiseleReduction,EiselePicard}.  Their prime-to-\(p\)
crossed-product finiteness is an antecedent of the extension method
below; the additional comparison here retains Sylow \(p\)-actions
and their specified inner multiplication factors.

\subsection{The two uniformity problems}

The general theorem requires both multiplicity control and control
of field of definition.  Bounds on Cartan entries alone do not
supply the latter.  A Cartan entry records a composition multiplicity
in a projective cover.  Their sum is the dimension of a basic algebra,
which represents the Morita class with all simple modules
one-dimensional.  Bounded dimensions do not by themselves confine
its structure constants to one finite field.  The Brauer--Feit
bound on ordinary characters also bounds the number of simple
modules in a block of bounded defect \cite{BrauerFeit}; thus a
Cartan-entry bound gives the required dimension bound.  If both
the dimension
and the size of a finite field of definition are bounded, only finitely many
multiplication tables can occur.

The multiplicity problem must also be uniform in parameters that
the defect does not bound.  In a group of Lie type, finite tori can
grow even when the chosen block has small defect; a bound for a
principal block with small Sylow subgroup does not address this
case.  The geometric construction below counts indecomposable
projective factors rather than their vector-space dimensions,
and estimates characters after projection to the chosen block.
Classical rank requires a separate reduction to bounded rank or
to a cuspidal series whose Harish--Chandra weight is bounded in
terms of the defect order.

Likewise, passing from quasisimple groups to
arbitrary groups needs more than a non-equivariant Morita--Frobenius
bound on each component.  Here coefficient Frobenius raises scalars
to their \(p\)th powers, and a Morita--Frobenius bound controls how
many iterations are needed to return to the Morita class.
A group extension also specifies how its homogeneous components
multiply: choices of coset representatives have products differing
by elements of the normal subgroup.  These inner factors remain
part of the crossed algebra throughout the proof.  We construct
Frobenius comparisons compatible with the actions of Sylow
\(p\)-subgroups in the crossed extensions and with those factors.
Scalar discrepancies can
then be removed; arbitrary central-unit discrepancies would not
permit this normalization.

\subsection{The proof and its new ingredients}

The proof first establishes a uniform Cartan bound for quasisimple
blocks of bounded defect.  It then reduces a general block to crossed
extensions of finitely many integral base orders and proves
finiteness of the remaining crossed data.  The Cartan estimate bounds the sizes of the basic algebras;
the comparison controls their multiplication and descent data.
Four constructions address the parameters that can still be unbounded.  The two main outputs
are Theorem~\ref{thm:quasisimple-cartan}, which supplies the Cartan
bound, and Proposition~\ref{prop:sylow-comparison}, which supplies
the equivariant comparison.  Figure~\ref{fig:proof-interfaces}
shows their distinct roles in the final extension argument.

\paragraph{Projective multiplicities in fixed root data.}
Section~\ref{sec:geometry} constructs projective characters from
tilting objects on the two-flag space, following Eteve's construction
\cite{Eteve}.  The essential estimate bounds
the sum of projective multiplicities in extraordinary stalks.
Regular Lang local systems and a Kummer-cohomology calculation remove
dependence on the size of the finite torus \(p\)-part.
An ordinary-coordinate projection then bounds the character norms
of projective indecomposable modules (PIMs).
This estimate is uniform in the field size and monodromy parameter,
but its root data remain fixed.

\paragraph{Classical rank and projective cones.}
Sections~\ref{sec:families}--\ref{sec:runners} deal with classical
groups of unbounded rank.  A single-cycle Jordan-label calculation
makes the required ordinary induction rules available beyond uniform
class functions.  Harish--Chandra moves then act on projected cones
of projective characters.  Most moves are exact permutations after
normalization.  Here the cones are the nonnegative spans of PIM
characters in selected ordinary-character coordinates.
The remaining moves have summable errors, including
a two-edge detour for the exceptional cohook configuration.
Determinant and cone-volume estimates convert this control into
bounds on the PIM columns.  These moves need not be Morita
equivalences.  The runner language belongs to the lineage of
Scopes's abacus methods, Jost's general-linear reductions, and
Hiss--Kessar's classical reductions
\cite{Scopes,Jost,HissKessarScopes,HissKessarScopesII}.
Here the transported object is a cone
of projective characters; its positivity and determinant bounds
carry a numerical estimate through all classical families.

\paragraph{The cuspidal endpoint.}
Runner reductions can leave a cuspidal label of unbounded rank:
a staircase partition in the unitary case, or a two-row symbol
with each row an initial interval of nonnegative integers in the
other classical types.
We call these labels cuspidal triangles.
Section~\ref{sec:endpoint} treats this endpoint directly by modular
Harish--Chandra induction.  We construct the intertwiners, remove
their extension obstruction, count the resulting Hecke-algebra
simple types, and prove that their heads cover the required simple
modules.  The bound depends on the Harish--Chandra weight above the
triangle, not on the triangle's rank.

\paragraph{Equivariant descent of crossed extensions.}
Sections~\ref{sec:extensions} and~\ref{sec:periodicity} address
arbitrary finite groups.  The generalized Fitting reduction leads
to base blocks equipped with an actual inner factor system.
We construct integral Morita comparisons with bounded Frobenius
period whose reductions respect the Sylow actions and the
specified inner factors.
Their multiplication error is scalar, which can be removed over
\(k\); an arbitrary central-unit error would not suffice.
Finite integral Picard groups then turn these comparisons into
Frobenius descent retaining the chosen base algebra.
Lang's theorem and a Sylow restriction
argument leave only finitely many crossed systems.

The role of the integral theory is confined to the base orders and
their comparisons.  The final removal of the large \(p'\)-kernel
and the final crossed-product finiteness are over \(k\).
The finite lists of integral identity-component orders and the
integral Morita bimodules with compatible projective transports
also serve the subsequent integral companion over \(W(k)\)
\cite{OpenAIIntegralDonovan2026}.  The field proof given here is
independent of that companion.
The proof invokes established structural, block-theoretic and
finite-reductive-group theorems with their hypotheses specified
where they are used.  The new uniformity, runner, endpoint and
equivariance arguments are proved below.

\input{figures/proof-interfaces}

Section~\ref{sec:preliminaries} fixes the coefficient conventions
and proves the transfer estimates used throughout.  The first
milestone is assembled in Section~\ref{sec:families}, using the
fixed-root estimate of Section~\ref{sec:geometry} and the classical
arguments developed in Sections~\ref{sec:labels}--\ref{sec:endpoint}.
Section~\ref{sec:extensions} states the precise comparison property
and proves that it suffices for finiteness.
Section~\ref{sec:periodicity} constructs that comparison with the
specified multiplication factors and proves finiteness over
\(k=\overline{\mathbb F}_p\).  Section~\ref{sec:coefficient-extension}
then transfers the finite list to the fixed algebraically closed field
\(K\), completing the proof of Theorem~\ref{thm:donovan}.
==== END Donovans-Conjecture-over-Algebraic-Closures-of-Prime-Fields//00_introduction.tex ====

==== BEGIN Donovans-Conjecture-over-Algebraic-Closures-of-Prime-Fields//01_preliminaries.tex ====
\section{Block-theoretic bounds and transfer}
\label{sec:preliminaries}

We first isolate the block-theoretic transfers used in the two
parts of the proof.  The ordinary-coordinate estimate recovers full
projective-character bounds from selected rows.  The central,
restriction and abelian-extension estimates then carry those
bounds between the groups in the quasisimple reduction.

Fix a prime \(p\), put \(k=\overline{\F}_p\), and let
\(\cO=W(k)\) be its Witt ring.  The letter \(M\) denotes an upper bound
on defect-group order.  Unless a further parameter is displayed, a
uniform bound may depend on \(p\) and \(M\), but not on the ambient
finite group.  When ordinary characters are used, we extend scalars to
a splitting characteristic-zero field.  This leaves the decomposition numbers unchanged and extends each
chosen integral Morita equivalence.  We do not infer descent of an
arbitrary equivalence from a splitting extension.

For a finite group \(G\), a block is a nonzero primitive central
idempotent \(b\) of \(kG\), or the corresponding summand \(kGb\).
The idempotent lifts uniquely to a block of \(\cO G\), denoted by the
same letter.  A defect group is a maximal \(p\)-subgroup \(D\) for which
\(\Br_D(b)\ne0\).  Here
\[
 \Br_D:(kG)^D\longrightarrow kC_G(D)
\]
deletes coefficients of group elements outside \(C_G(D)\).
Morita equivalences throughout are linear over the indicated
coefficient ring.

Write \(\Irr(b)\) for the ordinary irreducible characters in \(b\),
\(\IBr(b)\) for its irreducible Brauer characters, and
\(k(b)=|\Irr(b)|\), \(l(b)=|\IBr(b)|\).  If
\(\chi^0\) is the restriction of \(\chi\) to \(p\)-regular elements, then
\[
 \chi^0=\sum_{\varphi\in\IBr(b)}d_{\chi\varphi}\varphi .
\]
The nonnegative integral matrix
\(D_b=(d_{\chi\varphi})\) is the decomposition matrix.
Brauer reciprocity identifies its columns with the ordinary
characters \(\Phi_\varphi\) of projective indecomposable modules
(PIMs).  Thus the Cartan matrix is
\[
 C_b=D_b^{\mathsf T}D_b,\qquad
 (C_b)_{\varphi\psi}=\langle\Phi_\varphi,\Phi_\psi\rangle_G.
\]
In particular, a bound on PIM norms bounds every Cartan entry by
Cauchy--Schwarz.

We use the following standard facts of block theory
\cite{Linckelmann,BrauerFeit}.  The Brauer--Feit theorem bounds \(k(b)\)
in terms of defect-group order, and \(l(b)\le k(b)\).  The elementary
divisors of \(C_b\) divide \(|D|\); in particular
\[
 0<\det C_b\le |D|^{l(b)}.
\]
There is an ordinary character of height zero in every block.
If \(N\lhd G\) and \(B\) covers a block \(b\) of \(N\), defect groups
can be chosen so that \(D_b=D_B\cap N\).  Fong--Reynolds reduction
replaces the covering problem by the inertia group of \(b\), preserving
Morita type and defect.  If \(b\) is invariant and \(G/N\) is a
\(p\)-group, then \(b\) remains a block of \(G\) and
\[
 D_BN=G.
\]
These assertions will only be used in these stated forms.  We also
use Clifford theory for restriction to a normal subgroup, including
its projective multiplicity-space formulation.

\subsection{Central quotients and ordinary coordinates}

\begin{lemma}[Central transfer]
\label{lem:central-transfer}
Let \(Z\le Z(G)\) be a \(p\)-subgroup.  Blocks of \(G\) correspond to
blocks of \(G/Z\), every defect group contains \(Z\), and corresponding
defect orders differ by \(|Z|\).  Their simple modules correspond by
inflation.  With these identifications their Cartan matrices satisfy
\[
 C_b=|Z|\,C_{\bar b}.
\]
In particular, Cartan bounds transfer in either required direction
when \(|Z|\) is bounded.  For a central \(p'\)-subgroup \(Z'\), the
summand on which \(Z'\) acts trivially is the averaging-idempotent
summand and identifies with the group algebra of \(G/Z'\).
\end{lemma}

\begin{proof}
The block and defect assertions are the central-quotient theorem.
For the Cartan assertion put \(J=\rad(kZ)\).
The ideal \(JkG\) is nilpotent and its quotient is \(k(G/Z)\), so
primitive projective covers correspond under passage to coinvariants.
If \(P\) is such a cover, then \(P\) is projective, hence free, as a
\(kZ\)-module.  The canonical multiplication maps give
\[
 (J^i/J^{i+1})\otimes_k(P/JP)
       \ \simeq\ J^iP/J^{i+1}P .
\]
They are isomorphisms of \(G/Z\)-modules: centrality makes the action
on the first factor trivial, and changing a lift in \(G\) changes
the action only by the next radical layer.  The sum of the dimensions
of \(J^i/J^{i+1}\) is \(|Z|\).  Additivity of composition multiplicities
therefore gives the displayed equality.  For \(Z'\), use
\(|Z'|^{-1}\sum_{z\in Z'}z\).
\end{proof}

A set \(\mathcal B\subseteq\Irr(b)\) is an \emph{ordinary integral
basic set} if the characters \(\{\chi^0:\chi\in\mathcal B\}\)
are a \(\Z\)-basis of the Brauer character lattice.
The corresponding square submatrix of \(D_b\) then has determinant
\(\pm1\).  We also use basic sets for a direct sum of blocks.

\begin{lemma}[Bounded inverse projection]
\label{lem:row-projection}
For blocks of defect order at most \(M\), including unions of a
bounded number of such blocks, let \(V\) be the rational span of
their PIM columns in ordinary-character coordinates.
If a coordinate projection \(\pi\) is injective on \(V\), its inverse
\[
 (\pi|_V)^{-1}:\pi(V)\longrightarrow V
\]
has uniformly bounded Euclidean operator norm.  All full-column-size
minors of a decomposition matrix, before or after coordinate
projection, are uniformly bounded.

If a basic set for a union of blocks has bounded size and the
\(p\)-defects of all its ordinary degrees are bounded, then the number
and defect orders of those blocks, and hence the dimensions of their full ordinary-character coordinate
spaces, are bounded.  This bounds the numbers of ordinary irreducible
characters, not their degrees.
\end{lemma}

\begin{proof}
Let the PIM columns be \(v_1,\ldots,v_l\in\Z^n\).  Their exterior
product is a nonzero integral vector, and Cauchy--Binet gives
\[
 \|v_1\wedge\cdots\wedge v_l\|^2
    =\det(D_b^{\mathsf T}D_b)
    =\sum_{|I|=l}\det(D_{b,I})^2 .
\]
The preceding standard bounds control \(n,l\) and this determinant.
After padding the ambient coordinate spaces to one fixed dimension,
only finitely many exterior products occur.  Each determines the
subspace \(V\).  There are also only finitely many coordinate
projections in that dimension.  Restricting to the injective ones,
the maximum of their inverse norms is finite.  The same identity
bounds each displayed minor.  For a bounded block union take the
orthogonal direct sum.

A basic set partitions by block.  Indeed each ordinary character
has Brauer constituents in its own block, and the Brauer lattice is
the direct sum of the block lattices.  The basis property consequently
restricts to each summand, which must have at least one basic row.
It remains to recover its defect from those rows.
Fix a block \(b\) in the union and put
\(\mathcal B_b=\mathcal B\cap\Irr(b)\).
Write \(|G|_p=p^a\), and let \(b\) have defect \(p^d\).
Every ordinary degree in the block has \(p\)-valuation at least
\(a-d\).  Some degree has valuation exactly \(a-d\), by height zero.
If every basic-row degree had larger valuation, evaluating its
integral spanning expression at \(1\) would give the same larger
valuation for every ordinary degree, a contradiction.  Thus
\[
 d=\max_{\chi\in\mathcal B_b}
           v_p\bigl(|G|/\chi(1)\bigr).
\]
The asserted bounds now follow from the bound on basic-set size
and the Brauer--Feit theorem.
\end{proof}

This lemma bounds the projective \emph{subspace}, not a selected
nonnegative integral basis of it.  That distinction lets us first
estimate selected ordinary coordinates and only afterwards recover
the full PIM norms.

\subsection{Crossed algebras and upward Cartan transfer}

We record conventions for crossed algebras that will also be used in
Section~\ref{sec:extensions}.  A crossed algebra on a finite-dimensional
\(k\)-algebra \(R\), graded by a finite group \(A\), has the form
\[
 T=\bigoplus_{x\in A}Ru_x,\qquad
 u_xr=\alpha_x(r)u_x,\qquad u_xu_y=a_{xy}u_{xy},
\]
where each \(u_x\) is invertible, \(\alpha_x\in\Aut(R)\),
and \(a_{xy}\in R^\times\).  Associativity means
\[
 \alpha_x\alpha_y=\operatorname{ad}(a_{xy})\alpha_{xy},\qquad
 a_{xy}a_{xy,z}=\alpha_x(a_{yz})a_{x,yz}.
\]
In the second equality \(a_{xy,z}\) denotes the factor with first
index the group product \(xy\).  Replacing \(u_x\) by \(v_xu_x\),
\(v_x\in R^\times\), is called a change of homogeneous units.

If \(e\) is a full basic idempotent of \(R\), then
\(\alpha_x(e)\) is unit-conjugate to \(e\): both select one copy of
each indecomposable projective type.  Adjusting \(u_x\) by such units
makes it commute with \(e\).  Thus \(eTe\) is again crossed on \(eRe\),
and \(e\) is full in \(T\).  In particular
\[
 \dim_k(eTe)=|A|\dim_k(eRe).
\]
The same argument applies to block orders and integral basic
idempotents.

\begin{lemma}[Transfer across an abelian quotient]
\label{lem:abelian-transfer}
Suppose \(N\lhd G\) and \(G/N\) is abelian.  Among blocks \(B\) of
\(G\) of defect order at most \(M\), uniform Cartan bounds for their
covered blocks of \(N\) imply uniform Cartan bounds for \(B\).
No bound on the \(p'\)-part of \(|G/N|\) is required.
\end{lemma}

\begin{proof}
Apply Fong--Reynolds and assume the covered block \(b\) is invariant.
Let \(G_p/N\) be the Sylow \(p\)-subgroup of \(G/N\).  The unique
block of \(G_p\) above \(b\) has a defect group surjecting onto
\(G_p/N\); its order is bounded by a defect group of \(B\).
Consequently \(|G_p/N|\le M\).  A basic corner of the crossed
extension by this bounded group has bounded dimension, by the
preceding observation.  Its Cartan entries are therefore bounded.
We may replace \(N\) by \(G_p\), leaving a \(p'\)-quotient \(A\).
The identity block is invariant; alternatively one may take its
inertia group again.

Pass to a full basic corner, and write \(R\) for the identity
algebra and \(T\) for the resulting crossed algebra.
The dimension of \(R\) is bounded: for a basic algebra it is the
sum of its Cartan entries, and its number \(n\) of simple modules
is bounded by defect.  Put
\[
 d=\max_{i,j}\dim_k\Ext^1_R(S_i,S_j),\qquad
 (\rad R)^L=0.
\]
Both \(d\) and \(L\) are bounded.  Since \(\rad(R)\) is invariant,
\(\rad(R)T\) is nilpotent.  Maschke's theorem for the crossed
algebra on the semisimple quotient gives
\[
 \rad(T)=\rad(R)T,\qquad (\rad T)^L=0.
\]

The radical length is now controlled.  To bound the basic algebra
of the chosen block, it remains to bound the numbers of arrows in
its quiver, that is, first extensions between its simple modules.
Let \(X,Y\) be simple modules in the block of \(T\) under
consideration.  Their restrictions are semisimple over \(R\):
the nilpotent ideal \(\rad(R)T\) annihilates both modules.
Clifford theory describes the support of each restriction as one
\(A\)-orbit of simple \(R\)-types.  At each type its multiplicity
space is an irreducible projective representation of the stabilizer.
The scalar factor system accounts for the chosen intertwiners and
the inner factors \(a_{xy}\).  Simplicity makes that projective
representation irreducible.

A \(T\)-projective resolution restricts to an \(R\)-projective
resolution.  The group action on its \(R\)-linear Hom complex is
genuine, since the inner crossed factors cancel by \(R\)-linearity.
As \(|A|\) is invertible in \(k\), taking invariants is exact.
Thus \(\Ext^1_T(X,Y)\) is computed by invariant parts of the
extension spaces between these semisimple restrictions.
Decompose those spaces into orbits of pairs of simple types.
For a representative pair \(i,j\), write the corresponding
isotypic summands of \(X,Y\) as \(S_i\otimes V\) and
\(S_j\otimes W\), respectively.  Put \(H=A_i\cap A_j\) and
\(E=\Ext^1_R(S_i,S_j)\), with the induced projective
\(H\)-action.  Restrict \(V,W\) from their stabilizers to \(H\).
The corresponding summand has dimension
\[
 \dim\Hom_H(V,W\otimes E),
 \qquad \dim E\le d,
\]
with the compatible projective factors understood.

For completeness, these multiplicity spaces need not have bounded
dimensions.  Instead their normalized character norms give
\begin{equation}
 \dim\Hom_H(V,W\otimes E)
 \le \dim(E)\sqrt{[A_i:H][A_j:H]} .
 \label{eq:ext-stabilizer-bound}
\end{equation}
To see this, normalize the scalar factors to roots of unity of
\(p'\)-order and realize the projective representations on compatible
finite central \(p'\)-extensions.  Such normalization is possible
because scalar cohomology is killed by the group order.  These
semisimple representations lift to characteristic zero.
On the common scalar-character summands, the character formula,
\(|\chi_E(h)|\le\dim E\), and Cauchy--Schwarz reduce the estimate to
\[
 \frac1{|H|}\sum_{h\in H}|\chi_V(h)|^2\le[A_i:H],
 \qquad
 \frac1{|H|}\sum_{h\in H}|\chi_W(h)|^2\le[A_j:H].
\]
These follow by enlarging the sums to the respective stabilizers,
where the normalized squared norms are \(1\).
This proves \eqref{eq:ext-stabilizer-bound}.

The two indices are at most \(n\), since they count orbits of
simple types under a subgroup of \(A\); there are at most \(n^2\)
pair-orbit contributions.  Hence the coarse uniform estimate
\[
 \dim\Ext^1_T(X,Y)\le dn^3
\]
suffices.  The block of \(T\) corresponding to \(B\) has a bounded
number \(m\) of simple modules by its defect bound.
Its basic algebra has \(m\) vertices and at most \(m^2dn^3\)
arrows.  Lifts of a basis of its radical modulo its square generate
the radical, so paths of length less than \(L\) span that algebra.
Its dimension, and therefore its Cartan entries, are bounded.
\end{proof}

\subsection{Restriction and decomposition rows}

The preceding lemma transfers Cartan bounds upwards even across
an abelian quotient of unbounded $p'$-order.  The next lemma supplies
the complementary downward estimate: it controls decomposition
numbers through restriction multiplicities, without requiring a
bound on the index itself.

\begin{lemma}[Restriction of rows]
\label{lem:restriction-rows}
Let \(N\lhd G\), and fix a block \(b\) of \(N\).
Suppose that every block of \(G\) covering \(b\) has at most
\(L\) simple modules and decomposition numbers at most \(c\).
Suppose also that every simple module in these blocks restricts to
\(N\) with constituent multiplicities at most \(u\).
Then \(b\) has decomposition numbers at most
\(Lcu\).
If \(G/N\) is cyclic, one may take \(u=1\).
If \(C\le Z(G)\), one may instead take \(u=[G:NC]\).
\end{lemma}

\begin{proof}
Choose an ordinary irreducible character upstairs whose restriction
contains a given ordinary irreducible \(\chi\) downstairs.
Decomposition commutes with restriction.  On a fixed Brauer
constituent \(\varphi\) downstairs, the restriction of the
decomposition upstairs has multiplicity at most \(Lcu\).
Its other expression contains \(\chi^0\) with positive integral
multiplicity.  Nonnegativity therefore bounds every
\(d_{\chi\varphi}\) by \(Lcu\).

For the cyclic assertion, a simple \(N\)-type extends to its inertia
group when the quotient is cyclic.  Choose an intertwiner for a
cyclic generator; its power differs from the required action by a
scalar.  Rescale it using a root of that scalar in \(k\).
The remaining multiplicity space is a simple module for a cyclic
quotient, hence is one-dimensional even when \(p\) divides its order.
Clifford theory now gives multiplicity-free restriction.

For the central-factor assertion, let \(X\) be a simple
\(G\)-module and \(S\) a constituent of its restriction to \(N\).
The scalar character by which \(C\) acts on \(X\) extends \(S\)
to \(NC\): its restriction to \(N\cap C\) agrees with the
action on \(S\).  This extension is invariant under the inertia
group \(I\) of \(S\).  The Clifford multiplicity space is
therefore simple for a twisted group algebra of \(I/NC\).
Its dimension is at most \([I:NC]\le [G:NC]\), giving the
claimed restriction bound.
\end{proof}
==== END Donovans-Conjecture-over-Algebraic-Closures-of-Prime-Fields//01_preliminaries.tex ====

==== BEGIN Donovans-Conjecture-over-Algebraic-Closures-of-Prime-Fields//02_geometry.tex ====
\section{A Cartan bound for fixed root data}\label{sec:geometry}

In this section the characteristic of the algebraic group is denoted by
\(r\), and the coefficient characteristic is the fixed prime \(p\ne r\).
A root datum includes its character and cocharacter lattices, and hence
the central torus as well as the semisimple roots.

\begin{theorem}[Fixed-root Cartan bound]\label{thm:geometric-cartan}
Fix a finite collection \(\mathcal R\) of root data and a finite collection
of the usual pinned diagram automorphisms and exceptional diagram
isogenies on these data. There is a constant \(C=C(\mathcal R,p,M)\), also
depending on this fixed collection of maps, with the following property.
Let \(\mathbf G\) be a connected reductive group over
\(\overline{\F}_r\), with root datum in \(\mathcal R\), and let \(F\) be a
Steinberg endomorphism which, after an isomorphism of rational structures,
has the form
\[
 F=\theta\operatorname{Fr}_Q.
\]
Here \(Q\) is a power of \(r\), \(\operatorname{Fr}_Q\) is split
Frobenius, and \(\theta\) belongs to the specified finite collection;
exceptional isogenies are allowed only in their prescribed defining
characteristics. Put \(\Gamma=\mathbf G^F\).
For every subgroup \(Z\le Z(\Gamma)\) and every block \(b\) of
\(k[\Gamma/Z]\) with defect groups of order at most \(M\), all entries of
the Cartan matrix of \(b\) are at most \(C\).
\end{theorem}

The assertion places no bound on a Sylow \(p\)-subgroup of \(\Gamma\)
or on the \(p\)-part of a rational maximal torus. We will construct
projective modules using the horocycle correspondence, and bound their
characters only after projection to the prescribed quotient block.
The horocycle category and the passage from tilting sheaves to
projective representations follow Eteve's construction
\cite[Lemma~2.1.1 and Theorems~2.3.2, 3.1.1(ii), 3.2.4]{Eteve}.
Degree-zero projectivity of the corresponding tilting images was
also obtained independently by Zhu
\cite[Theorem~4.91]{ZhuTame}.  The uniform estimates needed here
are proved below: they must be independent of the rational torus
and its monodromy parameter.

We use the following standard parts of Deligne--Lusztig theory for
connected reductive groups with a Steinberg endomorphism, including its
isogeny form, in the large-parameter range used below: torus duality,
the partition into geometric Lusztig series,
the character formula, orthogonality, and the uniform Steinberg formula.
Their relevant forms are recalled when used; see
\cite[Theorems~4.2, 6.2 and 6.8, Proposition~5.22,
Corollaries~6.3 and 7.14, and Section~11]{DL}.
All constructible sheaves have coefficient characteristic different from
the geometric characteristic. Calculations with \(k\)-coefficients may
be descended to a sufficiently large finite subfield of \(k\), and
integral calculations to a finite extension of a complete splitting
discrete valuation ring. Tate twists do not affect any dimension below.

We use constructible descent and the six operations with finite
coefficients, and their adic counterparts, as in
\cite{LaszloOlssonFinite,LaszloOlssonAdic}.  For the finite local
coefficient algebras below, the assertions needed in
Lemma~\ref{geom:flat-costalk} are obtained after forgetting to a
finite coefficient field, as in its proof; no self-duality assertion
for an arbitrary finite local algebra is used.

\subsection{Strata and perfect extraordinary restrictions}

The first task is qualitative: the restrictions needed to glue
standard objects must be perfect complexes over their stabilizer
algebras.  Once this is established, a separate estimate will bound
the number of their indecomposable projective terms independently
of the stabilizer orders.

Work with one root datum. Choose an \(F\)-stable Borel pair
\(\mathbf T\subset\mathbf B=\mathbf T\mathbf U\), write \(W\) for the
Weyl group, and let \(\ell\) be its length function. A bounded range of
\(Q\) gives only finitely many groups for the specified data. It may
therefore be omitted until the end. In particular, assume \(dF=0\).
For \(x\in W\), choose \(\dot x\in N_{\mathbf G}(\mathbf T)\), and put
\[
 \mathbf G_x=\mathbf B\dot x\mathbf B,
 \qquad C_x=\mathbf G_x/\mathbf B,
 \qquad A_x=\mathbf T^{xF}.
\]
The torus Lang map is an \'etale isogeny, so \(A_x\) is a finite abelian
group of order prime to \(r\). It identifies with the rational points
of a corresponding \(F\)-stable maximal torus \(T_x\) of \(\mathbf G\).

Define
\[
 \mathcal H=
 \{(ut,u'F(t)):u,u'\in\mathbf U,\ t\in\mathbf T\},
 \qquad \mathcal X=[\mathbf G/\mathcal H],
\]
where \((b,b')\) sends \(g\) to \(bg(b')^{-1}\).
Its strata are
\(j_x:\mathcal X_x=[\mathbf G_x/\mathcal H]\hookrightarrow\mathcal X\).
The torus Lang isogeny makes this action transitive on each stratum.
The stabilizer of \(\dot x\) is
\[
 (\mathbf U\cap{}^{\dot x}\mathbf U)\rtimes A_x.
\]
Indeed its torus equation is \(t=xF(t)\), and its unipotent equation is
\(u={} ^{\dot x}u'\). The section of the torus quotient is
\(t\mapsto(t,F(t))\).

Consequently ordinary inflation, without a shift, identifies
\begin{equation}\label{geom:stratum-category}
 D^b_c(\mathcal X_x,k)\simeq D^b(kA_x\text{-mod}).
\end{equation}
Here and throughout, constructibility includes finite-dimensional
cohomology. To verify the identification, cohomology sheaves on the
classifying stack are representations of the component group \(A_x\).
The components in the nerve of the stabilizer are affine spaces, since
its connected unipotent radical is split. Their positive ordinary
\'etale cohomology vanishes. Descent therefore computes Ext by the
ordinary group-cohomology complex of \(A_x\). Truncation proves the
bounded derived assertion. This also explains why the equivalence
retains modular extensions even when \(p\mid |A_x|\); compare
\cite[Lemma 2.1.1]{Eteve}.

For a character \(\phi:A_x\to k^\times\), let \(L_{x,\phi}\) be its
one-dimensional module and let \(E_{x,\phi}\) be its projective cover.
Writing \(A_x=(A_x)_p\times(A_x)_{p'}\), we have
\[
 E_{x,\phi}=k[(A_x)_p]\otimes_k L_{x,\phi}.
\]
Set
\[
 \Delta_{x,\phi}=j_{x,!}E_{x,\phi}[\ell(x)],
 \qquad
 \nabla_{x,\phi}=Rj_{x,*}E_{x,\phi}[\ell(x)].
\]
A standard, respectively costandard, flag is a finite filtration by
extensions with factors of the displayed standard, respectively
costandard, objects, without additional shifts.

Pullback to \(\mathbf G\), followed by \([\dim\mathbf B]\), makes both
objects perverse. The cell has dimension
\(\dim\mathbf B+\ell(x)\), its coefficient is lisse, and its inclusion
is affine and quasi-finite: both \(\mathbf G_x\) and \(\mathbf G\) are
affine. Apply affine quasi-finite perverse exactness
\cite[Corollary 4.1.3]{BBD}. These assertions hold on invariant open
unions of cells as well, by base change.

We first prove the coefficient lemma that supplies perfection.

\begin{lemma}\label{geom:flat-costalk}
Let \(R\) be a finite-dimensional commutative local algebra over a
finite field of characteristic \(p\), with residue field \(k_0\).
Let \(Y\) be a variety over an algebraically closed field of
characteristic different from \(p\), and let \(\mathcal D\) be a
constructible sheaf of \(R\)-modules with projective geometric stalks.
For a locally closed immersion \(i:Z\hookrightarrow Y\) and a geometric
point \(z\in Z\), the complex \((Ri^!\mathcal D)_z\) is perfect over
\(R\). The same statement holds after extension to algebraic coefficient
fields when the data descend to finite coefficients.
\end{lemma}

\begin{proof}
Put \(P=(Ri^!\mathcal D)_z\). Forgetting the coefficient action commutes
with extraordinary restriction: the forgetful functor preserves
injectives because its left adjoint is exact. Constructible finiteness
therefore gives bounded finite \(R\)-cohomology for \(P\).
Projectivity on stalks says that \(\mathcal D\) is flat over \(R\).

For any bounded complex \(B\) of finite free \(R\)-modules, adjunction
with extension by zero gives the projection comparison
\[
 P\otimes_R B\xrightarrow{\sim}
 \bigl(Ri^!(\mathcal D\otimes_R B)\bigr)_z.
\]
It is an isomorphism for \(B=R\), hence for finite sums, shifts and
cones. To extend this comparison to the residue field, take a
bounded-above finite free resolution of \(k_0\), and let \(B_n\) be
its brutal truncation in degrees \([-n,0]\). There is a finite module
\(N_n\) such that
\(\operatorname{Cone}(B_n\to k_0)\simeq N_n[n+1]\).
Choose \(b\) with \(P\in D^{\le b}(R)\). Right \(t\)-exactness of
derived tensor gives
\[
 P\otimes_R^L N_n[n+1]\in D^{\le b-n-1}(R).
\]
Choose also an upper cohomological bound \(d\) for
\((Ri^!(\mathcal D\otimes_R k_0))_z\).
Every finite \(R\)-module has a finite filtration by \(k_0\).
Flatness and the long exact cohomology sequence show that
\((Ri^!(\mathcal D\otimes_R N))_z\in D^{\le d}(R)\)
for every finite module \(N\), independently of the filtration length.
Thus the error on the geometric side lies in
\(D^{\le d-n-1}(R)\). In every fixed degree both errors disappear for
large \(n\), giving
\begin{equation}\label{geom:coefficient-change}
 P\otimes_R^L k_0\simeq
 \bigl(Ri^!(\mathcal D\otimes_R k_0)\bigr)_z.
\end{equation}
The right-hand side is bounded. A minimal bounded-above free resolution
of \(P\) has differentials in \(\rad R\); tensoring with \(k_0\) kills
them. Boundedness in \eqref{geom:coefficient-change} forces all but
finitely many terms of that resolution to vanish. Hence \(P\) is
perfect. Extension of coefficients preserves the resulting finite
projective complex.
\end{proof}

\begin{lemma}\label{geom:cross-perfect}
For all \(v,x\in W\) and \(\phi\), the complex
\(j_v^!\Delta_{x,\phi}\) is perfect over \(kA_v\).
This remains true when the calculation is performed in an invariant
open containing the two strata.
\end{lemma}

\begin{proof}
Suppress shifts and put \(S=(A_v)_p\), embedded in \(\mathcal H\) as
above; it fixes \(\dot v\). It suffices to restrict to \(S\): since
\([A_v:S]\) is prime to \(p\), a complex of \(kA_v\)-modules is a
retract of the induction of its restriction to \(S\).
Changing equivariance from \(\mathcal H\) to \(S\) is smooth pullback.

Form the finite affine quotient \(\pi:\mathbf G\to Y=\mathbf G/S\).
The maps to the quotient and to the classifying stack give a
representable finite morphism
\[
 \Pi:[\mathbf G/S]\longrightarrow Y\times BS.
\]
Indeed its pullback along the trivial-torsor atlas
\(Y\to Y\times BS\) is \(\pi\). Bruhat strata descend to locally
closed subsets \(Y_x\). Give them reduced structure, which has no
effect on \'etale sheaves. Regard the finite pushforward
\(\mathcal D=\Pi_*(j_{x,!}E_{x,\phi})\) as a sheaf on \(Y\) of modules
over \(R=kS\). This is an ordinary sheaf: extension by zero is exact
and finite pushforward has no higher ordinary cohomology. The
identification with ring-valued sheaves follows from the finite
\'etale nerve \(Y\times S^\bullet\) of the displayed atlas.

Every geometric stalk of \(\mathcal D\) is projective over \(kS\).
Over \(Y_x\), a stalk is induced from the stabilizer in \(S\) of one
point of the corresponding orbit in \(\mathbf G_x\). Such a stabilizer
injects into \(A_x\): torus projection is unchanged by conjugation in
\(\mathcal H\), and its unipotent kernel has no nontrivial finite
\(p\)-subgroup since \(r\ne p\). The stalk coefficient is consequently
the restriction of \(E_{x,\phi}\) to a subgroup of \(A_x\), which is
projective; induction to \(S\) preserves projectivity. Stalks outside
\(Y_x\) are zero.

For \(i:Y_v\hookrightarrow Y\), proper base change and localization
commute \(Ri^!\) with \(\Pi_*\). At \(\pi(\dot v)\) the reduced
fiber is the single fixed point \(\dot v\), so the resulting stalk,
with its \(S\)-action, is exactly the restriction to \(S\) of the
desired cross-costalk. The algebra \(kS\) is commutative local.
Lemma~\ref{geom:flat-costalk} applies after finite descent and proves
perfection. All steps commute with restriction to an invariant open.
\end{proof}

\subsection{Uniformity without a bound on the finite torus}

We next bound the number of indecomposable projectives needed in the
cross-costalks. Their vector-space dimensions are not suitable for
this purpose, because \(\dim E_{x,\phi}=|(A_x)_p|\) can grow.

A rank-one multiplicative Kummer sheaf on a torus means a rank-one
summand of the direct image under an isogeny of degree prime to \(pr\),
with its multiplicative structure. For a fixed \(x\), pullback of the
stratum coefficients to \(\mathbf G_x\) has the following description.
Let
\[
 r_x:\mathbf G_x\longrightarrow\mathbf T,
 \qquad u\dot x b\longmapsto t_b,
\]
where \(u\in\mathbf U\cap{}^{\dot x}\mathbf U^-\) and \(t_b\) is the
torus component of \(b\). There are pairwise distinct multiplicative
Kummer sheaves \(\mathcal L^x_\phi\) on \(\mathbf T\), as \(\phi\)
varies, such that
\begin{equation}\label{geom:torus-coefficients}
 L_{x,\phi}|_{\mathbf G_x}=r_x^*\mathcal L^x_\phi,
 \qquad
 E_{x,\phi}|_{\mathbf G_x}
 =r_x^*(\mathcal L^x_\phi\otimes\mathcal P_x).
\end{equation}
Here \(\mathcal P_x\) is the direct image of the constant sheaf under
a torus isogeny with kernel \((A_x)_p\).
Indeed the torus-coordinate equation in the orbit map is a torus Lang
isogeny with kernel \(A_x\). Factoring it into its \(p\)- and
\(p'\)-parts proves \eqref{geom:torus-coefficients}. A character of the
kernel gives a trivial local system only when it is trivial, because
the covering torus is connected. This proves the asserted distinctness.

We record explicitly the geometric estimate for the rank-one part.

\begin{lemma}\label{geom:kummer-stalk}
Fix the root datum and \(v,x\in W\). Let \(V_v\subset\mathbf G/\mathbf B\)
be the translated opposite big cell through \(\dot v\mathbf B\), with
its natural section \(s:V_v\to\mathbf G\), and put
\(f=r_x\circ s\) on \(C_x\cap V_v\). For every rank-one multiplicative
Kummer sheaf \(\mathcal L\) on \(\mathbf T\) of order prime to \(pr\),
the sum of the dimensions of the stalk cohomology of
\[
 Rj_*(f^*\mathcal L),
 \qquad j:C_x\cap V_v\hookrightarrow V_v,
\]
at any point of \(C_v\) is bounded by a constant depending only on the
root datum. Its restriction cohomology sheaves on \(C_v\) are constant.
\end{lemma}

\begin{proof}
The group \(\mathbf U_v=\mathbf U\cap{}^{\dot v}\mathbf U^-\) acts
simply transitively on \(C_v\). It preserves \(V_v\), the section
satisfies \(s(uz)=us(z)\), and the right torus coordinate is invariant
under left \(\mathbf U\). Thus the entire construction is
\(\mathbf U_v\)-equivariant. Its restriction cohomology on \(C_v\) is
constant, and it suffices to work at \(\dot v\mathbf B\).

Choose a reduced expression \(x=s_1\cdots s_d\). The associated
Bott--Samelson variety \(\widetilde Z_x\) is smooth and proper over
the Schubert closure \(\overline C_x\), is an isomorphism over \(C_x\),
and has a simple normal-crossing boundary with \(d\) distinguished
divisors. In the gallery description these divisors are the conditions
that successive flags are equal. Restrict this proper morphism over
\(V_v\). Its open complement is \(C_x\cap V_v\); write \(\widetilde j\)
for this open immersion and \(\pi\) for the restricted proper morphism.
Then
\[
 Rj_*f^*\mathcal L=R\pi_*R\widetilde j_*f^*\mathcal L.
\]

We describe the cohomology sheaves of the star-extension on a boundary
stratum. A character of a split torus modulo an integer \(n\) prime
to \(pr\) defines its Kummer sheaf by taking an \(n\)-th root of a
torus monomial. The pullback monomial is a unit on the open complement.
In \'etale coordinates near a crossing it has the form
\[
 u\,z_1^{a_1}\cdots z_t^{a_t},
\]
where \(u\) is a unit and the \(z_i\) are parameters for the boundary
divisors. Take \(n\)-th roots of the parameters and of \(u\).
The constant-coefficient cohomology of the punctured crossing is the
exterior algebra on its \(t\) parameter loops. Coordinate deck
translations act trivially on this cohomology. Taking a character
summand is exact because \(p\nmid n\). It follows that the
star-extension vanishes on this stratum if any of its local monodromy
characters is nontrivial. Otherwise its cohomology sheaves are the
extending rank-one Kummer sheaf tensored with exterior powers of
\(k(-1)^t\): trivial local monodromy makes the relevant boundary
powers divisible by \(n\), so the finite Kummer system extends across
these divisors after removing the other boundary components.
The loop lines are constant on the stratum: their residues
are indexed by the globally distinguished divisors. In particular the
sum of their ranks is at most \(2^d\), independently of \(n\).

This bounds the ranks contributed by the boundary crossings.
We now bound the cohomology of these coefficients over the proper
fiber, using a filtration with uniformly boundedly many pieces of
the form \(\mathbb A^a\times\mathbb G_m^b\).
For proper base change, consider the fiber of \(\pi\) at
\(\dot v\mathbf B\). Write \(D_1,\ldots,D_d\) for the equality
divisors. Filter the fiber by the closed conditions that at least
\(m\) equalities hold, in descending order of \(m\). Each layer is a
finite disjoint union of its intersections with the exact strata
\[
 D_I^\circ=
 \Bigl(\bigcap_{i\in I}D_i\Bigr)
       \mathbin{\big\backslash}\Bigl(\bigcup_{j\notin I}D_j\Bigr).
\]
Deleting the steps in \(I\) identifies an exact stratum in the fiber
with the fiber of the uncompactified convolution for the remaining,
possibly nonreduced, simple-reflection word, with every remaining
step unequal. Subdivide further by recording the Bruhat position of
every intermediate flag.
Each nonempty piece is a product
\(\mathbb A^a\times\mathbb G_m^b\), with the number of pieces and
\(a+b\) bounded in terms of \(d\) and \(|W|\). Here is the elementary
gallery verification. Starting with the prescribed final flag, choose
predecessors backwards in the relevant minimal-parabolic projective
line. Its two Bruhat cells have dimensions differing by one. For an
endpoint in the shorter cell, the equal predecessor is a point and
the predecessors in the longer cell form \(\mathbb A^1\). For an
endpoint in the longer cell, the equal predecessor is a point, the
unequal predecessors in that cell form \(\mathbb G_m\), and the
predecessor in the shorter cell is a point. The unipotent section of
each Bruhat cell trivializes these alternatives algebraically as the
endpoint varies. Induction gives the displayed products. Recording
the initial position as the singleton cell \(C_1\) enforces the
initial flag without a further equation. To obtain a filtration on
each exact equality stratum, use the morphism recording intermediate
flags into a product of flag varieties. A linear ordering refining
the product Bruhat order has closed lower ideals. Their inverse
images give a filtration with the fixed-position records as its
successive differences. Combining it with the equality filtration
makes localization applicable, compatibly with the boundary strata.
This is the simple-reflection construction in
\cite[Theorem 1.1, Proposition 5.3, Lemma 6.2,
and Sections 6.2--6.3]{HainesPaving}.

On \(\mathbb A^a\times\mathbb G_m^b\), the Picard group is zero and
every invertible regular function is a constant times a Laurent
monomial. The Kummer exact sequence therefore expresses every
rank-one local system of order dividing \(n\) as a torus Kummer system.
A nontrivial such system has zero torus cohomology: pull back to its
finite prime-to-\(p\) torus cover and use the fact that torus
translations act trivially on cohomology. The constant system has
total compact Betti number \(2^b\), by K\"unneth and duality.
Thus all the extending rank-one systems on the pieces have total
compact Betti number at most \(2^b\).

Apply the cohomology-sheaf spectral sequence on each piece, followed
by the finite localization filtration of the proper fiber. The local
rank bound \(2^d\), the number of pieces, and their dimensions give a
bound depending only on the root datum. Proper base change proves
the claimed stalk bound, uniformly in \(r\), \(n\), and \(\mathcal L\).
More explicitly, there are at most \(3^d\) backward records and each
piece has \(a+b\le d\), so \(12^d\) is a sufficient bound.
\end{proof}

Lemma~\ref{geom:kummer-stalk} bounds the geometric contribution of
one rank-one coefficient.  We now sum over the possible characters
on the receiving stratum.  Character matching and the cohomology
of a covering torus will prevent the degree of the Lang cover
from entering this sum.

\begin{proposition}\label{geom:summed-ext}
For every integer \(J\ge0\), there is a constant \(D_J\), depending
only on the fixed root data and \(J\), such that for all \(v,x,\phi\)
and \(0\le j\le J\),
\begin{equation}\label{geom:ext-bound}
 \sum_{\psi\in\Hom(A_v,k^\times)}
 \dim_k\Ext^j_{\mathcal X}
       (j_{v,!}L_{v,\psi},j_{x,!}E_{x,\phi})\le D_J.
\end{equation}
The same bound holds, after increasing the constant if necessary, in
invariant open unions containing the relevant strata. Bounded shifts
of the two arguments are allowed by increasing \(J\).
\end{proposition}

\begin{proof}
First forget equivariance and compute on \(\mathbf G\). The chart
\(V_v\) contains the whole cell \(C_v\). Its section satisfies
\(r_v(s)=1\) on \(C_v\), and it gives a trivialization
\(\mathbf G|_{V_v}\simeq V_v\times\mathbf B\). Since the first
argument is extended by zero from this open, restriction to it
computes the same Ext. Write \(\rho:V_v\times\mathbf B\to V_v\)
for projection, and \(t_b\) for the torus coordinate of \(b\).
On \((C_x\cap V_v)\times\mathbf B\),
\[
 r_x(s(z)b)=f(z)t_b.
\]
By \eqref{geom:torus-coefficients}, the first coefficient is
\(\mathcal L^v_\psi(t_b)\). Tensor both arguments by its inverse and
use adjunction for \(\rho^*\) and \(R\rho_*\).

The module \(E_{x,\phi}\) has a filtration by copies of
\(L_{x,\phi}\). The corresponding rank-one graded terms, after this
twist, are external products with fiber coefficient
\[
 \mathcal L^x_\phi\otimes(\mathcal L^v_\psi)^{-1}.
\]
Their ordinary direct images are extensions by zero from
\(C_x\cap V_v\) of their fiber cohomology. To justify this assertion
for the nonproper projection, use Verdier duality on the smooth
fiber and compact-support K\"unneth for external products. This proves
both restriction base change and vanishing outside the immersed base
piece. The filtration transfers these two properties to the full
coefficient. Its possibly large length is used only for these
properties, not for a dimension estimate.

A nontrivial multiplicative rank-one torus system has zero ordinary
cohomology, by the torus-cover argument in the preceding proof.
Consequently all Ext groups vanish unless
\begin{equation}\label{geom:character-match}
 \mathcal L^v_\psi\simeq\mathcal L^x_\phi.
\end{equation}
There is at most one such \(\psi\), by the distinctness in
\eqref{geom:torus-coefficients}.

In the matching case the coefficient on the immersed product is
\[
 f^*\mathcal L^x_\phi\otimes\mathcal P_x(f(z)t_b).
\]
The algebraic change of variable \(t'_b=f(z)t_b\) identifies its
ordinary fiber integral with
\[
 f^*\mathcal L^x_\phi\otimes R\Gamma(\mathbf T'_x,k),
\]
where \(\mathbf T'_x\to\mathbf T\) is the torus isogeny defining
\(\mathcal P_x\); the unipotent part of \(\mathbf B\) is acyclic.
More explicitly, write this isogeny as \(h\).  The total cover
\(h(t')=f(z)t_b\) is the product of the immersed base with
\(\mathbf T'_x\), by
\((z,t')\mapsto(z,f(z)^{-1}h(t'),t')\).
Thus the fiber integral is constant as a derived complex, without
choosing a lift of \(f\) through the isogeny.
The covering torus has the same dimension as \(\mathbf T\). Therefore
the total dimension of this constant complex is
\(2^{\rk\mathbf T}\), independent of \(|(A_x)_p|\).
The direct image on all of \(V_v\) is its extension by zero, by the
base-change and vanishing just proved.

It remains to bound
\[
 R\Hom_{V_v}(a_!k,j_!f^*\mathcal L^x_\phi),
 \qquad a:C_v\hookrightarrow V_v.
\]
By adjunction this is
\(R\Gamma(C_v,a^!j_!f^*\mathcal L^x_\phi)\).
Verdier duality expresses its coefficient cohomology in terms of the
restriction to \(C_v\) of
\(Rj_*(f^*\mathcal L^x_\phi)^{\vee}\), with only the fixed smooth
dimension shifts. Lemma~\ref{geom:kummer-stalk} bounds these stalks
and makes their cohomology constant on \(C_v\).
Since \(C_v\) is affine space, ordinary cohomology of a constant
sheaf is concentrated in degree zero. The resulting spectral sequence
bounds the total Ext dimension by a root-datum constant.
This proves the summed estimate on \(\mathbf G\).

The estimate on \(\mathbf G\) is independent of the degree of
the Lang cover.  It remains to restore equivariance, retaining
only the bounded range of cohomological degrees in the statement.
Use the smooth atlas \(\mathbf G\to\mathcal X\).
Cohomological descent for Hom gives a spectral sequence whose term
in simplicial degree \(a\) is an Ext group on
\(\mathbf G\times\mathcal H^a\). Equivariance identifies the pulled
back arguments with projection pullbacks from \(\mathbf G\).
K\"unneth and smooth base change multiply the preceding Ext groups
by \(H^*(\mathcal H^a,k)\). The inputs before shifts are ordinary
sheaves, so these Ext degrees and the simplicial degrees are
nonnegative. Only \(a\le J\) can contribute to total degree at most
\(J\). As a variety, \(\mathcal H\) is a product of a fixed-dimensional
torus and affine space; its relevant Betti numbers, and hence those of
these finitely many powers, depend only on the root datum and \(J\).
Taking dimensions in the spectral sequence proves
\eqref{geom:ext-bound}. For invariant opens the first argument is
extended by zero to the ambient space, so open adjunction and base
change give the same calculation. Negative Ext degrees of ordinary
sheaves are zero, which also proves the assertion about bounded shifts.
For example, writing \(t=\rk\mathbf T\) and
\(d=\max_{w\in W}\ell(w)\), the generous choice
\[
 D_J=2^t12^d\sum_{a=0}^J2^{at}
\]
bounds the sum in every degree at most \(J\).
\end{proof}

\subsection{Gluing with a bounded number of standard factors}

We now combine perfection with the summed Ext estimate.
Perfection reduces each gluing obstruction to two projective terms;
the estimate bounds their multiplicities and hence the length of
the standard flag constructed at the next stratum.

\begin{proposition}\label{geom:bounded-tiltings}
There is a constant \(L\), depending only on the fixed root data,
such that for every \((x,\phi)\) there is an object
\(\mathcal T_{x,\phi}\) with both a standard and a costandard flag.
A specified standard flag has length at most \(L\), contains
\(\Delta_{x,\phi}\) once, and has all remaining factors on strictly
smaller Bruhat strata. Thus the standard multiplicity matrix is
triangular, with identity diagonal blocks.
\end{proposition}

\begin{proof}
Choose a total ordering refining descending Bruhat order. Its initial
unions are invariant opens. When the cell \(x\) is added, start with
the closed pushforward of \(E_{x,\phi}[\ell(x)]\), zero on the
previous cells. This has both flags. Suppose the object \(\mathcal M\)
has been constructed on an old open and \(v\) is the next stratum,
closed in the enlarged open. Write \(j\) for the old open immersion
and \(i\) for the closed stratum immersion, and put
\[
 K=i^!j_!\mathcal M[-\ell(v)].
\]
The standard flag of \(\mathcal M\) and Lemma~\ref{geom:cross-perfect}
make \(K\) perfect over \(kA_v\).
The object \(j_!\mathcal M\) is perverse after atlas pullback and the
fixed \([\dim\mathbf B]\) shift, because it has a standard flag.
Perverse cosupport on the smooth stratum gives \(K\in D^{\ge0}\).
Likewise \(Rj_*\mathcal M\) is perverse because it has a costandard
flag. The localization triangle and \(i^!Rj_*=0\) give
\[
 i^*Rj_*\mathcal M[-\ell(v)]\simeq K[1].
\]
Perverse support puts the left-hand side in \(D^{\le0}\).
Thus \(K\) has cohomology only in degrees \(0,1\).

The algebra \(kA_v\) is split symmetric, hence self-injective. Choose
a minimal bounded complex of projectives representing \(K\).
If its first nonzero term were below degree zero, vanishing of that
cohomology would make its differential injective; self-injectivity
would split this injection, contradicting minimality. If its last
term were above degree one, the preceding differential would be a
surjection and would split by projectivity. Consequently
\[
 K\simeq[P^0\xrightarrow{\delta}P^1],
\]
with the terms in degrees zero and one.
Let \(m^q_\psi\) be the multiplicity of \(E_{v,\psi}\) in \(P^q\).
The terms are also injective, so this complex computes
\(R\Hom_{kA_v}(L_{v,\psi},K)\). Its differential is zero on the
socle of every indecomposable injective summand: a map nonzero on
that essential simple socle would be injective, and hence split,
again contradicting minimality. Since the socle of \(E_{v,\psi}\)
is \(L_{v,\psi}\), it follows that
\begin{equation}\label{geom:multiplicity-ext}
 m^q_\psi=
 \dim_k\Ext^q_{kA_v}(L_{v,\psi},K),\qquad q=0,1.
\end{equation}

Let \(N\) be the old standard-flag length. Adjunction turns the
right-hand side of \eqref{geom:multiplicity-ext} into
\[
 \dim_k\Ext^{q-\ell(v)}
       (j_{v,!}L_{v,\psi},j_!\mathcal M).
\]
For a standard factor indexed by \((y,\eta)\), this is the Ext group
between unshifted ordinary sheaves in degree
\(q+\ell(y)-\ell(v)\).  If
\(d=\max_{w\in W}\ell(w)\), these degrees lie between \(-d\)
and \(d+1\).  Negative Ext degrees of ordinary sheaves vanish,
so we may use \(J=d+1\) in Proposition~\ref{geom:summed-ext}.
Taking long exact sequences along the standard flag gives a
constant \(D\) such that
\begin{equation}\label{geom:glue-size}
 \sum_\psi m^q_\psi\le DN\qquad(q=0,1).
\end{equation}
Computations inside the enlarged open agree with the previous
estimates by open extension adjunction.

We have bounded the two projective terms of the obstruction.
The next extension uses \(P^1\) to add standard factors and
\(P^0\) to supply the matching costandard restriction.
The triangle
\(K\to P^0\to P^1\to K[1]\) gives by adjunction a map
\(i_*P^1[\ell(v)]\to j_!\mathcal M[1]\). Let \(\mathcal M'\) be the
corresponding extension, so that
\[
 j_!\mathcal M\longrightarrow\mathcal M'
 \longrightarrow i_*P^1[\ell(v)]
 \longrightarrow j_!\mathcal M[1].
\]
Its restrictions satisfy
\(j^*\mathcal M'=\mathcal M\) and
\(i^*\mathcal M'=P^1[\ell(v)]\), whereas
\(i^!\mathcal M'=P^0[\ell(v)]\). The displayed triangle gives a
standard flag; the localization triangle
\[
 i_*P^0[\ell(v)]\longrightarrow\mathcal M'
 \longrightarrow Rj_*\mathcal M
\]
gives a costandard flag. By \eqref{geom:glue-size}, the new standard
length is at most \((1+D)N\).

There are at most \(|W|-1\) extension steps, so
\(L=(1+D)^{|W|-1}\) suffices. If the newly added stratum is outside
\(\overline{\mathcal X_x}\), its obstruction complex is zero and
nothing is added. Thus the support stays in
\(\overline{\mathcal X_x}\), and the original
factor on \(x\) occurs exactly once. Taking a maximum over the finite
collection of root data proves the proposition.
\end{proof}

\subsection{Projective representations from the correspondence}

The objects just constructed have bounded flags.  The next step
turns them into projective group representations: the two flags
force their character-functor images into degree zero, while
Rickard's cohomology complex supplies projectivity.  For the
degree bounds we first need affineness in a range uniform over
the fixed data.

The ample-boundary method goes back to Deligne--Lusztig
\cite[Sections~9.5--9.7]{DL}.  For ordinary Frobenius, an
affineness result for all field sizes is given in
\cite[Corollary~3.12]{BrosnanHongLee}; the argument here includes
the specified exceptional diagram isogenies and only needs a
uniform large-parameter range.

\begin{lemma}\label{geom:affineness}
For all sufficiently large \(Q\), uniformly over the specified root
data, maps \(\theta\), and \(x\in W\), the Deligne--Lusztig variety
\[
 X(x)=\{g\mathbf B:g^{-1}F(g)\in\mathbf G_x\}
\]
is smooth affine of dimension \(\ell(x)\).
\end{lemma}

\begin{proof}
Since \(dF=0\), the Lang map is \'etale. Its inverse image of
\(\mathbf G_x\) is a \(\mathbf B\)-bundle over \(X(x)\), which proves
smoothness and the dimension formula.

Here is the ample-line-bundle argument for affineness, including the
exceptional diagram isogenies. On \(\mathcal B=\mathbf G/\mathbf B\)
use the convention
\[
 \mathcal L_\lambda=
 \mathbf G\times^{\mathbf B}\overline{\F}_r{}_{-\lambda};
\]
strictly dominant characters define ample bundles. Choose such an
integral \(\lambda\), which exists for every root datum. On
\(\overline C_x\), the extremal-coordinate functional of weight
\(x\lambda\) in a highest-weight representation gives a
\(\mathbf B\)-semi-invariant section of \(\mathcal L_\lambda\), of
weight \(-x\lambda\), nonzero precisely on \(C_x\). To check its
vanishing on the boundary, along a saturated Bruhat step
\(y<ys_\alpha\) one has
\[
 y\lambda-ys_\alpha\lambda
 =\langle\lambda,\alpha^\vee\rangle y\alpha,
\]
a positive multiple of a positive root. Thus for \(y<x\),
\(y\lambda-x\lambda\) is a nonzero sum of positive roots. Positive
unipotents raise weights, so the coordinate of weight \(x\lambda\)
vanishes on \(C_y\) and is nonzero on \(C_x\). The same calculation
gives the claimed semi-invariance.

The closure of the diagonal relative-position orbit of \(x\) in
\(\mathcal B\times\mathcal B\) is
\(\mathbf G\times^{\mathbf B}\overline C_x\).
Twisting the preceding section by the first-factor line of weight
\(x\lambda\) makes it descend to a section of
\(\mathcal L_{-x\lambda}\boxtimes\mathcal L_\lambda\) whose nonzero
locus is the relative-position orbit itself. Pull back along the graph
\(u\mapsto(u,F(u))\). On the projective inverse image of the orbit
closure its line bundle is the restriction of
\[
 \mathcal L_{F^*\lambda-x\lambda},
 \qquad F^*\lambda=Q\theta^*\lambda.
\]
The pullback \(\theta^*\lambda\) is strictly dominant: the induced
map on flag varieties is finite, and finite pullback preserves
ampleness. This includes inseparable exceptional isogenies. Therefore
\(Q\theta^*\lambda-x\lambda\) is strictly dominant for all
sufficiently large \(Q\), with one threshold for the finitely many
possibilities. The nonzero locus of a section of an ample line bundle
on a projective scheme is affine, by passing to a very ample power.
Here it is \(X(x)\). For a torus the flag variety is a point.
Compare \cite[Sections 9.5--9.7]{DL}.
\end{proof}

Let \(\mathbf B_F^{\mathrm{gr}}\) and
\(\mathbf G_F^{\mathrm{gr}}\) denote the graphs of \(F\), and use
\[
 \mathcal X\xleftarrow{\ \beta\ }
 [\mathbf G/\mathbf B_F^{\mathrm{gr}}]
 \xrightarrow{\ \alpha\ }
 [\mathbf G/\mathbf G_F^{\mathrm{gr}}]\simeq B\Gamma.
\]
The equivalence on the right is Lang's theorem. The map \(\beta\)
is smooth, with unipotent fiber
\(\mathcal H/\mathbf B_F^{\mathrm{gr}}\simeq\mathbf U\), and
\(\alpha\) is proper with flag-variety fiber.
With our ordinary stratum inflation, put
\(\mathsf{Ch}=R\alpha_!\beta^*\), without an additional shift.
Pullback to a point of \(B\Gamma\) gives directly
\begin{equation}\label{geom:character-functor}
 \begin{aligned}
 \mathsf{Ch}(\Delta_{x,\phi})
   &=R\Gamma_c(X(x),\mathcal E_{x,\phi})[\ell(x)],\\
 \mathsf{Ch}(\nabla_{x,\phi})
   &=R\Gamma(X(x),\mathcal E_{x,\phi})[\ell(x)].
 \end{aligned}
\end{equation}
Here \(\mathcal E_{x,\phi}\) is the local system associated to
\(E_{x,\phi}\) on the finite \'etale \(A_x\)-torsor
\[
 Y(\dot x)=\{g\mathbf U:g^{-1}F(g)\in\mathbf U\dot x\mathbf U\}
 \longrightarrow X(x).
\]
For clarity, the flag fiber is stratified by the condition
\(g^{-1}F(g)\in\mathbf G_x\). Write
\(g^{-1}F(g)=c\dot x(c')^{-1}\) with \((c,c')\in\mathcal H\).
The stabilizer-component torsor maps to \(gc\mathbf U\), and
\((c')^{-1}F(c)\in\mathbf U\) gives the defining equation of
\(Y(\dot x)\). This identifies the coefficients and their left
\(\Gamma\)-action. Smooth base change for \(\beta\) handles star
extensions, and properness of \(\alpha\) gives both formulas in
\eqref{geom:character-functor}. Thus their normalization follows from
the fiber calculation itself.

By Lemma~\ref{geom:affineness} and Artin vanishing,
\[
 \mathsf{Ch}(\Delta_{x,\phi})\in D^{\ge0}(k\Gamma),
 \qquad
 \mathsf{Ch}(\nabla_{x,\phi})\in D^{\le0}(k\Gamma).
\]
Indeed a lisse coefficient shifted by \(\ell(x)\) is perverse on the
smooth affine \(X(x)\). Affine Artin vanishing gives the ordinary
cohomology bound, and duality gives the compact-support bound;
the modular coefficient form follows from the same theorem after
finite descent, as in
\cite[Section~4.0, Theorem~4.1.1 and Corollary~4.1.2]{BBD}.
Both flags therefore put
\(\mathsf{Ch}(\mathcal T_{x,\phi})\) in degree zero. Denote this module
by \(P_{x,\phi}\).

\begin{proposition}\label{geom:projective-images}
The modules \(P_{x,\phi}\) are projective. Let
\(\widehat A_x\) be the ordinary linear characters of \(A_x\), with
\(\xi\mapsto\bar\xi\) their modular reductions, and set
\begin{equation}\label{geom:torus-sum}
 \Psi_{x,\phi}=(-1)^{\ell(x)}
       \sum_{\substack{\xi\in\widehat A_x\\\bar\xi=\phi}}
              R_{T_x}^{\mathbf G}(\xi).
\end{equation}
If \(n_{x,\phi;y,\eta}\) are the multiplicities in the specified
standard flag, the ordinary lift character of \(P_{x,\phi}\) is
\begin{equation}\label{geom:projective-character}
 \chi_{P_{x,\phi}}=
       \sum_{y,\eta} n_{x,\phi;y,\eta}\Psi_{y,\eta},
 \qquad \sum_{y,\eta}n_{x,\phi;y,\eta}\le L.
\end{equation}
These projective characters and the \(\Psi_{x,\phi}\) have the same
linear span in characteristic zero, and that span contains the
regular character of \(\Gamma\).
\end{proposition}

\begin{proof}
At a geometric point \(g\mathbf U\) of \(Y(\dot x)\), the stabilizer
for left translation by \(\Gamma\) lies in
\(\Gamma\cap g\mathbf U g^{-1}\), an \(r\)-group, and hence has order
prime to \(p\). Rickard's equivariant cohomology theorem
\cite[Theorems 3.2 and 3.5]{RickardEtale} applies to this
quasi-projective variety with the action of \(\Gamma\times A_x\).
Restricting the resulting complex to \(\Gamma\), Mackey decomposition
gives summands of permutation terms induced from the intersections of point
stabilizers with \(\Gamma\). These intersections are the \(r\)-groups
just described. Thus the bounded complex is termwise projective
over \(\Gamma\), compatibly with integral, residue, and
characteristic-zero coefficients. The commuting right action of
\(A_x\) selects \(\mathcal E_{x,\phi}\) by the idempotent for the
specified character of \((A_x)_{p'}\). This idempotent is integral
over a splitting extension because its denominator is prime to
\(p\), and its direct summand remains perfect over \(k\Gamma\).
Its ordinary Euler character is exactly
\(\Psi_{x,\phi}\): all ordinary characters lifting \(\phi\) are
selected, and the shift in \eqref{geom:character-functor} supplies
\((-1)^{\ell(x)}\).

Thus every standard image is perfect, and a standard flag makes
\(P_{x,\phi}\) perfect. A finite module of finite projective dimension
over the self-injective algebra \(k\Gamma\) is projective. Lifting
projectives over a complete splitting discrete valuation ring
identifies the projective Grothendieck groups before and after
reduction. Taking the Euler classes of the standard flag proves
\eqref{geom:projective-character}; lifting the extension maps is
unnecessary. Proposition~\ref{geom:bounded-tiltings} makes its
multiplicity matrix triangular with identity diagonal blocks, so the
two collections have the same span.

Summing \eqref{geom:torus-sum} over \(\phi\) puts
\(R_{T_x}^{\mathbf G}(\operatorname{Reg}_{A_x})\) in this span for
every \(x\). In a uniform Steinberg formula
\cite[Corollary 7.14, formula (7.14.2), and Section 11]{DL}, replace
each trivial torus character by
\(\operatorname{Reg}_{A_x}/|A_x|\). The Deligne--Lusztig character
formula \cite[Theorem 4.2 and Section 11]{DL} shows that the resulting
class function agrees with \(\St_\Gamma\) on unipotents, since the
values of a torus-induced character there are independent of the
linear torus character. It vanishes on every element with nonidentity
semisimple part, since every corresponding torus evaluation is at a
nonidentity element. The Steinberg character vanishes on nonidentity
unipotents and has nonzero value at the identity. The resulting
function is therefore
\[
 \frac{\St_\Gamma(1)}{|\Gamma|}\operatorname{Reg}_\Gamma.
\]
This proves the final span assertion.
\end{proof}

\subsection{Projection to a block and the Cartan estimate}

The projective characters now span the regular character, so they
detect every projective indecomposable summand.  Their full norms
can still grow with the tori.  The remaining estimate therefore
uses only the ordinary coordinates in the chosen bounded-defect
block, including after a central quotient.

For a finite group, write \(\langle\ ,\ \rangle\) for the ordinary
character inner product and \(\|\chi\|^2=\langle\chi,\chi\rangle\).
Let \(K(M)\) be a Brauer--Feit bound for the number of ordinary
irreducible characters in any block with defect groups of order at
most \(M\).

\begin{lemma}\label{geom:projected-norm}
Let \(\mathcal I\) be any set of at most \(K\) ordinary irreducible
characters of \(\Gamma\), and let \(\operatorname{pr}_{\mathcal I}\)
be their orthogonal coordinate projection. Then
\[
 \|\operatorname{pr}_{\mathcal I}\Psi_{x,\phi}\|^2\le K|W|^3,
 \qquad
 \|\operatorname{pr}_{\mathcal I}\chi_{P_{x,\phi}}\|^2
       \le KL^2|W|^3.
\]
\end{lemma}

\begin{proof}
Deligne--Lusztig orthogonality gives
\(\|R_{T_x}^{\mathbf G}(\xi)\|^2\le |W|\)
for every linear torus character \(\xi\); see
\cite[Theorem 6.8 and Section 11]{DL}. Hence every individual
irreducible coefficient has absolute value at most \(\sqrt{|W|}\).
Fix an irreducible character \(\rho\). By geometric-series
disjointness and torus duality, the parameters \(\xi\) from a fixed
torus whose Deligne--Lusztig characters contain \(\rho\) have dual
semisimple elements in one geometric conjugacy class. The intersection
of that class with the fixed dual maximal torus is one Weyl orbit.
It has at most \(|W|\) elements. Thus at most \(|W|\) summands in
\eqref{geom:torus-sum} can contribute to the \(\rho\)-coordinate,
irrespective of how many ordinary characters lift \(\phi\).
Consequently
\[
 |\langle\Psi_{x,\phi},\rho\rangle|\le |W|^{3/2}.
\]
Summing squares over \(\mathcal I\) proves the first inequality.
Equation~\eqref{geom:projective-character} and its bound \(L\) on
the number of factors prove the second.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:geometric-cartan}]
First assume \(Q\) exceeds the uniform threshold used above.
Take central coinvariants
\[
 \overline P_{x,\phi}
   =k[\Gamma/Z]\otimes_{k\Gamma}P_{x,\phi}.
\]
These are projective \(k[\Gamma/Z]\)-modules, since tensoring an
exhibited summand of a free module gives a summand of a free quotient
module. The same construction on integral projective lifts commutes
with reduction. In characteristic zero its character is the
coordinate projection onto irreducibles trivial on \(Z\).
This observation applies also when \(p\mid |Z|\).
Project further to \(b\), obtaining actual projective modules
\(Q_{x,\phi}=b\overline P_{x,\phi}\).

The regular-character span in Proposition~\ref{geom:projective-images}
projects to the regular character of \(b\): the coordinates of
\(\operatorname{Reg}_\Gamma\) on characters inflated from
\(\Gamma/Z\) are exactly their degrees, as in
\(\operatorname{Reg}_{\Gamma/Z}\). Characters of projective
indecomposable modules are linearly independent, and every such
module in \(b\) occurs with positive multiplicity in its regular
module. Hence every projective indecomposable module of \(b\) occurs
as a summand of some \(Q_{x,\phi}\). Otherwise its coordinate would
vanish in the span of all their projective characters, contradicting
the regular-character assertion.

Inflate the set \(\Irr(b)\) to \(\Gamma\). It has at most \(K(M)\)
members, so Lemma~\ref{geom:projected-norm} gives
\[
 \|\chi_{Q_{x,\phi}}\|^2\le K(M)L^2|W|^3.
\]
If \(\Phi_i\) is an indecomposable projective character occurring in
\(\chi_{Q_{x,\phi}}\), nonnegativity of ordinary multiplicities gives
\(\|\Phi_i\|\le\|\chi_{Q_{x,\phi}}\|\).
The Cartan entries are
\(c_{ij}(b)=\langle\Phi_i,\Phi_j\rangle\), so Cauchy--Schwarz yields
\[
 c_{ij}(b)\le K(M)L^2|W|^3.
\]
The root data are fixed throughout this estimate; both \(|W|\) and
\(L\) depend only on their specified finite collection. Finally,
the omitted bounded range of \(Q\) contains only finitely many finite
groups and central quotients. Increase the constant by the maximum
of their relevant Cartan entries. This proves the theorem.
\end{proof}
==== END Donovans-Conjecture-over-Algebraic-Closures-of-Prime-Fields//02_geometry.tex ====

==== BEGIN Donovans-Conjecture-over-Algebraic-Closures-of-Prime-Fields//03_families.tex ====
\section{Cartan bounds for quasisimple groups}\label{sec:families}

We now reduce the quasisimple Cartan problem to the odd-prime classical
calculations of the next three sections.  Throughout this section, the
coefficient prime is \(p\), the defining characteristic of a finite
reductive group is \(r\), and \(q\) is its defining field parameter.
For a positive integer \(a\), write \(a_p\) for its \(p\)-part.  For an
ordinary irreducible character \(\chi\) of a finite group \(H\), put
\[
 \operatorname{def}_p(\chi)=\frac{|H|_p}{\chi(1)_p}.
\]
This is the degree defect, expressed as an order rather than an exponent.
If \(\chi\) belongs to a block with defect group \(D\), then
\(\operatorname{def}_p(\chi)\leq |D|\).

\begin{theorem}[Quasisimple Cartan bound]\label{thm:quasisimple-cartan}
For every prime \(p\) and positive integer \(M\), there is a constant
\(C_{\mathrm{qs}}(p,M)\) such that every Cartan entry of every block of
\(kS\), where \(S\) is quasisimple and its defect groups have order at
most \(M\), is at most \(C_{\mathrm{qs}}(p,M)\).
\end{theorem}

The proof uses Theorem~\ref{thm:geometric-cartan} for bounded root
data.  For unbounded rank its inputs are Proposition~\ref{prop:runner-reduction}
and Proposition~\ref{prop:triangle-cartan}, with the character and block
identifications proved in Propositions~\ref{prop:label-degrees},
\ref{prop:single-cycle}, and~\ref{prop:core-blocks}.
Those propositions concern explicit reductive groups and do not use
Theorem~\ref{thm:quasisimple-cartan}.

\subsection{Restriction and reductive-group conventions}

We record the consequence of restriction that will be used when a
covering block can have large defect.

\begin{corollary}\label{fam:restriction}
Let \(N\lhd H\) with cyclic quotient, and let \(b\) be a block of
\(N\).  Suppose that every block of \(H\) covering \(b\) has one
simple module and all its decomposition numbers are one.
Then every decomposition number of \(b\) is at most one.
No defect bound on the covering blocks is required.  If the
original block \(b\) has bounded defect, its Cartan entries are
therefore bounded.
\end{corollary}

\begin{proof}
Apply Lemma~\ref{lem:restriction-rows} with \(L=c=u=1\).
The Cartan assertion uses only the Brauer--Feit bound for the
number of ordinary rows of \(b\).
\end{proof}

Write \(J=\mathbf J^F\), where \(\mathbf J\) is a connected reductive
group over \(\overline{\F}_r\), and write \(\mathbf J^*\) for its dual
group with its dual endomorphism.  We suppress the star on the latter
endomorphism.  If \(s\in\mathbf J^{*F}\) is semisimple, then
\(\mathcal E(J,s)\) denotes its ordinary Lusztig series.  When \(s\)
has order prime to \(p\), set
\[
 \mathcal E_p(J,s)
  =\bigcup_t\mathcal E(J,st),
\]
where \(t\) runs over the \(p\)-elements of
\(C_{\mathbf J^*}(s)^F\), with the usual conjugacy identifications.
Here and in the reductive reductions below \(r\ne p\); defining
characteristic is treated at the end of the section.  The
Brou\'e--Michel theorem states that this is a union of blocks
\cite{BroueMichel1989,CEBook}.

We use the following forms of standard reductive-group character
theory.  They also specify the hypotheses at every later application.
If \(Z(\mathbf J)\) and \(C_{\mathbf J^*}(s)\) are connected, Jordan
decomposition identifies \(\mathcal E(J,s)\) with the unipotent
characters of \(C_{\mathbf J^*}(s)^F\).  Its degree formula is
\begin{equation}\label{fam:jordan-degree}
 \chi(1)=
  |\mathbf J^{*F}:C_{\mathbf J^*}(s)^F|_{r'}\,\psi(1),
 \qquad
 \operatorname{def}_p(\chi)
  =\frac{|C_{\mathbf J^*}(s)^F|_p}{\psi(1)_p}.
\end{equation}
The torus-pairing rule identifies the pairings with corresponding
unipotent Deligne--Lusztig characters, multiplied by
\(\varepsilon_{\mathbf J}\varepsilon_{C_{\mathbf J^*}(s)}\), where
\(\varepsilon_{\mathbf H}=(-1)^{\rk_{\F_q}(\mathbf H)}\).
We use these two rules from \cite{LusztigCharacters}; see also
\cite[Equation~(3.1) and Theorem~7.1]{DigneMichel1990}.
A compatibility
statement for arbitrary nonuniform Levi induction is neither part of
these rules nor assumed here.  The particular induction matrices needed
below are established in Proposition~\ref{prop:single-cycle}.

Unipotent character degrees and unipotent induction calculations are
unchanged by central isogenies, with the standard identifications.
Connected isogenous reductive groups have the same rational order.
For a product of factors permuted by \(F\), the calculation on one
factor uses the composite endomorphism around its orbit.
A \emph{packet} is a Frobenius orbit of classical factors of a
connected centralizer, calculated on one factor with this composite
endomorphism.

For the ordinary classical Frobenius structures used below, we
make the unipotent identifications through connected kernels.
Choose an $F$-equivariant regular embedding
$\mathbf H\hookrightarrow\widetilde{\mathbf H}$ with connected
center and the same derived group.  Put
$H=\mathbf H^F$, $\widetilde H=\widetilde{\mathbf H}^F$, and
$A=\widetilde H/H$, an abelian group.  Twisting an upstairs
unipotent character $\widetilde\chi$ by a nontrivial linear
character of $A$ changes its dual parameter from $1$ to a
nontrivial central element.  Series disjointness and Clifford
theory therefore give
\[
 \langle\Res_H\widetilde\chi,\Res_H\widetilde\chi\rangle
 =\sum_{\nu\in\Irr(A)}
       \langle\widetilde\chi,\widetilde\chi\nu\rangle=1.
\]
Restriction is thus irreducible.  Every unipotent character of
$H$ occurs in some $R_T^{\mathbf H}(1)$; the torus restriction
formula lifts it to an upstairs unipotent character.  Two
upstairs unipotent characters with the same restriction differ
by a quotient twist, so series disjointness makes them equal.
This gives a bijection preserving degrees.  Next,
$\widetilde{\mathbf H}\to\mathbf H_{\mathrm{ad}}$ has connected
central kernel.  Lang's theorem makes it surjective on rational
points, and unipotent characters are trivial on that kernel.
These two morphisms compare the forms through their common
adjoint group; the relevant character and Levi diagrams are
compatible by \cite[Proposition~2.5(i), Theorem~9.1 and
Corollary~9.2]{DigneMichel1990}, with corresponding Levi preimages.
Both identifications are bijections, so individual split type-$D$
labels remain distinct.  Rationally permuted factors again use
the composite Frobenius on one factor.  This argument does not
apply the connected-kernel statement to a finite disconnected
central kernel; exceptional diagram isogenies remain in the
separate bounded-root-data case.

Restriction through a regular embedding, and restriction to a connected derived
subgroup, sends a Lusztig parameter to its dual projection.  Geometric
series suffice for these restriction statements; all later Jordan
matrix calculations are made in groups with connected centers and
connected semisimple centralizers.

For a good prime \(p\) and connected \(Z(\mathbf J)\), the integral
basic-set theorem \cite{GeckHiss} says that
\(\mathcal E(J,s)\) is an ordinary basic set for
\(\mathcal E_p(J,s)\).  Thus its restrictions to \(p\)-regular elements
form an integral basis of the Brauer character lattice of that block
union.  Its intersection with each block is nonempty and is a basic
set for that block.  We use this theorem for all primes in type~A,
and for odd primes in types~B, C, and D.  We do not apply it to type~B,
C, or D at \(p=2\).

\begin{lemma}[Central quotients in the Levi equivalence]
\label{fam:br-central}
Let \(s\in\mathbf J^{*F}\) be a semisimple \(p'\)-element, and let
\(\mathbf L^*\) be an \(F\)-stable Levi subgroup containing the full
algebraic centralizer \(C_{\mathbf J^*}(s)\).  The
Bonnaf\'e--Rouquier equivalence matches the blocks in
\(\mathcal E_p(J,s)\) with those in the corresponding summand of
\(L=\mathbf L^F\), preserving their Cartan matrices.
If \(Z\leq Z(\mathbf J)^F\) is contained in \(L\), this equivalence restricts
to the categories on which \(Z\) acts trivially.  In particular, it
gives the corresponding equivalences after taking a common central
quotient, including a quotient by a central \(p\)-subgroup.
\end{lemma}

\begin{proof}
The full-centralizer containment allows us to apply the integral
theorem \cite[Theorem~B$'$, Section~11.4]{BR}.
Its Deligne--Lusztig cohomology bimodule gives the asserted Morita
equivalence.  On the defining variety, left multiplication by a common
rational central element agrees with right multiplication by that
element.  Hence \(zm=mz\) on the equivalence bimodule, for every
\(z\in Z\).  The equivalence and its inverse consequently carry the
full subcategory annihilated by all \(z-1\) to the corresponding full
subcategory on the other side.  These are exactly the module
categories for the quotient algebras.  This argument uses equality of
the central actions; it does not use exactness of ordinary coinvariants
on arbitrary modules.  It applies over a splitting modular system and
over \(k\).  When \(Z\) has a \(p'\)-part, only blocks in its trivial
central-character sector survive this quotient.  The Cartan matrices
of surviving matched blocks agree under Morita equivalence, and
their largest elementary divisors give the equality of defect orders.
\end{proof}

\subsection{Linear and unitary groups}

Use the notation
\(\GL_n^+(q)=\GL_n(q)\), \(\GL_n^-(q)=\GU_n(q)\), and similarly
\(\SL_n^+(q)=\SL_n(q)\), \(\SL_n^-(q)=\SU_n(q)\).
A block in \(\mathcal E_p(J,1)\) will be called a unipotent block;
its ordinary characters need not all be unipotent.

\begin{proposition}\label{fam:type-a}
Fix \(p\) and \(M\).  To bound Cartan entries for blocks of defect
order at most \(M\) of all central quotients of
\(\SL_n^\epsilon(q)\), with \(r\ne p\), it suffices to bound Cartan
entries for unipotent blocks of \(\GL_m^\eta(Q)\) of bounded defect
order, where the new bound depends only on \(p,M\).
At \(p=2\), these latter blocks have bounded rank.
\end{proposition}

\begin{proof}
Let \(b\) be the original block and put
\[
 b_q=(q-\epsilon)_p,\qquad
 h=(\gcd(n,q-\epsilon))_p.
\]
We use \(b_q\) to distinguish this integer from the block \(b\).
Lift \(b\) to \(S=\SL_n^\epsilon(q)\), and cover the lifted block
in \(J=\GL_n^\epsilon(q)\).  A central \(p'\)-kernel merely selects
an averaging summand; a central \(p\)-kernel increases the defect
order by its order, at most \(h\).  Since \(J/S\) is cyclic of
order \(q-\epsilon\), every covering block \(B\) satisfies
\begin{equation}\label{fam:a-cover-defect}
 |D_B|\leq Mhb_q\leq Mb_q^2.
\end{equation}
We first allow \(b_q\) to be unbounded.

Let \(s\) be the \(p'\)-parameter of \(B\).  Its algebraic centralizer
is a rational Levi with fixed points
\begin{equation}\label{fam:a-centralizer}
 C_{\mathbf J^*}(s)^F
   \cong\prod_i\GL_{m_i}^{\epsilon^{d_i}}(q^{d_i}),
 \qquad n=\sum_i d_i m_i.
\end{equation}
Here \(d_i\) is the length of the relevant eigenvalue orbit.  The
basic-set theorem supplies a character of \(B\cap\mathcal E(J,s)\).
If its Jordan labels are partitions \(\lambda_i\vdash m_i\), the
partition degree formula and \eqref{fam:jordan-degree} give
\begin{equation}\label{fam:a-hook-defect}
 \operatorname{def}_p(\chi)
  =\prod_i\prod_{u\in\mathsf H(\lambda_i)}
       (q^{d_i u}-\epsilon^{d_i u})_p,
\end{equation}
where \(\mathsf H(\lambda_i)\) is the multiset of hook lengths, one
for each box.

If \(b_q>1\), lifting the exponent, applied to
\(x=\epsilon q\), yields
\begin{equation}\label{fam:a-lte}
 (x^j-1)_p\geq (x-1)_p(j)_p=b_q(j)_p.
\end{equation}
For odd \(p\) this is the usual equality.  For \(p=2\) and odd \(j\)
it is also an equality.  For even \(j\), the formula
\[
 v_2(x^j-1)=v_2(x-1)+v_2(x+1)+v_2(j)-1
\]
implies the inequality, including the case \(v_2(x-1)=1\).
Absolute values are understood when \(x<0\).
Consequently, if \(t=\sum_i m_i\), then
\begin{equation}\label{fam:a-multiplicity}
 b_q^t\prod_i(d_i)_p^{m_i}
 \leq\operatorname{def}_p(\chi)
 \leq Mhb_q\leq Mb_q^2.
\end{equation}
For \(b_q>2M\), this forces \(t\leq2\).  If some \(m_i=2\), it
is the only factor and \(n=2d_i\).  Hence
\(h\leq(2d_i)_p\leq2(d_i)_p\), so the same inequality gives
\[
 b_q^2(d_i)_p^2\leq2M(d_i)_p b_q,
 \qquad b_q(d_i)_p\leq2M,
\]
a contradiction.  Thus every \(m_i=1\): the centralizer is a torus.

Lemma~\ref{fam:br-central} now identifies \(B\) with a block of a
finite torus.  Such a block has one simple module, and all its
decomposition numbers are one.  The integral equivalence gives the
same decomposition matrix, up to labels, for \(B\).  This argument
applies to every covering block, so Corollary~\ref{fam:restriction}
bounds all decomposition numbers of the lifted block of \(S\) by
one.  Passing to the original central quotient only selects inflated
ordinary rows and the corresponding simple modules: the central
\(p\)-kernel acts trivially on every simple module.  The original
block still has defect order at most \(M\), so its number of ordinary
rows is bounded by the Brauer--Feit bound.  Its Cartan entries, which
are sums of products of entries in those rows, are bounded.  No
bound on the row count of \(B\), or on its defect in this case, has
been used.

We may therefore assume \(b_q\leq2M\).  Then
\eqref{fam:a-cover-defect} bounds \(|D_B|\) by \(4M^3\).
Apply Lemma~\ref{fam:br-central} to
\eqref{fam:a-centralizer}.  In this Levi the parameter \(s\) is
central.  Tensoring with its associated linear character identifies
its \(s\)-summand with its unipotent summand.  A block of the product
is a tensor product of blocks, and its defect order is the product
of their defect orders.  At most \(\log_p(4M^3)\) factors can have
positive defect.  Factors of defect zero have Cartan matrix \((1)\)
and have no effect on a Cartan bound.  This proves the stated
reduction, followed by cyclic restriction and central descent.

Finally suppose \(p=2\).  Here every defining parameter \(Q\) is
odd, and every hook factor \((Q^u-\eta^u)_2\) is at least two.
Every basic row of a unipotent block of \(\GL_m^\eta(Q)\) therefore
has degree defect at least \(2^m\).  Bounded block defect bounds
\(m\), and Theorem~\ref{thm:geometric-cartan} applies.
\end{proof}

\subsection{Regular classical root data}

The linear and unitary reduction has isolated unipotent blocks.
For the other classical families we must retain both a central
quotient and actual product decompositions of the Levis used
later.  The next construction supplies those groups before any
character or runner calculation is made.

The regular embedding must supply three kinds of structure.
The connectedness assertions are used in Jordan decomposition and,
at odd \(p\), in the basic-set theorem.  Integral Levi splittings
identify the actual product group algebras used in Harish--Chandra
induction.  Control of the residual central tori then keeps track of
their contributions to
degree defects, including after repeated extractions.
We begin with the ordinary irreducible root systems, in particular
\(D_n\) with \(n\geq4\); smaller ranks already fall under
Theorem~\ref{thm:geometric-cartan}.  Rank-one orthogonal tori
are nevertheless retained as residual factors and centralizer packets.

\begin{lemma}\label{fam:regular-datum}
For simply connected groups of types~B, C, and D there are regular
embeddings \(\mathbf S=[\mathbf J,\mathbf J]\leq\mathbf J\) with
central torus rank at most three with the following properties.
\begin{enumerate}
\item The groups \(\mathbf J\), its dual, and all their Levi
subgroups have connected centers and simply connected derived groups.
\item Every extraction of linear blocks on split hyperbolic
coordinates gives an actual direct product of these general linear
groups with a residual reductive group.  This holds for nested
extractions and in graph-twisted type~D.
\item Every residual factor \(\mathbf R\) has an inherited central torus
\(\mathbf T_R\), the image of the original connected center under
the Levi product projection, of rank at most three.  Its full
connected center has rank at most four.  The two tori agree except
for the rootless orthogonal datum \(D_1\), whose additional torus
direction is retained as an \(\mathrm{SO}_2^\pm\) packet.
With bounded residual semisimple rank, only finitely many residual
root data and normalized Frobenius actions occur.
\end{enumerate}
\end{lemma}

\begin{proof}
We first construct the ambient lattice and its Frobenius action.
We then split off the extracted general linear factors integrally.
The remaining calculations identify the inherited central torus
and show that bounded residual rank gives only finitely many
residual root data.

Let \(V=\Q^n\) with orthogonal coordinate vectors \(e_i\).  Denote
the coroot lattice and coweight lattice of the chosen classical root
system by \(Q^\vee\subset P^\vee\).  Explicitly,
\[
\begin{array}{c|c|c}
 &Q^\vee&P^\vee\\ \hline
 B_n&\{a\in\Z^n:\sum a_i\text{ is even}\}&\Z^n\\
 C_n&\Z^n&\Z^n\cup((\tfrac12,\ldots,\tfrac12)+\Z^n)\\
 D_n&\{a\in\Z^n:\sum a_i\text{ is even}\}
     &\Z^n\cup((\tfrac12,\ldots,\tfrac12)+\Z^n).
\end{array}
\]
Introduce a central space \(\Q^h\), with \(h=1,1,3\), respectively,
and define an injective homomorphism
\(\gamma:P^\vee/Q^\vee\to\Q^h/\Z^h\) by the following table.
Put \(\omega=(\tfrac12,\ldots,\tfrac12)\).
\begin{equation}\label{fam:lattice-glue}
\begin{array}{c|c|c}
 &\gamma(e_i)&\gamma(\omega)\\ \hline
 B_n&\tfrac12&\text{not needed}\\
 C_n&0&\tfrac12\\
 D_n,\ n\text{ even}&(\tfrac12,\tfrac12,0)
                         &(\tfrac12,0,\tfrac12)\\
 D_n,\ n\text{ odd}&(\tfrac12,\tfrac12,0)
                         &(\tfrac14,-\tfrac14,\tfrac12).
\end{array}
\end{equation}
All central entries are read modulo \(\Z^h\).  In odd rank~D,
\(2\gamma(\omega)=\gamma(e_i)\); in even rank the two displayed
classes of order two are independent.  Thus these prescriptions
respect exactly the relations in \(P^\vee/Q^\vee\).
Set
\begin{equation}\label{fam:Y}
 Y=\{(a,z)\in P^\vee\oplus\Q^h:
                   z+\Z^h=\gamma(a+Q^\vee)\},
 \qquad X=\Hom(Y,\Z).
\end{equation}
Take the usual roots on \(a\), extended by zero on \(z\), and the
usual coroots with zero central coordinate.  They define a root datum
on \((X,Y)\).  Indeed roots take integral values on \(P^\vee\),
coroots belong to \(Y\), and the classical reflection formulas
preserve the congruence because they change \(a\) by a coroot.

Injectivity of \(\gamma\) gives
\(Y\cap(V\oplus0)=Q^\vee\).  Since \(Y\to P^\vee\) is onto,
\(X\cap V^*=Q\), the root lattice.  These are the two saturation
conditions for simply connected derived group and connected center;
they remain true on dualizing.  For a Levi subsystem, its simple
roots form a subset of a simple-root basis, and likewise for
coroots.  Intersecting with their rational spans preserves these
saturation conditions.  This proves the assertions about every
Levi and its dual.  In particular their semisimple centralizers are
connected, by the connected-centralizer theorem for simply connected
derived groups.

In type~D the nontrivial diagram automorphism changes the sign of
the last coordinate of \(a\).  Extend it on \(z\) by exchanging
\(z_1,z_2\).  For even \(n\), it sends
\(\gamma(\omega)\) to \(\gamma(\omega)-\gamma(e_n)\);
the same calculation holds for odd \(n\).  It fixes
\(\gamma(e_i)\).  Thus it preserves \(Y\), and defines the
graph-twisted Frobenius on this datum.  The action on the central
space, after the scalar \(q\) is removed, has order at most two.

\smallskip\noindent\emph{Integral Levi products.}
For the splitting assertion, orient the extracted hyperbolic
coordinates by signs \(\sigma_i\in\{1,-1\}\).  Let \(c\) be the
entry \(\gamma(e_i)\) in the table, with the displayed representative,
and set
\[
 f_i=(\sigma_i e_i,c),\qquad
 x_i(a,z)=\sigma_i a_i+\ell(z),\qquad
 \ell(z)=
 \begin{cases}
  0&\text{in type B},\\
  z_1&\text{in type C},\\
  z_3&\text{in type D}.
 \end{cases}
\]
Then \(f_i\in Y\), and \(x_i\in X\): the possible half-integral
part of \(a_i\) is canceled by the indicated central coordinate.
Moreover \(\ell(c)=0\), so
\begin{equation}\label{fam:split-lattice}
 x_i(f_j)=\delta_{ij},\qquad
 Y=\bigoplus_{i\in I}\Z f_i\ \oplus\
               \bigcap_{i\in I}\ker x_i.
\end{equation}
Within an extracted block, its roots are \(x_i-x_j\) and its
coroots are \(f_i-f_j\).  They are exactly the root datum of the
corresponding general linear group.  The residual roots and
coroots lie on the complementary summand.  Thus
\eqref{fam:split-lattice} is a product decomposition of root data,
not merely of rational root spaces.

For a rational split hyperbolic extraction, conjugate its split
cocharacters into standard position on a maximal split torus.
The construction is then \(F\)-compatible.  In nonsplit type~D,
the extracted directions are in split hyperbolic planes; the
remaining diagram involution acts on the residual coordinates and
on \(z\) as above.  The same construction applies after each
extraction, proving the nested assertion.

\smallskip\noindent\emph{Residual center and root data.}
The general linear factors have now been split off.  We retain
the image of the original center separately from any extra torus
direction in the residual group.
To identify this inherited center, let \(m=|I|\).  The projection
of an original central vector \((0,z)\) to the residual summand is
\((0,z)-\sum_{i\in I}x_i(0,z)f_i\).  Its remaining
\(a\)-coordinates are zero, its discarded coordinates are
\(a_i=-\sigma_i\ell(z)\), and its central coordinate is
\[
 z'=(\id-mc\ell)z.
\]
Since \(\ell(c)=0\), the central map has inverse \(\id+mc\ell\)
and determinant one.  It is \(F\)-equivariant, so its image
\(\mathbf T_R\) is a torus of rank \(h\) with the same rational
Frobenius representation as the original center.  The central
lattices are commensurable with the coordinate lattice only at
the prime two, by the graph congruences.  Together with the
determinant-one map this shows that the isogeny onto
\(\mathbf T_R\) has kernel of 2-power order.  For odd \(p\)
it therefore identifies the Sylow \(p\)-subgroups on rational
points.  The original center may also project to the general
linear centers; it need not lie solely in the residual factor.

If the residual root system spans the remaining coordinate
space, then \(\mathbf T_R=Z^\circ(\mathbf R)\).  In the
rootless datum \(D_1\), the remaining coordinate instead adds
one central torus dimension with Frobenius eigenvalue \(q\)
or \(-q\).  Thus the full residual central rank is at most
\(h+1\leq4\).  This extra torus is the rank-one orthogonal
packet, separate from the inherited central contribution.

Finally, on the residual summand the equations \(x_i=0\) express
every discarded coordinate as \(a_i=-\sigma_i\ell(z)\).
Substitution into \eqref{fam:Y} leaves congruences with uniformly
bounded denominators, dividing four.  Independently of how many
coordinates were discarded, the resulting lattice lies in a
fixed small-denominator coordinate lattice and contains a fixed
power-of-two multiple of the integral coordinate lattice.  In
bounded remaining dimension there are only finitely many intermediate
lattices.  The residual signed-coordinate and diagram actions are
also drawn from a finite set in that dimension.  This proves the
last assertion, including the rank-one orthogonal residual tori.
\end{proof}

\subsection{Reduction to the two sign eigenspaces}

For an original regular ambient group of
Lemma~\ref{fam:regular-datum}, before any split extraction, put
\[
 T_0=Z^\circ(\mathbf J)^F,\qquad Z_p=O_p(T_0).
\]
Here \(O_p(T_0)\) denotes its unique Sylow \(p\)-subgroup.  A block
of \(J/Z_p\) with defect order at most \(M\) lifts to a block of
\(J\) with defect order at most \(M|Z_p|\).  We call this a relative
defect bound.  It will always be accompanied by the requirement
that the final Cartan bound is for the block of \(J/Z_p\).
For odd \(p\), the basic rows at a semisimple \(p'\)-parameter
\(s\) descend to this quotient.  Indeed each such row occurs in
a Deligne--Lusztig character \(R_T^{\mathbf J}(\theta)\) for a torus
character \(\theta\) dual to \(s\).  Torus duality gives
\(|\theta|=|s|\), and \(Z(\mathbf J)^F\) acts on every constituent
by the restriction of \(\theta\) to \(Z(\mathbf J)^F\)
\cite[Corollary~1.22 and Section~5.21]{DL}.
Their central characters therefore have \(p'\)-order and are trivial
on \(Z_p\).  Simple modules also factor through \(J/Z_p\), by
Lemma~\ref{lem:central-transfer}.  Thus the descended rows form the
same integral basic set on the corresponding quotient block union,
and their degree defects there are
\(\operatorname{def}_p(\chi)/|Z_p|\).

For a later residual factor, the central contribution in its
packet degree formula will be \(\mathbf T_R\), rather than its
possibly larger full connected center.

We also fix the spectral convention in the following reduction.
Project a semisimple \(p'\)-parameter \(s\) to the adjoint dual
group and represent it in natural symplectic or orthogonal
eigenvalue coordinates.  The passage to a natural isometry group
requires at most a central 2-isogeny.  We call this eigenvalue
multiset the \emph{normalized natural spectrum}; its ambiguity,
and the ambiguity in its rational Frobenius action, is at most a
common sign.  In particular, containment in \(\{1,-1\}\) is
independent of this choice.

\begin{proposition}\label{fam:central-classical}
To prove Theorem~\ref{thm:quasisimple-cartan} for the
cross-characteristic families of types~B, C, and D, it suffices to
prove the following assertion, together with the type~A bounds:
for the groups of Lemma~\ref{fam:regular-datum}, blocks of
\(J/Z_p\) of bounded defect have bounded Cartan entries whenever
their \(p'\)-parameter has normalized natural spectrum contained
in \(\{1,-1\}\).
\end{proposition}

\begin{proof}
The centers on points of the simply connected covers of types~B, C, and~D have
order at most four.  By Lemma~\ref{lem:central-transfer}, it
suffices to treat these covers and their quotients by central
\(p\)-subgroups, with defect bounds changed by a factor at most four.
Embed \(S=\mathbf S^F\) into \(J\) as in
Lemma~\ref{fam:regular-datum}, and put \(K=S\cap Z_p\).
The central isogeny
\(\mathbf S\times Z^\circ(\mathbf J)\to\mathbf J\), followed
by Lang's theorem, gives
\[
 [J:S T_0]\leq4,\qquad
 [J/Z_p:S/K]_p\leq4.
\]
Thus every block of \(J/Z_p\) covering a given block of \(S/K\)
has bounded defect.  A simple module of \(J/Z_p\) restricts to
\(S/K\) with bounded multiplicity: the central group \(T_0/Z_p\)
acts by scalars on an inertia type, and the remaining inertia
quotient has order at most four.  Lemma~\ref{lem:restriction-rows}
therefore transfers a Cartan bound for these covering blocks
down to \(S/K\).  Central transfer then recovers \(S\) and its
required central quotients.

Consider one such covering block, with \(p'\)-parameter \(s\).
Eigenvalues different from \(1,-1\) occur in
reciprocal pairs and give linear root subsystems in the connected
centralizer.

There is an \(F\)-stable Levi \(\mathbf M^*\) containing this full
centralizer and retaining the full ambient classical subsystem on
the \(\{1,-1\}\)-coordinates.  To construct it, take independent
torus directions with a scalar on each non-sign eigenvalue space,
its inverse on the reciprocal space, and zero direction on the
remaining coordinates, and centralize this torus.  The span of
these directions is stable under Frobenius, which permutes the
pairs and may invert them or multiply all normalized eigenvalues
by a common sign.  The full centralizer is contained because its
roots do not join different reciprocal packets.  The centralizer
is connected by Lemma~\ref{fam:regular-datum}.

Apply Lemma~\ref{fam:br-central}, also modulo \(Z_p\), and let
\(\mathbf M\) be the corresponding primal Levi.  Its connected
derived subgroup has simply connected factors.  The normal subgroup
\[
 N=[\mathbf M,\mathbf M]^F/
       ([\mathbf M,\mathbf M]^F\cap Z_p)
       \ \lhd\ M/Z_p
\]
has abelian quotient.  The intersection removed here lies in
\(S\), so has order at most four.  Restrict a block of \(M/Z_p\)
to \(N\), and lift across that bounded central intersection.
The covered blocks on \([\mathbf M,\mathbf M]^F\) have bounded
defect.  This group is a direct product of simply connected
linear or unitary groups and possibly one residual group of type~B, C, or~D.
Only boundedly many of its block factors can have positive defect.

On the residual factor, dual projection keeps precisely the roots
on the sign coordinates.  Projection commutes with taking the
prime-to-\(p\) part of a semisimple element.  The residual block
therefore has a geometric \(p'\)-parameter with normalized spectrum
in \(\{1,-1\}\).  Embed this factor regularly again as above;
all lifted parameters retain this adjoint-spectrum property.
The stated special assertion bounds its blocks.  The type~A
bounds handle the other positive-defect factors.  Tensor products,
central transfer, and Lemma~\ref{lem:abelian-transfer} then bound
the blocks of \(M/Z_p\), and hence the original block.
If no non-sign eigenvalue is present, no Levi reduction is made.
Otherwise the residual classical rank is strictly smaller.  In
either case this argument invokes only the stated special
assertion, not the general quasisimple conclusion.
\end{proof}

\subsection{The prime two}

\begin{proposition}\label{fam:two}
At \(p=2\), the blocks in the special assertion of
Proposition~\ref{fam:central-classical} have bounded ambient
semisimple rank.  They consequently have bounded Cartan entries.
\end{proposition}

\begin{proof}
Here \(q\) is odd.  The adjoint projection of the odd-order element
\(s\) has a natural lift with spectrum in \(\{1,-1\}\).  Choose
the lift of odd order.  Its spectrum is then \(\{1\}\), so \(s\)
is central.  Tensoring by the corresponding central linear
character removes \(s\) from the calculation.

Let \(B\) be the lifted block of \(J\), whose defect order is at
most \(M|Z_2|\), and choose any \(\chi\in\Irr(B)\).  Its
semisimple parameter is \(st\), where \(t\) is a 2-element.
Put \(\mathbf C=C_{\mathbf J^*}(t)\), which is connected.
The degree formula identifies \(\operatorname{def}_2(\chi)\)
with the degree defect of its unipotent correspondent \(\psi\)
in \(\mathbf C^F\).  Unipotent characters are trivial on the
center: they occur in Deligne--Lusztig characters induced from
trivial torus characters, on which that center acts trivially.
In particular the central subgroup
\[
 A=\langle t,O_2(Z^\circ(\mathbf J^*)^F)\rangle
\]
lies in the kernel of \(\psi\), and hence
\[
 |A|\leq\operatorname{def}_2(\chi)\leq M|Z_2|.
\]
The two ambient central tori have the same rational Frobenius
eigenvalues under duality and isogeny, so
\(|O_2(Z^\circ(\mathbf J^*)^F)|=|Z_2|\).
It follows that the order of \(t\) modulo the ambient central
torus is at most \(M\).

Its adjoint image therefore has bounded order.  A lift to the
natural isometry group has order at most twice this bound.  Its
normalized eigenvalues belong to a set of roots of unity of
bounded order.  Both the number of distinct eigenvalues and
the lengths of their Frobenius orbits are consequently bounded;
the possible common-sign ambiguity only enlarges these bounds
by a factor of two.

Up to central isogeny, the rational root datum of \(\mathbf C\)
is the product of its ambient central torus and its eigenvalue
packets.  A reciprocal non-sign packet gives a linear or unitary
factor, and a sign packet gives a factor of type~B, C, or~D.  Factors permuted
by Frobenius use their composite field parameter.  Rank-one
orthogonal tori \(\mathrm{SO}_2^\pm\) are included as rank-one
type~D packets.  Thus rational orders and unipotent degrees give
\begin{equation}\label{fam:two-product}
 \frac{\operatorname{def}_2(\chi)}{|Z_2|}
   =\prod_i\frac{|H_i|_2}{\psi_i(1)_2}\leq M,
\end{equation}
where \(\psi_i\) is unipotent and \(H_i\) is the corresponding
packet group.  This equality is an order and degree calculation;
no direct-product assertion about the finite centralizer is needed.

A linear or unitary packet of dimension \(n\) contributes at least
\(2^n\), by the hook formula.  For completeness, the symbol formula
gives a similarly explicit estimate for every packet of type~B, C, or~D
and positive rank \(n\).  Write its two strictly increasing beta rows
as \(X,Y\), let \(u=|X|+|Y|\), and let \(c=|X\cap Y|\).
The rank formula is
\[
 n=\sum_{a\in X}a+\sum_{a\in Y}a
                  -\left\lfloor\frac{(u-1)^2}{4}\right\rfloor.
\]
Count all hooks and cohooks of positive length: these are pairs
of a bead and a strictly lower vacant position in its own row
or the opposite row.  Before vacancies are imposed there are
\(2\sum a\) possible pairs.  The number ending at occupied
positions is \(\binom u2-c\).  Therefore their number is
\begin{align*}
 H&=2\left(n+\left\lfloor\frac{(u-1)^2}{4}\right\rfloor\right)
                 -\binom u2+c\\
  &=2n-\lfloor u/2\rfloor+c.
\end{align*}
The unipotent symbol degree formula \cite[Section~3.3]{LMT}
expresses the 2-defect as the product of
\((q_i^j-1)_2\) over hooks and \((q_i^j+1)_2\) over cohooks,
times a power of two whose exponent is at least
\(\lfloor(u-1)/2\rfloor-c\).
For unequal rows this exponent is exactly the displayed one;
for equal rows in degenerate type~D the exponent is zero, which
is larger.  Every positive-length hook or cohook contributes
at least two, whence
\[
 v_2\!\left(\frac{|H_i|}{\psi_i(1)}\right)
 \geq 2n-\lfloor u/2\rfloor+\lfloor(u-1)/2\rfloor
 \geq 2n-1.
\]
This argument allows a negative displayed prefactor exponent;
it is the total exponent that is estimated.  The degenerate
case gives at least \(2n\).  For rank-one orthogonal tori the
defect is directly \((q_i-1)_2\) or \((q_i+1)_2\), and is at
least two.  Empty packets contribute one.

Equation~\eqref{fam:two-product} now bounds the effective rank
of every packet and the number of nonempty packets: each
positive-rank packet contributes at least \(2^{n_i}\), so
\(\sum_i n_i\leq\log_2 M\).  Their
Frobenius orbit lengths were already bounded, so it also bounds
the actual ambient semisimple rank.  The central rank is at
most three.  Lemma~\ref{fam:regular-datum} gives finitely many
root data, and Theorem~\ref{thm:geometric-cartan}, applied after
the central quotient, proves the result.
\end{proof}

\subsection{Odd-prime packets and the classical reduction}

The prime-two case is now reduced to fixed root data.  At an odd
prime the rank can remain unbounded; we instead express the
degree defect as a product of contributions from at most two
classical packets and a controlled central torus.  These are the
inputs for the label, runner and endpoint arguments that follow.

The next lemma describes the sign packets and their split Levi
factors, on which the subsequent label, runner, and endpoint
arguments operate.

\begin{lemma}\label{fam:packets}
Suppose that \(p\) is odd and the normalized spectrum of the
\(p'\)-parameter \(s\) is contained in \(\{1,-1\}\), in a
regular group of Lemma~\ref{fam:regular-datum}.
Apart from the ambient central torus, its connected dual
centralizer has at most two classical factors.  In natural dual
type~C they are of types C and C; in natural dual type~B they are
of types B and D; in natural dual type~D they are of types D
and D.  Empty factors are omitted and rank-one type~D is toral.
The factors have parameters \(q_i=q\), unless two like factors
are exchanged by Frobenius, in which case there is one packet
with parameter \(q_i=q^2\).

If split hyperbolic blocks of sizes \(b_j\) are extracted in
one packet with parameter \(q^\delta\), \(\delta\in\{1,2\}\),
the corresponding primal split Levi has actual factors
\(\GL_{\delta b_j}(q)\).  On each such factor the parameter
has a single eigenvalue orbit of degree \(\delta\) and
multiplicity \(b_j\).  The residual parameter retains the
same sign-packet description.  These assertions persist under
nested extractions with earlier factors held fixed.
\end{lemma}

\begin{proof}
Inspect the classical roots on the natural coordinate spaces.
In symplectic type, the roots evaluating trivially on \(s\)
are the type~C roots within its two sign eigenspaces.  In odd
orthogonal type, the positive eigenspace contains the distinguished
odd coordinate, giving a B factor, and the negative eigenspace
gives a D factor.  In even orthogonal type both eigenspaces
have even dimension, giving two D factors.  Their form signs
determine their split or graph-twisted rational structures and
must be retained.  In defining characteristic two the two signs
coincide, leaving a single packet.  Frobenius can exchange only
like factors; hence its orbit lengths here are at most two.

On a chosen split hyperbolic block, take a cocharacter that is
constant on its polarized half and on the Frobenius translates
of that half, has the opposite value on the reciprocal halves,
and is zero elsewhere.  Independent choices for the extracted
blocks define a split torus.  Its centralizer in the ambient
dual group is a split Levi, and its intersection with
\(C_{\mathbf J^*}(s)\) is exactly the desired split Levi of
the packet.  A packet coordinate has \(\delta\) ambient
coordinates in its Frobenius orbit, so the ambient linear
block has size \(\delta b_j\).

By Lemma~\ref{fam:regular-datum}, dualizing this construction
gives an actual product with \(\GL_{\delta b_j}(q)\) on the
primal side.  The centralizer roots on such a block form one
linear system of rank \(b_j-1\) on each of its \(\delta\)
Frobenius translates.  Equivalently its eigenvalues have one
orbit of degree \(\delta\) and multiplicity \(b_j\).
On the unextracted coordinates the original root evaluations
are unchanged, proving the residual assertion.  The independent
cocharacters can be chosen with all earlier extracted coordinates
fixed, which proves the assertion for a chain of extractions.
\end{proof}

The order and degree calculation used at \(p=2\) also gives,
for a basic row \(\chi\in\mathcal E(J,s)\),
\begin{equation}\label{fam:packet-defect}
 \frac{\operatorname{def}_p(\chi)}{|Z_p|}
       =\prod_i\frac{|H_i|_p}{\psi_i(1)_p}.
\end{equation}
Here the groups \(H_i\) are the packets of
Lemma~\ref{fam:packets}; their orthogonal signs are part of the
data.  This identity follows by decomposing the rational
reflection space into the packet spaces and the ambient central
space, then using central-isogeny invariance of orders and
unipotent degrees.  The partition and symbol formulas for its
right-hand side are made explicit in
Proposition~\ref{prop:label-degrees}.
For a residual factor \(\mathbf R\), the corresponding identity is
\[
 \operatorname{def}_p(\chi_R)
  =|\mathbf T_R^F|_p
        \prod_i\frac{|H_i|_p}{\psi_i(1)_p}.
\]
The product includes its toral \(D_1\) packet if present;
using the inherited torus ensures that this packet is counted once.

\begin{lemma}\label{fam:central-order}
For odd \(p\), the original central subgroup and every inherited
residual central torus satisfy
\[
 |Z_p|,\ |\mathbf T_R^F|_p
       \leq\bigl((q-1)_p(q+1)_p\bigr)^3.
\]
The full residual connected center satisfies the analogous bound
with exponent four.  Let \(q_i=q^\delta\), \(\delta\in\{1,2\}\),
be an original packet parameter,
\(e_i=\ord_p(q_i)\), and put
\[
 (d_i,\eta_i)=
 \begin{cases}
  (e_i,1)&e_i\text{ odd},\\
  (e_i/2,-1)&e_i\text{ even}.
 \end{cases}
 \qquad
 \alpha_i=(q_i^{d_i}-\eta_i)_p.
\]
Then \(|Z_p|\) and \(|\mathbf T_R^F|_p\) are at most
\(\alpha_i^3\), while \(|Z^\circ(\mathbf R)^F|_p\) is at most
\(\alpha_i^4\).  Thus a bound on \(\alpha_i\) bounds both
inherited and full residual central \(p\)-orders.
\end{lemma}

\begin{proof}
On the inherited central rational space Frobenius is \(q\theta\),
where \(\theta^2=1\), and the dimension is at most three.
The full residual center adds at most one coordinate with
eigenvalue \(q\) or \(-q\).  Its rational order is therefore
\((q-1)^a(q+1)^b\), with \(a+b\leq3\) for the inherited
torus and \(a+b\leq4\) for the full center.  These orders
are unchanged by isogeny of their integral lattices.
If either \(p\)-part is nontrivial, then \(p\mid q-1\) or
\(p\mid q+1\), and for odd \(p\) at most one occurs.
For \(\delta=1\), the relevant \(\alpha_i\) is exactly the
nontrivial one of these two \(p\)-parts.  For \(\delta=2\),
we have \(e_i=1\) and
\(\alpha_i=(q^2-1)_p=(q-1)_p(q+1)_p\).
If both \(p\)-parts are trivial the estimates are immediate;
otherwise the preceding calculation proves them.
\end{proof}

We now identify precisely how the later results finish this
section.  For a block of relative defect at most \(M\),
\eqref{fam:packet-defect} and
Proposition~\ref{prop:label-degrees} bound the total relevant
hook or cohook weight.  A positive weight bounds the corresponding
cyclotomic \(p\)-part \(\alpha_i\), so
Lemma~\ref{fam:central-order} bounds the ambient central
\(p\)-part whenever it is needed.  At weight zero, the core
arguments handle the central quotient directly.  The parameter
on each Levi is the specified rational parameter, rather than
a sum over its several possible preimages from the ambient
group.  Propositions~\ref{prop:single-cycle} and
\ref{prop:core-blocks} establish, respectively, the required
ordinary matrices on these labels and the fact that the
projectors used are projectors to unions of blocks.

Proposition~\ref{prop:runner-reduction} then transfers Cartan
bounds from two kinds of endpoints.  The first has bounded
active rank.  Its extracted factors are of order prime to
\(p\), split off by Lemma~\ref{fam:regular-datum}, and contribute
only defect-zero blocks.  Its residual root data form a finite
set by that lemma, so Theorem~\ref{thm:geometric-cartan} applies.
The second endpoint has bounded ordinary Harish--Chandra rank
above a possibly arbitrarily large cuspidal triangle.
Proposition~\ref{prop:triangle-cartan} gives the required bound
there.  The same two endpoint arguments cover unipotent
\(\GL\) and \(\GU\) blocks in
Proposition~\ref{fam:type-a}.  Thus these later propositions
prove both of the assertions left open by
Propositions~\ref{fam:type-a} and~\ref{fam:central-classical}
for every odd \(p\).

\subsection{Completion over all quasisimple families}

\begin{proof}[Proof of Theorem~\ref{thm:quasisimple-cartan}]
We use the classification of finite simple groups and the standard
Schur multiplier descriptions \cite{Wilson,MalleTesterman}.
Sporadic groups and the finitely
many exceptional multiplier stems form a bounded collection of
finite groups; all their blocks may be included in the bound.

For alternating groups, Scopes' theorem \cite{Scopes} gives
finitely many Morita classes of symmetric-group blocks of each
fixed weight.  Bounded defect order bounds this weight, since
the defect group is a Sylow \(p\)-subgroup of the symmetric
group on \(pw\) letters for a block of weight \(w\).
For the double covers, Kessar's finiteness theorem
\cite{KessarSymmetric} supplies the same bounded-defect
conclusion for spin blocks at odd primes; the nonspin blocks
factor through the symmetric group.  At \(p=2\), quotient by
the central involution and use central transfer.  Every block
of a double cover of an alternating group is covered in the
corresponding double cover of the symmetric group.  Its
covering defect increases by at most a factor two, and
restriction across this cyclic quotient has multiplicity one.
Lemma~\ref{lem:restriction-rows} therefore bounds its decomposition
numbers and Cartan entries.  The same argument without the
central cover treats the alternating group itself.

Consider groups of Lie type in defining characteristic
\(p=r\).  For their standard simply connected covers, the
defining-characteristic block theorem \cite{HumphreysDefining}
says that every positive-defect block has full defect.  The
standard central kernels on rational points have order prime
to \(r\) \cite[Proposition~3.8]{BMO}.
Consequently a positive-defect block on a central
quotient has defect equal in order to a Sylow \(r\)-subgroup
of the cover.  Bounded Sylow \(r\)-order bounds both its
defining field parameter and its rank, and hence its group
order.  The bounded list of small exceptions is already harmless.
\mbox{Defect-zero blocks have Cartan matrix \((1)\).}

For \(r\ne p\), every bounded-rank Lie-type family, including
all exceptional types, Suzuki and Ree groups, and triality,
is covered by Theorem~\ref{thm:geometric-cartan}.  Its hypotheses
allow the finitely many root data, diagram actions, and
exceptional diagram isogenies that occur, as well as their
central quotients.  Thus only the ordinary unbounded-rank
classical families remain.

For these, Proposition~\ref{fam:type-a} gives the type~A
reduction and handles the large central lift without imposing
a bounded defect upstairs.  The bounded-lift case at \(p=2\)
has bounded rank.  Proposition~\ref{fam:central-classical}
reduces types~B, C, and D to the regular sign-spectrum case,
which Proposition~\ref{fam:two} handles at \(p=2\).
For odd \(p\), Propositions~\ref{prop:runner-reduction} and
\ref{prop:triangle-cartan}, combined with the packet estimates above,
supply the remaining bounds for general linear, unitary, and regular
groups of types~B, C, and~D.  The restriction,
central-quotient, product, and abelian-extension arguments
already proved then return these bounds to every quasisimple
group in the corresponding families.  All changes in the
defect bound depend only on \(p,M\), completing the proof.
\end{proof}
==== END Donovans-Conjecture-over-Algebraic-Closures-of-Prime-Fields//03_families.tex ====

==== BEGIN Donovans-Conjecture-over-Algebraic-Closures-of-Prime-Fields//04_labels.tex ====
\section{Ordinary labels, single cycles, and block cores}\label{sec:labels}

Throughout this section $p$ is odd and different from the defining
characteristic.  We work with the remaining groups of
Section~\ref{sec:families}: unipotent series of general linear or unitary
groups, and the regular classical groups of
Lemma~\ref{fam:regular-datum}, with the sign packets of
Lemma~\ref{fam:packets}.  The letter $\mathbf H$ also allows their
successive split Levis.  Previously extracted linear factors can be
retained as fixed factors; their ordinary parameters and labels are
fixed throughout an extraction.  In the runner applications those
factors have order prime to $p$.

We first specify the ordinary character theory being used.  We then
prove the compatibility with ordinary Jordan labels that is needed
below.  Finally we show that the core projections in the runner
argument are projections by block idempotents.  All three statements
concern a fixed rational semisimple parameter in each Levi: we do not
sum over different Levi classes that fuse in the ambient group.

For a finite group $H$ and $\chi\in\Irr(H)$ put
\[
 \operatorname{def}_p(\chi)=\frac{|H|_p}{\chi(1)_p}.
\]
This is the degree defect, expressed as an order.  If a central
$p$-subgroup $Z$ is in the kernel of $\chi$, its degree defect as a
character of $H/Z$ is $\operatorname{def}_p(\chi)/|Z|$.

\begin{proposition}[Labels and ordinary branching]\label{prop:label-degrees}
The following conventions and formulas apply to the unipotent factors
of the ordinary Jordan labels in the groups just specified.
\begin{enumerate}
\item For $\GL_n^\epsilon(q)$, where $\epsilon=1$ or $-1$,
unipotent characters are indexed by partitions $\nu$ of $n$, and
\[
 \operatorname{def}_p(\chi_\nu)
   =\prod_{h\in\operatorname{Hook}(\nu)}
       \bigl((\epsilon q)^h-1\bigr)_p.
\]
Here the $p$-part of a negative integer means the $p$-part of its
absolute value.
\item In types $B,C,D$ a symbol is a pair $(A,B)$ of strictly
increasing finite sets of nonnegative integers.  Simultaneously
replacing both rows $A,B$ by $\{0\}\cup(A+1),\{0\}\cup(B+1)$
does not change the symbol.  The rows are unordered, subject to the
usual two separate labels for equal rows in split type $D$ of positive
rank.  If $u=|A|+|B|$, the rank is
\[
 \sum_{a\in A}a+\sum_{b\in B}b
       -\left\lfloor\frac{(u-1)^2}{4}\right\rfloor.
\]
The difference $|A|-|B|$ is odd in $B,C$, is $0$ modulo $4$ in
split $D$, and is $2$ modulo $4$ in nonsplit $D$.  We use one
empty-group label in rank zero, and the toral convention for
$\mathrm{SO}_2^\pm$.

A hook is a bead $a$ and a smaller gap $b$ in the same row; a cohook
uses a gap in the other row.  Their lengths are $a-b>0$.
For a packet with field parameter $q_i$, its degree defect at the
odd prime $p$ is the product of
$(q_i^h-1)_p$ over hooks and $(q_i^c+1)_p$ over cohooks.
The inherited central torus contribution must also be included.
For an original regular group this is its full connected center;
for a residual factor $\mathbf R$, it is the torus $\mathbf T_R$
of Lemma~\ref{fam:regular-datum}.  The product over packets,
including a toral $D_1$ packet, multiplied by $|\mathbf T_R^F|_p$,
is the degree defect of the ordinary Jordan correspondent.
Thus the toral packet is counted exactly once.
\item Lusztig induction from a single positive cycle torus of odd length
$d$ and a residual classical group adds $d$-hooks to symbols;
a negative cycle adds $d$-cohooks.  The coefficients have the usual
leg signs, including the common extra minus sign for a cohook
in type $D$.  Each removal contributes absolute value one before the
type-$D$ identifications.  The individual split labels are retained.
Removing all cycles of the relevant length packs the runners and
gives a core.  Every unipotent label with that core occurs with
nonzero coefficient in induction from the core and its cycle tori,
with the appropriate residual orthogonal sign and, when necessary,
the appropriate choice of split label.
\item Ordinary split Harish--Chandra restriction removes $1$-hooks
for $\GL$, $2$-hooks for $\GU$ with a
$\GL_1(q^2)$ extraction, and same-row $1$-hooks for symbols.
On an ordinary Harish--Chandra series this is Young branching on
partitions or bipartitions.  In the unitary case the $2$-core stays
fixed; for symbols the row-length difference stays fixed up to
exchange.  The relative Weyl group is $W(B_b)$, except for the
zero-difference series in type $D$, where it is $W(D_b)$ and the
branching keeps the two labels attached to equal bipartitions.

At the first sign wall over a cuspidal symbol, the ratio of the two
ordinary rank-one degrees is a power of $q_i$, up to inversion.
Over the unitary cuspidal staircase of size $t(t+1)/2$ the ratio is
$q^{2t+1}$, up to inversion.
\end{enumerate}
\end{proposition}

\begin{proof}
These are the ordinary partition and symbol parametrizations, degree
formulas, and Harish--Chandra theory of
Lusztig~\cite{LusztigCharacters}.  In the degree formulas the factors
that have been omitted are powers of the defining characteristic,
signs, and, for symbols, powers of $2$; none changes the asserted
odd $p$-part.  The degree rule for Jordan decomposition cancels the
ambient-to-centralizer index, leaving exactly the centralizer degree
defect.  Central isogenies preserve the rational order and unipotent
degrees, so these calculations apply to our regular forms and to
Frobenius orbits of factors, using $q_i=q^\delta$.

The single-cycle formulas are Asai's hook and cohook formulas
\cite{Asai}; see also \cite[Theorem~3.2]{LubeckMalle}, where a
degenerate symbol initially denotes the sum of its two characters.
The individual normalization needed here is spelled out in
Lemma~\ref{lab:sparse} below.  The assertion about a full packed core
is the ordinary unipotent cyclotomic Harish--Chandra series rule
\cite[Theorem~3.2 and Section~3.A]{BMM}.
This is a statement about ordinary unipotent characters.
For split Harish--Chandra series the ordinary endomorphism algebras
give the indicated Weyl-group branching.  The rank-one degree
ratios are the ordinary rank-one parameters, equivalently the
ratios in the same partition and symbol degree formulas.
\end{proof}

Here and below the $e$-part of a torus means its connected
$\Phi_e$-subtorus, defined by the $\Phi_e$-isotypic subspace of its
rational cocharacter space.  Taking the full lattice in that subspace
defines the actual subtorus.  Set
\[
 e=\ord_p(q),\qquad e_i=\ord_p(q_i),\qquad
 (d,\eta)=
 \begin{cases}
  (e_i,+1),&e_i\text{ odd},\\
  (e_i/2,-1),&e_i\text{ even}.
 \end{cases}
\]
The core operation on a packet removes positive $d$-cycles when
$\eta=+1$ and negative $d$-cycles when $\eta=-1$.
The ambient cyclotomic index remains $e$, including when
$q_i=q^2$.

\subsection{Uniform tests and a minimum principle}

For a fixed target character, the coefficients of an induction matrix
form an integral vector on the source characters.  Torus pairings
need not determine this vector.  For the unipotent single-cycle
formulas of Proposition~\ref{prop:label-degrees}, we will show that
the hook or cohook coefficient vector is the unique integral vector
of minimum norm with its prescribed pairings.  The transport proof
in the next subsection will then identify the ordinary induction
rows by comparing the sums of their squared norms.

A uniform function is a linear combination of Deligne--Lusztig torus
characters.  The torus almost characters provide convenient tests
for its scalar products with irreducible unipotent characters.
We record precisely the portion of the classical Fourier formula
that will be used.  Within a symbol family, the entries occurring
twice are fixed, as is the set $\Omega$ of entries occurring once.
Write $h=|\Omega|$.  A row of the Fourier matrix is indexed by a
permitted distribution $B\subseteq\Omega$ between the symbol rows.
For types $B,C$ we orient the symbol to have row-length difference
$1$ modulo $4$; the resulting subsets have one fixed parity.
For type $D$ the permitted differences are even, and complementary
subsets represent row exchange.  The uniform columns are balanced
distributions $U$, of cardinality $\lfloor h/2\rfloor$ or the
complementary cardinality.  In type $D$ complementary columns are
identified.  For graph-twisted $D$ the columns use extensions of the
nondegenerate, equal-row-length Weyl labels to the twisted coset.
Equal row lengths here refer to symbol defect, not to block defect.

Write $f_B(U)$ for the scalar product in this rectangle.  The
classical Fourier formula~\cite{LusztigCharacters}, in the
subset conventions of
\cite[Section~6.1 and Proposition~6.3]{DigneMichel1990}, says that
pairings between different families vanish, that all entries in
one rectangle have the same nonzero absolute value, and that
\begin{equation}\label{lab:fourier-ratio}
 \frac{f_B(U)f_{B'}(U')}{f_B(U')f_{B'}(U)}
   =(-1)^{|(B\mathbin\triangle B')\cap
                    (U\mathbin\triangle U')|}.
\end{equation}
Phases attached to entire test columns have no effect on this
identity.  Degenerate split symbols have singleton families and
are separately uniform, with their own degenerate Weyl tests.
Type $A$ is uniform.  Products of packets use tensor products of
these test matrices.

The classical formulas hold over every finite field by
\cite[Corollary~4.3]{AsaiSmall}.  The following separation assertion
is the classical case of Digne--Michel
\cite[Proposition~6.3]{DigneMichel1990}.
It concerns ordinary characters only and is independent of the
coefficient prime $p$.  In particular, it will also apply at $p=2$
in Section~\ref{sec:periodicity}.
We recall the subset argument to fix the conventions used below;
the marked-diagram consequence is then applied to our Levi data.

\begin{lemma}[Separation by uniform tests]\label{lab:fingerprints}
The uniform pairing vectors of two ordinary unipotent labels in
these groups are proportional only when the labels coincide.
Consequently ordinary Jordan labels are functorial under rational
conjugacies of marked torus and Levi diagrams.
\end{lemma}

\begin{proof}
Different families have disjoint nonzero test supports.  In one
nondegenerate family, the differences $U\triangle U'$ of all
balanced subsets span the even-weight subspace of $\F_2^\Omega$:
for any distinct $i,j$, choose a balanced subset containing $i$
and not $j$, and replace $i$ by $j$.  These differences generate
all $\{i,j\}$.  Thus proportionality and
\eqref{lab:fourier-ratio} imply that
$B\triangle B'$ is either empty or all of $\Omega$.
In types $B,C$, $h$ is odd and the fixed parity convention excludes
the latter possibility.  In type $D$ it is precisely row exchange.
The cases $h=1$, or $h=2$ in type $D$, have only one relevant row;
degenerate labels have their separate tests.  Tensor factors may be
varied separately, proving the product assertion.

The ordinary Jordan torus-pairing rule is invariant under transport
of the full marked dual torus diagram.  Such transport therefore
preserves every uniform pairing of the prospective label.  The
first assertion removes the only possible character permutation
ambiguity.
\end{proof}

\begin{lemma}[The separated slice]\label{lab:slice}
Fix a nondegenerate Fourier rectangle and two positions
$P=\{x,y\}\subset\Omega$.  Suppose an integral vector $v$ on its
rows has two nonzero entries, both of absolute value one, at the
distinct labels $B$ and $B\triangle P$.  Suppose its uniform
pairing vector vanishes when $|U\cap P|\ne1$ and has twice the
single-entry absolute value when $|U\cap P|=1$.
Among integral vectors with this pairing vector, $v$ is the unique
one of minimum squared Euclidean norm, namely two.
The same conclusion holds after tensoring with fixed rows on
other packets.
\end{lemma}

\begin{proof}
Let $c>0$ be the common absolute value in the rectangle and put
$\mathcal S=\{U:|U\cap P|=1\}$.  This set is nonempty.
A vector of squared norm at most one is zero or one signed unit
vector, so its pairings have absolute value at most $c$.
It cannot give the required value $2c$ on $\mathcal S$.
An integral vector of squared norm two has exactly two distinct
signed unit entries.  On every column in $\mathcal S$, equality
in the triangle inequality forces each of these entries to give
the same value as each summand of $v$.

We determine which signed row can agree with a given signed row
on $\mathcal S$.  Regard subsets as vectors over $\F_2$ and let
$E(P^c)$ denote the even-weight subspace supported on
$P^c=\Omega\setminus P$.  If $h\ge3$, then
\begin{equation}\label{lab:slice-span}
 \left\langle U\triangle U':U,U'\in\mathcal S\right\rangle
     =\langle\mathbf1_P\rangle\oplus E(P^c).
\end{equation}
Indeed one may switch the chosen element of $P$.  In $P^c$ the
chosen subsets have size one less than the balanced size.  For
$h\ge4$ that size lies strictly between zero and $|P^c|$, so the
same single exchange argument as above generates $E(P^c)$.
For $h=3$ the complement has one element and its even subspace
is zero, giving the same formula.  In type $D$, lift balanced
columns to both complementary representatives before making this
calculation.  Even row differences ensure that their pairing
ratios agree on the two representatives.

The annihilator of the space in \eqref{lab:slice-span} consists
of vectors constant on $P$ and constant on $P^c$.  In odd type,
the fixed row parity excludes toggling $P^c$, whose size is odd.
In type $D$, toggling $P^c$ is the same as toggling $P$ modulo
full complement.  Thus the only possible rows are $B$ and
$B\triangle P$.  Their signs are fixed by a single column in
$\mathcal S$; the two distinct entries are consequently exactly
those of $v$.  The case of two distinct rows does not occur for
$h\le2$.  Varying test columns independently in every other
packet and applying Lemma~\ref{lab:fingerprints} fixes the inactive
rows and proves the tensor-product assertion.
\end{proof}

\begin{lemma}[Sparse single-cycle rows]\label{lab:sparse}
Fix a target unipotent label and a source family for one of the
single-cycle inductions of Proposition~\ref{prop:label-degrees}.
The list of coefficients on that family is zero, one signed unit
entry, or two distinct signed unit entries.  In the last case it
satisfies Lemma~\ref{lab:slice}, with $P$ the two moved positions.
In all cases this list is the unique integral list of minimum
squared norm having its uniform pairings.
\end{lemma}

\begin{proof}
Pad the two beta rows by simultaneous shifts so that the row-length
conventions agree in all symbols being compared.  The two family
multisets determine the starting and finishing positions
$a,a-d$ of a removal.  Away from split degeneracy there is at
most one removal in each of the two rows.  Two removals occur
only when the target has two beads at $a$ and no bead at $a-d$.
Their source distributions differ by toggling
$P=\{a,a-d\}$.  If these two removals gave the same unordered
source label, the target rows would agree away from these two
positions and also at them; that is exactly a degenerate target.
Thus, in the present case, the two source labels are distinct.

We verify their uniform support; this also fixes their relative
sign.  On Weyl-group tests, restriction at either a positive or
a negative signed cycle is same-row hook removal on bipartitions.
This follows from the induced-character description of signed
permutation characters and the symmetric-group
Murnaghan--Nakayama rule: the sign of the cycle changes the sign
of one bipartition contribution, but does not change which row
loses the hook.  A balanced source distribution can reach the
target duplicate at $a$ and vacancy at $a-d$ only when it
separates those two positions.  Hence torus transitivity forces
the two-entry pairing to vanish off $\mathcal S$.
Formula~\eqref{lab:fourier-ratio} says that the two row ratios are
constant on each slice and opposite on the two slices.  The two
rows are not proportional by Lemma~\ref{lab:fingerprints}.
Consequently their signed sum has absolute value $2c$ on
$\mathcal S$.  Both slices occur in every relevant two-removal
family.  Lemma~\ref{lab:slice} applies.

Here is the normalization at the type-$D$ boundary.  A degenerate
active character is uniform, and a contributing family on the
other side has two singles: deleting one bead from equal rows,
and inserting it at a previously vacant position, leaves just
these two single positions.  There is one ordinary row and one
uniform column in that nondegenerate rectangle.  The relevant
cycle classes do not split under restriction from $W(B)$ to
$W(D)$.  A positive cycle here has odd length, so flipping all
its coordinate signs is an odd-sign element of its centralizer;
a negative cycle itself supplies such an element.  Thus either
half of an equal-bipartition Weyl character has half the $W(B)$
value on the cycle classes in question.  Its two row removals
combine to give absolute value at most one on the residual
unordered Weyl label.  Conversely, from a fixed orientation of
a nondegenerate bipartition, only one hook reaches the equal
rows, and its coefficient on each split label has absolute value
one.  On the twisted coset the same calculation uses the
extensions of the nondegenerate Weyl characters; switching the
extension changes an overall sign, not the absolute value.
It therefore never identifies the two split labels.

For completeness, these checks also cover the ends of the rank
range.  A toral $D$ packet has its single unipotent character.
For a rank-zero residual packet, evaluate the Weyl character at
the full signed cycle.  In type $D$ only the single-hook
bipartitions just considered contribute.  In types $B,C$ their
symbol families have one or three singles.  In the three-single
case the full hook can leave either packed empty row; there are
at most two balanced contributions, each of Fourier absolute
value $1/2$, so again an individual coefficient has absolute
value at most one.  The one-single case has one test.  These
computations are also the individual-character interpretation
of the hook/cohook formulas when the degenerate-symbol notation
in \cite[Theorem~3.2]{LubeckMalle} is used.

Finally, a zero list is the unique vector of norm zero.  For one
signed entry, any integral competitor of norm at most one must
itself be one signed entry, and Lemma~\ref{lab:fingerprints}
identifies it.  Inactive factors are fixed and tensor with the
active calculation.  This proves the assertion in every case.
\end{proof}

\subsection{Transport of a single cycle}

The minimum-norm statements now reduce single-cycle transport to
an equality of total norms.  We prove that equality by Mackey's
formula, retaining the specified rational parameter.
We adapt the norm-comparison and minimum-norm argument of
Enguehard~\cite[Section~5.3, equations~(5.3.3)--(5.3.8) and
Lemma~5.3.9]{EnguehardJordan}.  We give the single-cycle argument
explicitly, retaining the rational Levi parameter, individual split
labels, toral and empty packets, and fixed inactive factors.

Let $s\in\mathbf H^{*F}$ be the fixed semisimple parameter and
put $\mathbf C=C_{\mathbf H^*}(s)$.  This centralizer is connected.
We allow any fixed additional linear packets, including parameters
with nontrivial $p$-parts on those inactive packets.  The active
packet is one of the classical packets just described.
For this subsection we also allow ambient Levis obtained by earlier
removals of disjoint cycles of the indicated kind.  The removed
cycles are fixed torus packets in their semisimple centralizers;
on the ambient side they may lie in linear factors.  These groups
are still Levis of the original regular group, so both centers
remain connected and both derived groups remain simply connected.
Their active centralizer is the smaller residual classical packet.
Thus this enlarged class is closed under removing another disjoint
cycle and under the intersection diagrams used in the proof below.

In a rational maximal torus of $\mathbf C$ choose $\mathbf V$
supported on one cycle only: either a split hyperbolic coordinate,
or the primitive $\Phi_e$-part of one of the signed $d$-cycles.
Put
\begin{equation}\label{lab:cycle-levi}
 \mathbf M^*=C_{\mathbf H^*}(\mathbf V),\qquad
 \mathbf C_M=C_{\mathbf M^*}(s)=C_{\mathbf C}(\mathbf V).
\end{equation}
The second group is, up to central isogeny, the residual classical
factor times the cycle torus and the inactive factors.  To see this
on the root datum,
in each absolute component the nonzero coordinate characters on
$\mathbf V$ are distinct up to sign.  On the primitive part they
are the successive powers of a primitive $e_i$th root, with the
prescribed signed closing.  All other coordinates vanish on
$\mathbf V$.  Precisely the desired residual roots vanish.
For a packet with $q_i=q^\delta$, its full signed orbit has length
$\delta d$ before folding.  The split-coordinate construction
gives an ordinary split Levi and a split parabolic.

Write $J_{H,s}$ and $J_{M,s}$ for the ordinary Jordan bijections
to the unipotent characters of $\mathbf C^F$ and
$\mathbf C_M^F$.  Let $\varepsilon_{\mathbf K}$ denote the usual
split-rank sign of an $F$-group $\mathbf K$.

\begin{proposition}[Single-cycle Jordan transport]\label{prop:single-cycle}
With the fixed markings in \eqref{lab:cycle-levi}, the matrix of
$R^{\mathbf H}_{\mathbf M}$ on
$\mathcal E(\mathbf M^F,s)$, in ordinary Jordan labels, is
\begin{equation}\label{lab:single-cycle-matrix}
 A=\epsilon B,\qquad
 B=\left[R^{\mathbf C}_{\mathbf C_M}\right]_{\mathrm{unip}},
 \qquad
 \epsilon=
 \varepsilon_{\mathbf H}\varepsilon_{\mathbf M}
 \varepsilon_{\mathbf C}\varepsilon_{\mathbf C_M}.
\end{equation}
For the split hyperbolic extraction the sign is positive.
This includes individual split labels, toral and empty residual
packets, and products with fixed inactive packets.  Successive
split extractions consequently give ordinary partition,
bipartition, and folded type-$D$ Harish--Chandra branching on
these Jordan labels.
\end{proposition}

\begin{proof}
All matrix entries in $A$ are integers, since Lusztig induction
is the alternating character of a cohomology complex.  Series
disjunction puts its image in the stated ambient series.  On
uniform inputs, the equality \eqref{lab:single-cycle-matrix}
already follows from transitivity of torus induction and the
ordinary Jordan torus-pairing rule.  More explicitly, apply
transitivity to each torus character in $\mathbf M$ with
parameter $s$ and pair with a fixed target character.  The
Jordan torus pairings on the two sides give its predicted
uniform pairing against every source family.  By
Lemma~\ref{lab:sparse}, that target row of $\epsilon B$ has
the unique minimum norm among integral rows with these pairings.
Summing over source families and then target rows gives
\begin{equation}\label{lab:norm-min}
 \|A\|_{\mathrm{HS}}^2\ge\|B\|_{\mathrm{HS}}^2,
\end{equation}
and equality forces $A=\epsilon B$.  Here the square of the
Hilbert--Schmidt norm is the sum of the squares of all matrix
entries.  It remains to prove equality of these total norms.

\smallskip\noindent\emph{Matching the rational terms.}
We use the Lusztig-induction Mackey formula, which holds for
these classical components for every $q$
\cite[Theorem~3.8(4)]{BonnafeMichel}.
The argument is by induction on the rank of the active packet.
Apply Mackey to
$({R^{\mathbf H}_{\mathbf M}})^*R^{\mathbf H}_{\mathbf M}$
and take its trace on $\mathcal E(\mathbf M^F,s)$.
We describe the rational terms and their markings, since an
unmarked Weyl-double-coset count would not suffice.
The object to match is an intersection together with its two
rational embeddings into the Levi.  Both embeddings are needed
to impose the chosen $s$-series at both ends of each term.

Fix dual $F$-stable maximal tori $\mathbf T\subseteq\mathbf M$
and $\mathbf T^*\subseteq\mathbf M^*$, and retain their full
character and cocharacter lattices.  Put
$W=W(\mathbf H,\mathbf T)$ and $W_M=W(\mathbf M,\mathbf T)$.
The algebraic double positions in the Mackey formula are indexed
by $W_M\backslash W/W_M$.  If $n\in N_{\mathbf H}(\mathbf T)$
represents $w\in W$, their
intersection is
$\mathbf K=\mathbf M\cap{}^n\mathbf M$; it is the centralizer
of the torus generated by $Z^\circ(\mathbf M)$ and
${}^nZ^\circ(\mathbf M)$, and hence is connected.  Its roots
are $\Phi_M\cap w\Phi_M$.  The stabilizer for the double action
of $\mathbf M\times\mathbf M$ is the graph of this connected
intersection.  Lang's theorem therefore gives exactly one
rational double orbit for each $F$-stable algebraic double orbit.
This assertion does not identify the different rational conjugacy
classes of maximal tori in an intersection.

We record the descent of both embeddings, since an $F$-stable
Weyl double coset need not have an $F$-fixed representative.
Choose $u,v\in W_M$ with $F(w)=uwv^{-1}$, lift $u$ to
$a\in N_{\mathbf M}(\mathbf T)$, and set
$b=F(n)^{-1}an\in N_{\mathbf M}(\mathbf T)$.
Thus $F(n)=anb^{-1}$ and $b$ lifts $v$.  The maps
\[
 j_1:\mathbf K\hookrightarrow\mathbf M,\qquad
 j_2=\operatorname{Ad}(n^{-1}):\mathbf K\longrightarrow\mathbf M
\]
commute with the respective Frobenius maps
\[
 F_1=\operatorname{Ad}(a^{-1})F,\qquad
 F_2=\operatorname{Ad}(b^{-1})F,\qquad F_K=F_1|_{\mathbf K}.
\]
Choose $x,y\in\mathbf M$ with $F(x)=xa^{-1}$ and
$F(y)=yb^{-1}$.  Then $g=xny^{-1}\in\mathbf H^F$.
Conjugating $\mathbf K$ by $x$ gives
$\mathbf M\cap{}^g\mathbf M$ with its two rational embeddings,
inclusion and $\operatorname{Ad}(g^{-1})$.

Here is the dual Frobenius check on the full lattices.
Choose dual bases of $X(\mathbf T)$ and
$X(\mathbf T^*)=Y(\mathbf T)$.
Write $Q$ for the pullback of $F$ on $X=X(\mathbf T)$, with
Weyl elements acting on $X$ in the usual root-datum convention.  Then
\[
 F(w)=Q^{-1}wQ,\qquad F_1^*=Qu,\quad F_2^*=Qv,
 \qquad wQv=Quw.
\]
Write $F^\vee$ for the Frobenius on the dual group, also
denoted by $F$ elsewhere.  On its character lattice
$Y=X(\mathbf T^*)$ put
$Q^\vee=Q^t$ and $w^\vee=w^{-t}$.  Transposing gives
\[
 w^\vee v^tQ^\vee=u^tQ^\vee w^\vee.
\]
Consequently, for
$U=(Q^\vee)^{-1}u^tQ^\vee$ and
$V=(Q^\vee)^{-1}v^tQ^\vee$,
\[
 F^\vee(w^\vee)=Uw^\vee V^{-1}.
\]
Apply the same normalizer-lift and Lang construction on the dual
side.  Its two twisted Frobenius pullbacks are
$Q^\vee U=(Qu)^t$ and $Q^\vee V=(Qv)^t$, and its second
embedding has pullback $w^\vee$.  Its intersection roots are
$\Phi_M^\vee\cap w^\vee\Phi_M^\vee$.
Thus the construction dualizes both embeddings and their
Frobenius maps, including the central lattices.

For completeness, fix $n$ and compare two choices $(a,b)$ and
$(a',b')$.  There is $k\in N_{\mathbf K}(\mathbf T)$ with
\[
 a'=ak,\qquad b'=b(n^{-1}kn).
\]
The new intersection Frobenius is
$F'_K=\operatorname{Ad}(k^{-1})F_K$.  Choose
$z\in\mathbf K$ with $F_K(z)=zk^{-1}$.
Conjugation by $z$ on the first endpoint and by $n^{-1}zn$
on the second identifies the two twisted diagrams, and hence
their descents.  Indeed, one may take $x'=xz$ and
$y'=y(n^{-1}zn)$, which gives $x'ny'^{-1}=xny^{-1}$.
Replacing
$n$ by $lnr^{-1}$ for $l,r\in N_{\mathbf M}(\mathbf T)$
changes only the two endpoint markings: take
$a'=F(l)al^{-1}$ and $b'=F(r)br^{-1}$.
Different Lang solutions $x,y$ differ on the left by elements
of $\mathbf M^F$, so the resulting rational double orbit is
unchanged.  The same argument applies in the connected dual
intersection.  Lattice duality makes the construction reversible.
We have therefore matched the rational Levi-intersection diagrams
with both embeddings.  The auxiliary torus used for the lattice
calculation is discarded after descent; no rational conjugacy
of arbitrary choices of common maximal torus is asserted.

For a dual rational representative $g\in\mathbf H^{*F}$,
refine its term by rational semisimple classes in the intersection
that map to the class of $s$ on both sides.
Series disjunction is exactly this refinement of the projected
primal Mackey term.  Every such matching class can be written
\[
 {}^{m_1}s={} ^{g m_2}s,
       \qquad m_1,m_2\in\mathbf M^{*F}.
\]
Changing the two markings gives
$h=m_1^{-1}g m_2\in\mathbf C^F$.
The remaining changes are the left and right actions of
$\mathbf C_M^F$.  Conversely a double coset with a
representative $h\in\mathbf C^F$ gives this matching class.
Changing $g$ while returning to the same double position
conjugates the parameter by the intersection, exactly as in
the refinement.  Thus the refined terms are in bijection with
the Mackey terms for $(\mathbf C,\mathbf C_M)$.
All centralizers involved are connected, since the relevant
derived groups on the dual side are simply connected.
Lemma~\ref{lab:fingerprints}, applied to the transported torus
diagrams, identifies the Jordan labels under every conjugation
in this correspondence.

\smallskip\noindent\emph{Comparing the contributions.}
The rational Mackey terms and their parameter markings have now
been matched.  We next compare their contributions: equal supports
give the identity terms, whereas disjoint supports reduce to a
single-cycle calculation on a smaller active packet.
There is an explicit description of the intersections in this
bijection.  For $h\in\mathbf C^F$, the intersection of
$C_{\mathbf H^*}(\mathbf V)$ and
$C_{\mathbf H^*}({}^h\mathbf V)$ contains a maximal torus
if and only if $\mathbf V$ and ${}^h\mathbf V$ commute.
For the forward implication, both tori belong to the connected
centers of their respective Levis and hence to every maximal
torus in them.  The converse follows by placing the commuting
tori in a maximal torus.  The same condition in $\mathbf C$
gives its Mackey intersection
$C_{\mathbf C}(\mathbf V,{}^h\mathbf V)$.

Take a rational common maximal torus in this connected
intersection.  The full signed-cycle torus containing
$\mathbf V$ belongs to the connected center of
$C_{\mathbf C}(\mathbf V)$: within each absolute component
the restricted coordinate characters are nonzero and distinct
up to sign, so no root involving a cycle coordinate vanishes
except the roots of the inactive factors.  Every maximal
torus of this centralizer therefore contains that full cycle
torus.  Its active support is consequently preserved when
we pass to the common maximal torus, and the same holds for
${}^h\mathbf V$.

On a packet of $\delta$ components, $F$ permutes the
components transitively and $F^\delta$ cyclically permutes
the distinct within-component restricted characters with
the prescribed signed closing.  The support is one signed
$F$-orbit, even though restrictions on different components
can coincide.  The two supports are therefore equal or
disjoint.  For a cyclotomic cycle the characteristic
polynomial on the full orbit is $X^{\delta d}-\eta$,
so $\Phi_e$ occurs with multiplicity one.  For a split
extraction it is $X^\delta-1$, with $\Phi_1$ occurring
once.  Thus the indicated primitive or split subspace
is unique on its support.  Equal supports thus
mean equal subtori.  Conjugation in $\mathbf C$ cannot add
central coordinates or move a support to a different
eigenvalue packet.

If the subtori are equal, the maps in the corresponding
intersection term are identities followed by the matched
conjugation.  If they are disjoint, both induction maps in
that term remove a single cycle of the same kind from a
strictly smaller residual packet.  The cycle already removed
is now an inactive torus factor.  This remains true if
coincident coordinates from distinct sign packets formed
an ambient linear factor in $\mathbf M^*$: its intersection
with $\mathbf C$ is the indicated cycle torus.  By induction
on active rank, both maps in the term agree in Jordan labels
with the centralizer maps.  To check the signs, write
$\mathbf K^*=C_{\mathbf H^*}(\mathbf V,{}^h\mathbf V)$
and $\mathbf D=C_{\mathbf C}(\mathbf V,{}^h\mathbf V)$.
Each smaller map has relative sign
$\varepsilon_{\mathbf M}\varepsilon_{\mathbf K}
 \varepsilon_{\mathbf C_M}\varepsilon_{\mathbf D}$;
conjugating the marking preserves these split ranks.
The two signs therefore multiply to one.  The equal-support terms start
the induction, including rank zero and the toral $D$ case;
an ambient identity extraction needs no induction.

The trace of the projected ambient Mackey formula is
therefore the trace of the unipotent Mackey formula for
$(\mathbf C,\mathbf C_M)$.  These traces are respectively
$\|A\|_{\mathrm{HS}}^2$ and $\|B\|_{\mathrm{HS}}^2$.
Equality in \eqref{lab:norm-min} proves
\eqref{lab:single-cycle-matrix}.  For split extraction both
sides are ordinary Harish--Chandra induction matrices, with
nonnegative entries and a nonzero entry, so the relative sign
is positive.  Transitivity proves the last assertion.
\end{proof}

\begin{corollary}[Projected split induction]\label{lab:projected-hc}
Let $s$ be a $p'$-parameter and fix its rational class in a
split Levi $\mathbf M$ as above.  On characters in
$\mathcal E_p(\mathbf M^F,s)$, the coordinates in
$\mathcal E(\mathbf H^F,s)$ after split induction depend only
on the coordinates in $\mathcal E(\mathbf M^F,s)$.
On these coordinates the matrix is the nonnegative branching
matrix of Proposition~\ref{prop:single-cycle}, iterated when
several size-one factors are extracted.  The assertion remains
valid after a common central $p$-quotient.
\end{corollary}

\begin{proof}
A row with parameter $st$, where $t\ne1$ is a commuting
$p$-element, can induce only to that geometric parameter.
It cannot induce to the $p'$-parameter $s$, because its
$p$-part stays nontrivial under conjugacy.  Thus its projection
onto $\mathcal E(\mathbf H^F,s)$ is zero.  For $t=1$ use
Proposition~\ref{prop:single-cycle} and transitivity, always
retaining the chosen Levi class.  The assertion for a common
central quotient follows because its left and right actions
on the induction variety agree.  Ordinary split induction
and restriction are exact and preserve projectives, so the
same coordinate rule applies to projective characters and
to their block summands.
\end{proof}

\subsection{Core collections are block collections}

We have determined the ordinary matrices, including the
nonuniform rows.  To use them on projective modules, the core
projections must also be realized by block idempotents.  This is
the separate modular assertion proved in this subsection.

We use a restricted form of the good-prime block theorem of
Cabanes--Enguehard.  The relevant statements are
\cite[Theorems 2.5 and 4.1]{CE99}.
For the connected reductive groups at hand, ordinary
$e$-cuspidal pairs in $p'$-series have assigned blocks; all
constituents of their Lusztig induction lie in the assigned
block, and rationally nonconjugate pairs give different blocks.
An $e$-split Levi is a centralizer of a $\Phi_e$-torus; ordinary
$e$-cuspidality means vanishing of adjoint Lusztig induction to
every proper $e$-split Levi.  This is the general connected
reductive statement of the theorem; a central torus need not be
split.  In our application the semisimple components are
classical of type $B,C,D$ or $A_1$, $p$ is odd and good, both
ambient centers are connected, and the relevant center and
fundamental-group orders have no $p$-part.  There is no
triality component.  This includes $p=3$; see also the
classical small-prime discussion in \cite[Remark~5.2]{CE99}
and \cite[proof of Theorem~3.14(e)]{KessarMalle2015}.
We apply this theorem directly to these connected groups.
For the connected reductive form used here, including its possibly
nonsplit central torus, see also
\cite[Assumption~2.1.2 and Propositions~2.1.6--2.1.7]{EnguehardJordan}.

\begin{lemma}[The packed inducing pair]\label{lab:packed-pair}
Fix target Jordan labels on the active classical packets,
and let $\kappa$ be their collection of packed cores.
Take the signed cycles removed in packing these labels
to $\kappa$; their number in each packet is determined by
its target rank and core rank.  Centralize their independent
primitive $\Phi_e$-subtori in $\mathbf H^*$, obtaining
$\mathbf L^*$.  Then $s$ belongs
to $\mathbf L^{*F}$, and its dual Levi $\mathbf L$ is
$e$-split.  The ordinary character $\lambda$ of
$\mathbf L^F$ whose Jordan label is $\kappa$, trivial on
the removed cycle tori and with the fixed inert torus data,
has central $p$-defect and is ordinary $e$-cuspidal.
If there are separate split core labels the assertion applies
to the corresponding choices individually.
\end{lemma}

\begin{proof}
Independent subtori are obtained by giving each of these removed
signed cycles its own coordinates and making its cocharacters
zero elsewhere.  Their centralizer is an $e$-split Levi and
contains $s$.  The permitted torus positions and the residual
orthogonal sign are exactly those of the unipotent core
rule; a negative cycle switches the type-$D$ defect congruence.
That rule also supplies the inducing label when the residual
rank is zero.  We check the degree and cuspidality assertions
on this actual Levi.

A packed core has no hook or cohook divisible by the relevant
cyclotomic step, so its degree defect contributes no $p$-part
beyond its torus data.  The $p$-part of every removed cycle
torus is already contained in its primitive $\Phi_e$-part.
Indeed a factor $\Phi_j(q)$ can be divisible by $p$ only for
$j=e p^a$ with $a\ge0$ (with the usual $e=1$ convention).
The signed absolute cycle length is $\delta d$, with
$\delta\le2$ and $d\mid(p-1)$ or $2d\mid(p-1)$.
It has no $p$-factor, so $a>0$ cannot occur.  These primitive
tori and the ambient central torus are central in
$\mathbf L^*$ by construction.  No residual core torus
has an additional noncentral $\Phi_e$-part: such a toral
$D$ remainder would still admit a removable cycle.  Frozen
extracts have order prime to $p$.

Explicitly, a split $D_1$ packet has order $q_i-1$.
If its $p$-part is nontrivial, then $e_i=1$, and its label
admits a removable $1$-hook.  A nonsplit $D_1$ packet has
order $q_i+1$; a nontrivial $p$-part forces $e_i=2$ and
a removable $1$-cohook.  Neither can remain in a packed
core.  Thus every residual $D_1$ core torus has order
prime to $p$, even when it enlarges the full residual
center.  The $p$-part of the full center in
\eqref{lab:central-defect} is consequently the inherited
central contribution together with the removed primitive-cycle
torus contributions.

There is no odd-prime isogeny loss in this calculation.
Within each absolute packet the primitive-cycle coordinate
characters are distinct up to sign.  Equalities between
different sign packets can therefore tie at most two
coordinates and create at most $A_1$ systems.  The residual
semisimple systems are $B,C,D$, and frozen extractions add
only the stated rank-one systems.  Their standard center
orders are powers of $2$.  The signed equality lattices and
the regular central dressing likewise have at worst
$2$-primary indices on both root-data sides.  Thus rational
torus orders, computed on the rational spaces, give the
correct $p$-parts also for the actual centers.  Dual centers
have the same rational order.  The degree rule of
Proposition~\ref{prop:label-degrees} consequently gives
\begin{equation}\label{lab:central-defect}
 \operatorname{def}_p(\lambda)=|Z(\mathbf L)^F|_p.
\end{equation}
One may also see the lower bound in this equality directly:
a character in a $p'$-series is trivial on the central
$p$-subgroup.

We have established the actual-center degree identity.
To prove ordinary $e$-cuspidality, we now combine projectivity
modulo that center with the larger central $p$-subgroup in every
proper $e$-split Levi.
Put $Z=O_p(Z(\mathbf L)^F)$.  Equation
\eqref{lab:central-defect} says that $\lambda$, viewed on
$\mathbf L^F/Z$, is an ordinary defect-zero character,
hence the character of a projective module.  Let
$\mathbf K$ be a proper $e$-split Levi of $\mathbf L$.
Its adjoint Lusztig restriction
$\psi={}^*R^{\mathbf L}_{\mathbf K}\lambda$ is a virtual
projective character of $\mathbf K^F/Z$.
Here is the quotient justification.  In a Deligne--Lusztig
induction variety the left and right actions of the common
finite central subgroup agree, and these translations are
free.  After division by $Z$, stabilizers for either group
action have prime-to-$p$ order: they come from unipotent
stabilizers, with the harmless prime-to-$p$ central kernels.
The projective-complex theorem for Rickard cohomology
\cite{RickardEtale} therefore sends projectives to virtual
projectives.  Over characteristic zero, taking the quotient
variety selects precisely the $Z$-invariant summand.  This
proves the assertion about $\psi$ without imposing modular
compatibility on Jordan decomposition.

Every parameter occurring in $\psi$ is conjugate to the
$p'$-element $s$.  Thus $\psi$ is trivial on
$O_p(Z(\mathbf K)^F)$.  This group is strictly larger than
$Z$.  Indeed the centers of these Levis are connected, and
proper $e$-split centralization adds a nonzero
$\Phi_e$-subspace to the center.  The quotient of the
centers is a torus whose order has a factor
$\Phi_e(q)$.  Lang's theorem on the connected kernel
$Z(\mathbf L)$ makes the sequence on rational centers
surjective, so the quotient adds a nontrivial $p$-part.
Choose a central element of order $p$ in its
image in $\mathbf K^F/Z$.

A virtual projective character vanishes on $p$-singular
elements.  For any $p$-regular element $x$ of
$\mathbf K^F/Z$, multiplying $x$ by this central element
gives a $p$-singular element, with the same value of $\psi$
because the central element is in its kernel.  Hence
$\psi(x)=0$ as well, and $\psi=0$.  Since $\mathbf K$
was arbitrary, $\lambda$ is ordinary $e$-cuspidal.
\end{proof}

\begin{remark}\label{lab:jordan-cuspidal}
The same pairs also have the centralizer description often
called $e$-Jordan cuspidality.  The $\Phi_e$-center of
$C_{\mathbf L^*}(s)$ is central in $\mathbf L^*$, and the
unipotent core is cuspidal for the corresponding cyclotomic
index.  If a packet is represented over $q^2$, projection of
an ambient $\Phi_e$-subtorus gives its cyclotomic subtorus
for $\ord_p(q^2)$; thus folding does not change which
cuspidality test is made.  Moreover
$\mathbf L^F/(Z(\mathbf L)^F[\mathbf L,\mathbf L]^F)$
and the intersection in this central product have
prime-to-$p$ order.  Clifford restriction across this
abelian quotient preserves degree $p$-parts: inertia
multiplicities are degrees of projective representations
of a $p'$-group, realized over a finite $p'$-central
extension.  Equation~\eqref{lab:central-defect} therefore
gives defect zero on the derived group after the central
part is removed, which is the quasi-central defect
condition.  The direct ordinary cuspidality proof above
is what is needed for the stated theorem of Cabanes--Enguehard.
\end{remark}

\begin{proposition}[Core block projections]\label{prop:core-blocks}
In each remaining type-$A$ unipotent target, and in each
regular classical target with fixed $p'$-parameter $s$,
the following holds.  Partition the basic rows in
$\mathcal E(\mathbf H^F,s)$ by their packed core
collections, using the actual unordered-symbol and
type-$D$ identifications.  Each part is the intersection
of that basic set with a union of blocks in
$\mathcal E_p(\mathbf H^F,s)$.  Distinct core collections
cannot occur in the same block.  Split sign choices
within one underlying core may be merged.

This remains true in every intermediate split Levi with
the specified frozen extractions, and after the common
central $p$-quotient.  Consequently all these core
projections are projections by sums of block idempotents
on projective modules.
\end{proposition}

\begin{proof}
For the general linear and unitary unipotent targets this
is the ordinary block/core rule of
Fong--Srinivasan~\cite{FongSrinivasan}, with partition step
$\ord_p(\epsilon q)$.

Consider a regular classical target.  Choose a basic row
with a given core collection and apply
Lemma~\ref{lab:packed-pair} to its labels.
The resulting pair is ordinary $e$-cuspidal, belongs to a
$p'$-series, and has central defect.  The preceding
root-lattice calculation verifies all the stated
classical hypotheses of the Cabanes--Enguehard theorem
on $\mathbf H$ and on the intermediate Levis: no
exceptional or triality component occurs, all relevant
odd primes are good, both centers are connected, and
the extra equality components are at most $A_1$.
Thus the theorem assigns a block to this pair.
By Proposition~\ref{prop:single-cycle}, transitivity,
and the ordinary unipotent core rule, every basic row
with the given core occurs in the induction of one
of its packed inducing characters.  Constituent
inclusion puts it in the corresponding assigned block.

We must distinguish the pairs for different cores
under rational conjugacy.  Suppose two such primal
pairs are rationally conjugate.  Transport a rational
torus in their Levis, with its full ambient marking.
Duality and the descent of the marked diagram used in the
proof of Proposition~\ref{prop:single-cycle} transport
the dual Levi diagram, and the series rule transports
its parameter.  Adjusting inside the target dual Levi
then makes this parameter exactly $s$.  The resulting
transport lies in $\mathbf C^F$.  Its action on the
residual unipotent labels is determined by the torus
pairings, hence by Lemma~\ref{lab:fingerprints}.

Inside $\mathbf C^F$ these are the ordinary
cycle-Levi and core data on the fixed eigenvalue
packets.  The central primitive-cycle factors record
the number of removed cycles in each nontoral packet;
a residual toral core has no additional noncentral
$\Phi_e$-part.  A conjugation matching the cycle data
acts on the residual packed label by its actual
symbol identifications.  It cannot change it to a
different row pair, nor can inner conjugation in the
connected centralizer exchange different eigenvalue
packets.  A wholly toral packet has only its single
unipotent label.  The allowed type-$D$ graph and sign
identifications give exactly the already stated
core equivalence.  Equivalently, this is the rational
nonconjugacy assertion for the packed data in the
ordinary unipotent core rule.  Therefore different
core collections give nonconjugate inducing pairs.
Uniqueness up to conjugacy in the Cabanes--Enguehard theorem puts
their rows in different blocks.

Every block in $\mathcal E_p(\mathbf H^F,s)$ meets
$\mathcal E(\mathbf H^F,s)$, and that intersection
is an integral basic set for the block.  Since the
basic rows have just been partitioned by disjoint
assigned block collections, each core collection
is exactly the basic-row intersection of its union
of blocks.  No assertion about modular Jordan
decomposition is involved.  The argument applies
verbatim with the frozen factors and their markings.
Finally a central $p$-quotient gives the usual
bijection of blocks, preserving their basic rows
with trivial central action.  Sums of the resulting
block idempotents preserve projectives, proving
the last assertion.
\end{proof}
==== END Donovans-Conjecture-over-Algebraic-Closures-of-Prime-Fields//04_labels.tex ====

==== BEGIN Donovans-Conjecture-over-Algebraic-Closures-of-Prime-Fields//05_runners.tex ====
\section{Runner reductions and projective cones}\label{sec:runners}

We now reduce the unbounded ranks in the odd-prime classical problem.
The operations will be split Harish--Chandra induction and restriction,
followed by projections onto unions of blocks.  Their matrices on the
basic rows are particularly simple.  It is these matrices, together
with positivity on projectives, that will transfer bounds.

Runner rearrangements connect this argument with the abacus
reductions of Scopes and Jost and the classical unipotent
equivalences of Hiss--Kessar
\cite{Scopes,Jost,HissKessarScopes,HissKessarScopesII}.
Here the transported objects are cones of projective characters;
the moves need only transfer numerical bounds, without providing
Morita equivalences.

Throughout this section \(p\) is odd and does not divide \(q\).
We use the ordinary labels and
degree formulas of Proposition~\ref{prop:label-degrees}, the branching
rule of Proposition~\ref{prop:single-cycle}, and the block projections
of Proposition~\ref{prop:core-blocks}.  The type \(A\) series considered
here is unipotent.  In the regular classical groups of
Lemma~\ref{fam:regular-datum}, a \emph{packet} means a Frobenius orbit
of classical factors of the connected centralizer of the fixed semisimple
\(p'\)-parameter whose normalized spectrum is contained in
\(\{1,-1\}\).  We calculate on one representative factor with the
composite endomorphism around the orbit.  There are boundedly many
packets, with field
parameters \(q_i=q^\delta\), \(\delta\leq2\), as in
Lemma~\ref{fam:packets}.  The parameter is retained on every Levi:
induction always starts in its prescribed rational Levi conjugacy
class.

\begin{proposition}\label{prop:runner-reduction}
Fix \(p\) and a defect-order bound \(M\).  Consider either a unipotent
block of \(\GL_n(q)\) or \(\GU_n(q)\) of defect order at most \(M\), or a
block in a sign-parameter series of an original regular classical
group above, before extraction, whose defect order modulo its
central \(p\)-subgroup is at most \(M\).
There are constants depending only on \(p,M\) with the following
properties.

The required Cartan bound reduces to Cartan bounds for groups with
root data in a finite set, and for the following block-union
projections: labels in a fixed ordinary cuspidal triangle series,
with a bounded number of split Harish--Chandra rank-one steps above
the triangle.  The latter projections have boundedly many basic
rows and bounded block defects, although the triangle rank can be
unbounded.  The transfer constants are uniform.

Consequently Theorem~\ref{thm:geometric-cartan} and
Proposition~\ref{prop:triangle-cartan} give the required Cartan bounds
for all these blocks.
\end{proposition}

We first bound the runner weights and calculate the move matrices,
including their type-D foldings.  We then control the complete
reduction in two stages: a map of bounded condition number on the
full coordinate space, followed by the symmetric stage.  Bounds
return through these stages in reverse order, using the two forms
of the cone lemma below.
All dimensions and norms
below are ordinary Euclidean ones; changing to a coordinate
\(\ell^1\)-norm changes constants only by the bounded dimension.

\subsection{Charged runners and bounded weight}

A charged bead set is a subset \(B\subset\Z\) containing every
sufficiently negative integer and no sufficiently positive integer.
Its charge is
\[
 c(B)=|B\cap\Z_{\geq0}|-|\Z_{<0}\setminus B|.
\]
List its elements in decreasing order as \(b_1>b_2>\cdots\).
There is a unique partition \(\lambda\) such that
\[
 b_a=c(B)-a+\lambda_a.
\]
Its weight \(w(B)=|\lambda|\) is also the number of pairs
\((b,g)\) with \(b\in B\), \(g\notin B\), and \(g<b\).
Indeed the bead \(b_a\) has exactly \(\lambda_a\) gaps below it.
The packed set of charge \(c\) is
\(B(c)=\{k\in\Z:k<c\}\).

\begin{lemma}\label{run:localization}
If \(w(B)\leq W\), then \(B\) agrees with \(B(c(B))\) outside
\([c(B)-W,c(B)+W)\).  For charged sets \(B,B'\) of total weight at
most \(W\), put \(m=c(B)-c(B')\).  If \(m>2W\), then
\(B'\subset B\) and \(|B\setminus B'|=m\).  In general
\begin{equation}\label{run:availability}
 |B\setminus B'|\leq \max(m,0)+2W.
\end{equation}
There are at most as many charged sets of fixed charge and weight
\(w\) as partitions of \(w\).
\end{lemma}

\begin{proof}
For \(a>W\) we have \(\lambda_a=0\), and
\(b_1\leq c(B)-1+W\).  This proves the asserted strip.
One can obtain \(B\) from its packed set by \(w(B)\) unit bead
moves; consequently each of the numbers of added beads and removed
beads is at most \(w(B)\).
When \(m>2W\), every gap of \(B\) is above every possible bead
of \(B'\) not already in the common filled negative part, giving
the inclusion.  The difference of charges is then the size of the
set difference.  Comparing both sets with their packed sets proves
\eqref{run:availability}.  The partition description proves the
last assertion.
\end{proof}

To pass from the finite labels to charged bead sets, use
\(\{\lambda_a-a:a\geq1\}\) for a partition \(\lambda\);
this set has charge zero.  For a symbol \((A,B)\), first extend
its rows to \(A\cup\Z_{<0}\) and \(B\cup\Z_{<0}\), of charges
\(|A|\) and \(|B|\).  Translate both sets by
\(-\lfloor(|A|+|B|)/2\rfloor\).  Their charge sum is then zero
or one.  A simultaneous symbol shift translates both extended
sets by one and increases this floor by one, so the normalized
charged rows are independent of the finite representative.

Here are the runner conventions, including the offsets at their
ends.  Each runner records the integers \(k\) for which the
displayed position is occupied in these charged sets.
Write \(\epsilon=1\) in the linear case and \(\epsilon=-1\)
in the unitary case, and set \(e=\ord_p(\epsilon q)\).
Partition positions \(j+ek\) form runner \(j\), with bead set \(B_j\)
in the coordinate \(k\) and packed threshold \(C_j=c(B_j)\).
Extend the indices by
\begin{equation}\label{run:a-offset}
 B_{j+e}=B_j-1,\qquad C_{j+e}=C_j-1.
\end{equation}
A split rank-one extraction moves \(j\) to \(j-1\) for
\(\GL\), and \(j\) to \(j-2\) for \(\GU\), at unchanged \(k\).
Thus the unitary move is a \(2\)-hook removal.

For a symbol, put \(Q=q_i\).
If \(\ord_p(Q)=d\) is odd, use in each row the runners
\(j+dk\), with \(C_{j+d}=C_j-1\).  This is the \emph{hook case}.
If \(\ord_p(Q)=2d\), use positions \(j+dk\) in the first row
when \(k\) is even and in the second row when \(k\) is odd.
Taking \(0\leq j<2d\) lists every position in the two rows once.
Now
\begin{equation}\label{run:cohook-offset}
 \begin{aligned}
 B_{j+2d}=B_j-2,\qquad C_{j+2d}=C_j-2,\qquad
 \\
 (\tau B)_j=B_{j+d}+1,\quad
 (\tau C)_j=C_{j+d}+1,
 \end{aligned}
\end{equation}
where \(\tau\) exchanges the two symbol rows.
This is the \emph{cohook case}.  In both symbol cases a split
rank-one extraction is \(j\longrightarrow j-1\).
All translations in these formulas translate the whole bead set.

Thus partitions have total charge zero, and the two physical symbol
rows have charge sum zero or one.  In types \(B,C\) orient the
rows so that their length difference is \(1\) modulo \(4\).
Packing cohook runners can change this difference by \(2\) modulo
\(4\) at each removal.  The oriented packed core is therefore
determined by the core class and the weight.  Type \(D\) cores are
instead taken up to \(\tau\).  A core is called \emph{symmetric}
if it is fixed by \(\tau\), with the offsets in
\eqref{run:cohook-offset} included.

The sum \(W=\sum_jw(B_j)\), over all the relevant runners and
packets, is the cycle weight.  Packing gives exactly the cores in
Proposition~\ref{prop:core-blocks}.  A hook or cohook at a contributing
distance is an inversion on one of these runners.

\begin{lemma}\label{run:defect-bounds}
For the starting blocks in Proposition~\ref{prop:runner-reduction},
the cycle weights are bounded in terms of \(p,M\).
If all weights vanish, the block is of defect zero after the
indicated central quotient.  Otherwise the relevant basic
cyclotomic \(p\)-parts, and in the regular classical case the actual
central \(p\)-order, are bounded.

In the positive-total-weight case, every endpoint core collection
retaining these weights has boundedly many basic rows and bounded
block defects upstairs.
These bounds also hold for a union of a bounded number of such
collections.
\end{lemma}

\begin{proof}
Every contributing inversion supplies a factor divisible by \(p\)
in the degree-defect formula of
Proposition~\ref{prop:label-degrees}.  In the regular classical
case divide first by the original central \(p\)-order \(|Z_p|\).  Hence
\(p^W\leq M\).
If \(W=0\), every basic-row degree has relative defect one;
the integral-basic-set degree criterion of
Lemma~\ref{lem:row-projection} gives defect zero in the quotient.
This case is completed before any runner extraction.  The toral
\(D_1\) core boundary contributes no hidden \(p\)-factor, by
the explicit packed-core check in Section~\ref{sec:labels}.

If a packet has positive weight, its degree defect is divisible by
the basic factor
\[
 \alpha=
 \begin{cases}
 ((\epsilon q)^e-1)_p,&\text{type }A,\\
 (Q^d-1)_p,&\text{hook case},\\
 (Q^d+1)_p,&\text{cohook case}.
 \end{cases}
\]
Thus the relevant \(\alpha\) is bounded.  All classical packets
have the same \(Q\).  Lemma~\ref{fam:central-order} then bounds
the central \(p\)-order, so that from now on we can work upstairs.
Subsequent residual degree calculations use the inherited torus
and retain any \(D_1\) packet separately, as in
Proposition~\ref{prop:label-degrees}.
Lemma~\ref{fam:central-order} also bounds the full residual
centers, with exponent at most four.  All further steps are
performed upstairs; they do not require a product decomposition
of a quotient by a diagonally acting central subgroup.
Packets of weight zero contribute no such factor.

If an inversion has runner distance \(h\), it accounts for at
least \(h\) inversions: between its gap \(g\) and bead \(b=g+h\),
each intermediate position is either a bead paired with \(g\)
or a gap paired with \(b\).  Therefore \(h\leq W\).
Lifting the exponent, for odd \(p\), bounds its factor by
\(\alpha h_p\); the same statement for a negative cycle follows
by applying it to odd powers in \(Q^d+1\), with the even powers
recorded as hooks on the same alternating runner.
There are at most \(W\) factors.  Thus every basic-row degree
defect is bounded at a retained-weight endpoint.
The integral-basic-set criterion bounds the block defects of the
whole collection, not just the defects of its displayed rows.

There are boundedly many runners, because \(e,d\leq p-1\), and
boundedly many packets.  Lemma~\ref{run:localization} bounds the
number of labels of total weight \(W\), including the at most two
split signs on a type \(D\) label.  The number of blocks is no
greater than the number of basic rows.  The block-union version
of Lemma~\ref{lem:row-projection} now gives all the stated bounds.
\end{proof}

The zero-total-weight case is now complete, so henceforth \(W>0\).
For a retained-weight core collection \(\mathcal C\), let
\(D_{\mathcal C}\) be the matrix whose columns are its PIM
characters projected onto its ordinary basic rows.  The lemma
makes this a square, full-rank nonnegative integral matrix of
bounded size and bounded determinant.  Its column cone is
\[
 \mathcal P_{\mathcal C}
   =D_{\mathcal C}\mathbb R_{\geq0}^{\,l(\mathcal C)},
\]
where \(l(\mathcal C)\) is the number of simple modules in the
block collection.  We will use induction to compare these cones
and thereby bound the entries of \(D_{\mathcal C}\).
Lemma~\ref{lem:row-projection} then bounds the full projective
characters and their Cartan inner products.

\subsection{Ordered moves and their realization}

An edge means one of the allowed moves \(j\to j'\).
Its positive difference is \(C_j-C_{j'}\), using the displayed
offsets if an index passes an end.  If this difference is
\(m>2W\), the corresponding bead sets are nested.  Transfer the
\(m\) beads in \(B_j\setminus B_{j'}\) across the edge.
This interchanges the two bead sets, with their coordinate offsets.

\begin{lemma}\label{run:ordered-swap}
Suppose the ordered core convention is allowed at both ends.
The projection of \(m\) successive rank-one extractions onto the
swapped core has matrix \(m!P\), where \(P\) is the bijection
interchanging the runner bead sets.  The reversed induction matrix
is its transpose.  The same conclusion holds for two disjoint
edges, each with difference \(m\), with coefficient \((2m)!\)
and the simultaneous-swap bijection.
\end{lemma}

\begin{proof}
Let \(v_j\) count the moves across the edge from \(j\).
Their effect on the charges is the incidence vector \(\partial v\);
on a step-one cycle,
\[
 (\partial v)_j=v_{j+1}-v_j.
\]
Changes of charges have no offset.  The kernel of this incidence
map consists precisely of vectors constant on each directed cycle.
The prescribed charge change is \(\partial(me_j)\).
The difference between any other flow and \(me_j\) is therefore
constant on each cycle.  Each cycle has an unused edge, so
nonnegativity forces every such constant to be nonnegative.
The total number \(m\) of extractions forces every constant to be
zero.  For two disjoint opposite edges use
\(me_j+me_{j^*}\) instead.  In the cases where we use this
argument each cycle again has an unused edge.

Thus a bead can move only across the prescribed edge or edges.
It can cross each edge at most once.  Every surplus bead must
cross, and any order of the \(m\), respectively \(2m\), distinct
transfers is legal.  This gives the factorial and the unique
output.  The total runner weight is unchanged by permuting the
bead sets.  Conversely every target label has the same bounded
weight, so Lemma~\ref{run:localization} supplies the reverse
nesting and reconstructs its unique source.  Path reversal proves
the assertion for induction.
\end{proof}

The cycles omitted by the unused-edge assertion cannot have a
large positive difference: for a length-one cycle the difference
is the negative period offset.  In particular this covers
\(e=1\) in \(\GL\), \(e\in\{1,2\}\) in \(\GU\), and \(d=1\)
in hook symbols.

Each individual transfer is an actual hook removal on a partition
or a same-row hook removal on a symbol, and therefore an actual
split Harish--Chandra step.  By Lemma~\ref{fam:packets}, a
classical packet step extracts a factor \(\GL_\delta(q)\);
the type \(A\) extracts are \(\GL_1(q)\) and \(\GL_1(q^2)\).
All these factors have order prime to \(p\) whenever a large
move occurs.  In the hook case a large move requires \(d>1\);
in the cohook case \(p\nmid Q-1\).  If \(\delta=2\), the former
condition also excludes \(p\mid q^2-1\), and the latter does so
directly.  The type \(A\) exclusions above likewise give the claim.
These extracted factors can be retained as independent frozen
factors in the Levi, each in its fixed defect-zero block.

There is no additional rank or sign constraint to impose after a
legal path has been specified.  Ordinary Harish--Chandra branching
realizes each of its labels in the appropriate residual classical
group.  The physical row lengths, and thus the symbol defect
congruence, are preserved by each symbol step.  A nonsplit
orthogonal residual group cannot reach a rank-zero label.
The independent hyperbolic coordinates in
Lemma~\ref{fam:packets} realize the whole chain of extractions,
including the fixed other packets.

Induction and restriction are exact and preserve projectives:
the unipotent radical has order prime to \(p\), so its averaging
idempotent defines an exact direct summand.  Projection onto a
union of blocks also preserves projectives.  All resulting
projective maps are therefore nonnegative in PIM bases.
By the ordinary parameter rule in
Proposition~\ref{prop:single-cycle}, rows with a nontrivial
semisimple \(p\)-part cannot contribute to the basic rows with
trivial \(p\)-part.  Consequently their projected coordinate maps
depend only on the displayed basic coordinates.  This remains
true with intermediate core projections.  Possible other Levi
parameter classes inducing to the same ambient class are
irrelevant, since the source has been projected onto the
prescribed class.

For an unnormalized composite functor \(F\) between retained-weight
core collections \(\mathcal C\) and \(\mathcal C'\), let
\(A_F\) be this map on basic coordinates.  Let \(N_F\) record
the multiplicities of target PIMs in the images of source PIMs.
The two descriptions of each image give
\[
 A_FD_{\mathcal C}=D_{\mathcal C'}N_F,
 \qquad N_F\geq0,
 \qquad
 A_F\mathcal P_{\mathcal C}\subseteq\mathcal P_{\mathcal C'}.
\]
The entries of \(N_F\) are integers, but no bound on them is
assumed.  Dividing \(A_F\) by a positive scalar preserves the
cone inclusion, with the same division applied to \(N_F\).
The bounded sizes and determinants concern the retained-weight
endpoints; no bound on intermediate projection dimensions is needed.

\subsection{The type D folding}

The ordered swap calculation is now available as an actual
projective-preserving operation.  In type $D$ its ordinary matrix
must still be checked after row exchange, because an unordered
target can receive extra paths and an equal-row label has two
distinct characters.

\begin{lemma}\label{run:folding}
In a type \(D\) ordinary Harish--Chandra series with equal row
lengths, folding ordered bipartitions sends a nonsymmetric
bipartition to its unordered character and an equal bipartition
to the sum of its two split characters.  With twice the single
character as the folding convention at relative rank zero, this
map intertwines induction through any positive number of marked
rank-one extractions and every underlying-core projection.
Such induction annihilates the difference of two split source
characters.
\end{lemma}

\begin{proof}
Write \(B_b=C_2\wr S_b\) and let \(D_b\) be its subgroup of even
signed permutations.  The asserted folding at positive rank is
\(\Res^{B_b}_{D_b}\), by the elementary index-two character
description of these groups.  If \(1\leq j<b\), embed \(B_j\)
on the old indices and fix every new index.  An odd sign change
on an old index belongs to \(B_j\), so
\[
 D_bB_j=B_b,\qquad D_b\cap B_j=D_j.
 \]
The Mackey formula, with its single double coset, gives
\[
 \Res^{B_b}_{D_b}\Ind^{B_b}_{B_j}
   =\Ind^{D_b}_{D_j}\Res^{B_j}_{D_j}.
 \]
At \(j=0\) there are two double cosets instead, and the left
side is twice \(\Ind^{D_b}_{D_0}\); this proves the stated
rank-zero convention.

An odd sign change on an old index interchanges the two split
characters.  Multiplying it by a sign change on a new index
gives an element of \(D_b\) inducing the same conjugation on
\(D_j\).  Their induced characters are thus equal, proving
the assertion about their difference.
Core projectors retain precisely a row-exchange orbit and both
split signs if present.  They commute with folding.  Iteration
proves the assertion with intermediate projectors.
\end{proof}

For nonzero row-length difference orient by the length and use
ordinary ordered branching.  For zero difference
Lemma~\ref{run:folding} shows that, between nonsymmetric cores,
the coefficient is the count of ordered paths from one fixed
orientation to either orientation of the target.  There are no
equal-row labels above a nonsymmetric core.  Reversing paths and
exchanging both orientations gives the identical rule for
induction.

\begin{lemma}\label{run:nonsymmetric}
Suppose \(d>1\), and let \(j^*\) be the edge paired with \(j\)
by row exchange.  They are disjoint.  A sufficiently large swap
from a nonsymmetric core to a nonsymmetric core has exactly the
matrix of Lemma~\ref{run:ordered-swap}, also on unordered labels.
If \(d\geq3\) and a sufficiently large swap would give a symmetric
core, there is a different large edge whose swap stays
nonsymmetric.
\end{lemma}

\begin{proof}
Let \(m=C_j-C_{j'}\), and put
\(m_1=C_{j^*}-C_{(j')^*}\).
If \(v\) is a flow to the exchanged target, applying row
exchange to its charge equation and adding gives
\[
 \partial(v+\tau v)=\partial(me_j+me_{j^*}).
 \]
Every relevant cycle has a missing edge because \(d>1\)
(in the hook case \(d\) is odd, hence at least three).
Nonnegativity and the total length \(2m\) therefore give
\(v+\tau v=me_j+me_{j^*}\).
In particular \(v\) is supported on these two disjoint edges.
Their charge equations give
\begin{equation}\label{run:exchanged-flow}
 v_j=\frac{m-m_1}{2},\qquad
 v_{j^*}=\frac{m+m_1}{2}.
\end{equation}
If \(|m_1|\leq2W\), the second expression is larger than the
availability bound \eqref{run:availability} for sufficiently
large \(m\).  If \(m_1>2W\), nesting gives
\(v_{j^*}\leq m_1\), whereas \(v_j\geq0\) gives \(m_1\leq m\).
Equality \(m_1=m\) is forced, so \(v_j=0\); the full charge
equation then says that the source is symmetric.  If
\(m_1<-2W\), no move at \(j^*\) is available, so
\(v_{j^*}=0\); the target is symmetric.  Both possibilities
are excluded.

Now suppose that the intended target is symmetric.  The source
and its exchange then differ exactly on the two pairs of
vertices incident to the edge and its opposite.  In the hook
case, put the thresholds on one edge equal to \(L,H\), with
\(m=H-L\).  The other row has \(H,L\) at these vertices.
Along the remaining \(d-1\) edges of that other row, the
threshold rises sum to \(m-1\), the subtraction of one coming
from the period offset.  One of these other edges consequently
has positive difference at least \((m-1)/(d-1)\).

In the cohook case number the vertices so that the four
exceptional thresholds are
\[
 C_0=L,\quad C_1=H,\quad C_d=H-1,\quad C_{d+1}=L-1.
 \]
At every other pair, \(C_{i+d}=C_i-1\).
Put \(\Delta_i=C_i-C_{i-1}\).
For \(d\geq3\), the identity
\[
 \Delta_{d+2}+\sum_{i=3}^{d}\Delta_i=m-1
 \]
again supplies a different positive difference at least
\((m-1)/(d-1)\).
Choose the original threshold large enough that this exceeds
the inclusion threshold.  The new edge is incident to a
different pair of row-exchange vertex pairs.  Its charge
change cannot cancel the original asymmetry, which is supported
on exactly the original two pairs.  Its target is therefore
nonsymmetric.
\end{proof}

\begin{lemma}\label{run:symmetric}
For a symmetric core and \(d>1\), simultaneous swaps on two
opposite large edges have coefficient \((2m)!\) on ordered
labels.  The output bijection commutes with row exchange and
preserves equal-row labels.
After factorial normalization, the composite of any sequence
of these inductions has bounded rational entries and bounded
denominators.  Its image is the subspace in which coordinates
on each affected split pair agree, with the full coordinates
retained on unaffected packets.
\end{lemma}

\begin{proof}
Symmetry makes the two edge differences equal.  The flow proof
of Lemma~\ref{run:ordered-swap} gives the unique simultaneous
flow and the \((2m)!\) possible orders.  Both the operation and
its inverse commute with row exchange, so they preserve fixed
points as well as two-element orbits.

Use a basis consisting of the nonsplit labels and the sums of
split pairs.  Lemma~\ref{run:folding} says that normalized
induction is the resulting bijection on this basis, except
for its factor of two at rank zero.  For an individual split
source its image is half the image of the sum, because the
difference is annihilated as soon as a new index is added.
At each subsequent step the sum, rather than either individual
label, is transported bijectively.  Thus the factor \(1/2\)
occurs only once.  Rank zero is passed at most once in an
induction chain.  These observations also prove that the image
is the entire indicated subspace.  Tensoring over the bounded
number of packets preserves the bounded-entry and
bounded-denominator assertions.  Packets on which no such
induction occurs retain their full space.
\end{proof}

\subsection{The two exceptional cohook moves}

Only cohook \(d=1,2\) remain.  Their normalized matrices need not
be permutations, but the errors form convergent series.  For
\(d=1\), repeated moves of the same bead give an exponentially
small extra contribution.  For \(d=2\), a two-edge path avoids
the symmetric core and gives a quadratic error bound.  Both
estimates will control products of arbitrarily many moves.

\begin{lemma}\label{run:d-one}
For cohook \(d=1\) and sufficiently large positive difference
\(m\), both endpoints of the swap are nonsymmetric.
After division by \(m!\), its unordered matrix is \(P+E_m\),
where \(P\) is the ordered-swap permutation and, for constants
depending only on the weight,
\[
 \|E_m\|\leq K\,2^{-m/2}.
 \]
The next positive difference is \(m-2\).
\end{lemma}

\begin{proof}
Write \(C_0=L,C_1=H,C_2=L-2\), with \(m=H-L\).
The ordered target is \((H,L)\), whereas its exchange is
\((L+1,H-1)\).  A path of length \(m\) to the latter has
exactly \((m+1)/2\) moves on \(1\to0\) and \((m-1)/2\)
on the other edge, by its charge equations.  If these are not
integers, there is no such path.

Below \(L-W-2\) both runners are full.  No bead in this region
can ever move, since a move goes downwards and would require
a gap below it.  There are at most \(K_0=K_0(W)\) beads
originally on the low runner above this region.  Every move
on the other edge is therefore a repeated move of a bead
already moved earlier, except possibly for these \(K_0\)
beads.  At least \((m-1)/2-K_0\) moves repeat a bead.

Fix the two oriented endpoint labels.  Within each physical
row beads cannot pass each other: all these moves have
physical length one.  Matching them in their order fixes
each bead's number \(a_t\) of moves.  The number of possible
paths is at most
\[
 \frac{m!}{\prod_t a_t!}.
 \]
For \(a\geq1\), \(a!\geq2^{a-1}\).  Hence
\(\prod_ta_t!\geq2^{(m-1)/2-K_0}\).
This bounds every extra normalized entry exponentially.
The endpoint dimensions are bounded by
Lemma~\ref{run:defect-bounds}, giving the claimed operator
bound.

The source is symmetric only for \(m=-1\), and the swapped
target only for \(m=1\), so neither large endpoint is symmetric.
After the swap the other edge has
positive difference \(H-(L+2)=m-2\), as claimed.
\end{proof}

\begin{lemma}\label{run:d-two}
In cohook \(d=2\), suppose a large swap from a nonsymmetric
core would give a symmetric core.  Number its thresholds as
\[
 [C_0,C_1,C_2,C_3]=[L,H,H-1,L-1],\qquad m=H-L.
 \]
There is a projected extraction of length \(2m-1\) to the
nonsymmetric core
\([H,H-1,L,L-1]\).
After division by \(m!(m-1)!\), its matrix has the form
\[
 (1-1/m)P+E_m,\qquad
 \|E_m\|\leq \frac{K(W)}{m(m-1)},
 \]
where \(P\) is a bijection preserving total runner weight.
Reindexing the target from the old vertex \(3\) gives the
same exceptional pattern with parameter \(m-1\).
\end{lemma}

\begin{proof}
Use the ordered edge sequence
\begin{equation}\label{run:detour}
 (1\to0)^{m-1},\quad (2\to1),\quad
 (1\to0),\quad (2\to1)^{m-2},
\end{equation}
projecting onto its core after every individual step.
After \(u,v\) steps on the two edges the thresholds are
\[
 [L+u,H-u+v,H-1-v,L-1].
 \]
The symmetry equations are \(u+v=m\) and \(u-v=m\);
thus symmetry occurs exactly at \((u,v)=(m,0)\).
The path \eqref{run:detour} avoids it.

A single step between these nonsymmetric cores cannot reach
the opposite orientation.  Indeed adding its flow to its
row exchange leaves total length two, supported on that edge
and its opposite by the missing-edge argument.
Choosing the same edge for the exchanged target would make
the target symmetric; choosing the opposite edge would make
the source symmetric.  Both are excluded.  Thus the actual
unordered projections force exactly the ordered trajectory,
even when a particular label has equal row lengths.

The initial nesting gives \(B_0\subset B_1,B_2\).
Set
\[
 A=B_1\setminus B_0,\qquad C=B_2\setminus B_0,\qquad
 |A|=m,\quad |C|=m-1,\quad \rho=|C\setminus A|.
 \]
Here \(\rho\leq W\): every bead of \(B_2\setminus B_1\)
is either a hole below the threshold of \(B_1\) or an extra
bead above the threshold of \(B_2\), and these holes and
extra beads are bounded by the two runner weights.

The first segment transfers all of \(A\) except one choice
\(a\).  The next step transfers a choice
\(b\in C\setminus\{a\}\).
If the following step transfers \(a\), the last segment must
transfer all remaining elements of \(C\).  Its output is
the rotation
\[
 [B_0,B_1,B_2]\longmapsto[B_1,B_2,B_0].
 \]
For each such pair \((a,b)\), the first and last long
segments can be ordered in
\((m-1)!(m-2)!\) ways.  The number of pairs is
\(m(m-1)-|A\cap C|\).  The exact rotation coefficient is
therefore
\begin{equation}\label{run:rotation-count}
 m!(m-1)!
 \left(1-\frac{|A\cap C|}{m(m-1)}\right).
\end{equation}

The only alternative is to transfer \(b\) at the third
segment.  It can enter runner \(0\) only if \(b\notin A\).
Completing all \(m-2\) final transfers is then possible only
if \(a\notin C\); otherwise one fewer bead can enter runner
\(1\).  Conversely these two conditions suffice, and give
exactly the same number \((m-1)!(m-2)!\) of paths.
There are \(\rho(\rho+1)\) such pairs, since
\(|A\setminus C|=\rho+1\).
Also \(|A\cap C|=m-1-\rho\).  Thus the column \(\ell^1\)-error
from \((1-1/m)P\), after normalization, is exactly
\[
 \frac{\rho+\rho(\rho+1)}{m(m-1)}
   =\frac{\rho(\rho+2)}{m(m-1)}
   \leq\frac{W(W+2)}{m(m-1)}.
 \]
The bounded endpoint dimension gives the operator estimate.

The rotation is a bijection: at the target its low runner
is still nested in both high runners, so the inverse rotation
recovers every endpoint label.  It permutes runner
partitions and hence preserves their total weight.  Each
transfer uses the same physical row at its source and
target; physical row lengths are preserved.
Finally the period offset gives, on reindexing from \(3\),
\[
 [L-1,H-2,H-3,L-2],
 \]
which has parameter \(m-1\).  No bound on the dimensions of
the intermediate projections was used.
\end{proof}

\subsection{Termination and transfer of bounds}

All individual move matrices have now been calculated.  We must
choose a terminating sequence and control its cumulative effect:
bounded condition number handles the invertible stage, while a
separate argument handles the split-pair identifications.

Choose all large-difference thresholds in terms of \(p,W\),
larger than the inclusion thresholds above.  At a nonsymmetric
type \(D\) core use an exact swap when its target is nonsymmetric.
For \(d\geq3\), Lemma~\ref{run:nonsymmetric} replaces a forbidden
swap by a different large swap.  Stop once every positive
difference is below the resulting fixed threshold.
For cohook \(d=1\) use Lemma~\ref{run:d-one}; for cohook \(d=2\),
use exact swaps until stopping or first reaching the exceptional
pattern, then use Lemma~\ref{run:d-two} repeatedly.
All other ordered packets use Lemma~\ref{run:ordered-swap}.
Perform these full-space reductions before reducing symmetric
packets by the paired moves of Lemma~\ref{run:symmetric}.

Every move removes positive actual rank, so the process terminates.
This also proves termination when replacing a forbidden edge
by a different one.  The special cohook sequences have strictly
decreasing parameters \(m\), by two or by one, respectively.
There is at most one such sequence of each kind per packet:
once the exceptional pattern is reached its own rule continues
to the stopping threshold.  Exact moves between any other
stages have permutation matrices after normalization.

Number the basic-coordinate spaces at the starting collection,
the end of the full-space stage, and the final endpoint by
\(0,1,2\), and write \(n_j\) for their respective dimensions.
Reversing the extractions gives the normalized induction maps
\[
 \mathbb R^{n_2}\xrightarrow{\ A\ }\mathbb R^{n_1}
              \xrightarrow{\ T\ }\mathbb R^{n_0}.
\]
Bounds will return along these arrows.  The full-space map \(T\)
will be controlled by its condition number.  The map \(A\) can
annihilate split-pair differences.  Its bounded entries and
denominators, together with its image containing the vector of
ordinary character degrees, will instead control the PIM columns
at stage \(1\).

\begin{lemma}\label{run:condition}
The normalized ordinary-coordinate induction from the endpoint
of the full-space stage to its starting point is invertible,
with uniformly bounded condition number.
\end{lemma}

\begin{proof}
Normalize an exact move by its factorial.  Normalize a
cohook-\(1\) move in the same way, and a cohook-\(2\) move
by \(m!(m-1)!(1-1/m)\).
Every matrix is then a permutation plus an error of norm
\(\varepsilon_m\), where the sums of all errors are bounded
uniformly: they are bounded by a constant times
\(\sum_{m\geq m_*}2^{-m/2}\) or
\(\sum_{m\geq m_*}m^{-2}\), where \(m_*\) is the stopping cutoff.
Take \(m_*\) large enough that every \(\varepsilon_m<1/2\).
The singular values of a permutation plus that error lie
between \(1-\varepsilon_m\) and \(1+\varepsilon_m\).
Consequently the condition number of the product is at most
\[
 \prod_m\frac{1+\varepsilon_m}{1-\varepsilon_m}
 \leq \exp\!\left(4\sum_m\varepsilon_m\right).
 \]
The bounded number of packets gives a uniform bound.
\end{proof}

Here is the precise linear algebra that converts these maps to
bounds on projective characters.  It is useful that a scalar
normalization, however small, preserves projective positivity.

\begin{lemma}\label{run:cone-transfer}
Let \(V_-,V_+\) be nonnegative integral square invertible
matrices of bounded size, with
\(|\det V_+|\leq\Delta\), and suppose the entries of \(V_-\)
are bounded.

If an invertible map \(T\) of bounded condition number sends
the cone generated by the columns of \(V_-\) into the cone
generated by those of \(V_+\), the entries of \(V_+\) are bounded.

Alternatively let \(A\), possibly rectangular, have bounded
rational entries with a common bounded denominator.  Suppose
\[
 V_+^{-1} A V_-\geq0
 \]
entrywise, and the image of \(A\) contains a vector
\(V_+d\) with every coordinate of \(d\) strictly positive.
Then the entries of \(V_+\) are bounded.
\end{lemma}

\begin{proof}
Write \(\mathcal C_\pm\) for the two cones.
There are only finitely many possible bounded integral
matrices \(V_-\) in each of the bounded dimensions.
Thus the volume of \(\mathcal C_-\) in the unit ball has a
positive uniform lower bound.  Rescale \(T\) so its largest
singular value is one.  If its condition number is at most
\(\kappa\), its determinant in absolute value is at least
\(\kappa^{-n}\).  Its image of this part of
\(\mathcal C_-\) lies in \(\mathcal C_+\) and in the unit
ball.  The corresponding volume for \(\mathcal C_+\)
therefore also has a positive uniform lower bound.
Since all coordinates in the latter cone are nonnegative,
its cut by \(\sum x_i\leq1\) has a positive uniform
volume lower bound as well.

If \(s_j\) is the coordinate sum of column \(j\) of \(V_+\),
the linear change of variables \(x=V_+t\) gives exactly
\[
 \operatorname{vol}
 \{x\in\mathcal C_+:\textstyle\sum_i x_i\leq1\}
   =\frac{|\det V_+|}{n!\prod_j s_j}.
 \]
Every \(s_j\) is a positive integer.  The upper bound
\(\Delta\) for the numerator bounds their product and
hence each \(s_j\), proving the first assertion.

For the second, put \(M_0=V_+^{-1}AV_-\).
Because \(V_-\) is invertible, its image is
\(V_+^{-1}\operatorname{im}A\), which contains \(d\).
No row of \(M_0\) can therefore be zero.
Nonnegativity makes every entry of
\(\lambda=M_0\mathbf1\) positive.
If \(Q\) is a common denominator for \(A\), Cramer's rule
shows that the denominator of each entry of \(\lambda\)
divides \(Q\det V_+\).  In particular
\(\lambda_j\geq1/(Q\Delta)\).
The vector \(y=AV_-\mathbf1\) has bounded entries, and
\[
 y=V_+\lambda
 \]
is a nonnegative combination of every column of \(V_+\)
with these uniformly positive coefficients.  It bounds
each column entry, proving the claim.
\end{proof}

Apply the second assertion first, to transfer a bound from the
end of the symmetric stage to its beginning.  Its matrix \(A\)
is the one in Lemma~\ref{run:symmetric}.  Its image contains
the vector of ordinary degrees on the basic rows, since
those degrees agree on split signs by
Proposition~\ref{prop:label-degrees}.  This vector is the
projection of the regular character and equals
\(V_+d\), where \(d\) lists the positive dimensions of the
simple modules.  Thus all hypotheses of the second assertion
hold.  The bounded determinant is supplied by
Lemma~\ref{run:defect-bounds}; no bound on \(V_+^{-1}\)
is assumed.

Next apply the first assertion to the full-space stage, using
Lemma~\ref{run:condition}.  Projective positivity gives the
required inclusion of cones.  Thus a bound at the final
endpoint gives a bound at the original core collection.
Lemma~\ref{lem:row-projection} then gives the full
projective-character and Cartan bounds.  These arguments
require no equivalence of module categories associated to a
runner move.

\subsection{The endpoint and its triangle series}

At the stopping point, every positive edge difference is
bounded.  On each directed cycle the sum of differences is
its fixed period offset, with a minus sign.  A lower bound
for each difference follows by summing the upper bounds
for the others.  Thus all thresholds in a cycle have bounded
spread.

If the runner graph has a single cycle, fixed total charge
then bounds every threshold.  The packed configurations have
bounded rank, and the bounded runner weights give bounded
rank for every endpoint label.  This applies to \(\GL\),
to \(\GU\) with \(e\) odd, and to cohook symbols.
In the regular classical case the packets are all of the
same hook or cohook kind, because their \(q_i\)'s agree.
The accumulated frozen factors split off by
Lemma~\ref{fam:packets}.  The remaining root data belong to
a finite set by Lemma~\ref{fam:regular-datum}: restricting
the regular lattice after extraction retains its uniform
bounded-denominator coordinate description, now in bounded
dimension.  The geometric theorem applies to that finite
set, while every frozen block is of defect zero.

There are two remaining graphs.  If \(e=2d\) in \(\GU\),
the step-two graph has separate parity cycles.  In the hook
symbol case there is one cycle in each physical row.
Their mean thresholds can differ without bound.  This
difference records precisely the ordinary cuspidal triangle:
the \(2\)-core staircase of a unitary partition, or the
symbol obtained by packing its two rows separately at
step one, taken up to row exchange in type \(D\).

\begin{lemma}\label{run:triangle-endpoint}
At these two-cycle endpoints the number \(m_i\) of ordinary
split Harish--Chandra rank-one steps above the triangle in
each packet is uniformly bounded.
The complete collection at this triangle and this \(m_i\)
is a union of the core block projections of
Proposition~\ref{prop:core-blocks}.  It has boundedly many
basic rows and bounded block defects, independently of
the triangle ranks.
\end{lemma}

\begin{proof}
Consider one physical row in the hook case, or one parity
of positions in the unitary case.  Translate all its runner
thresholds by a common integer until one is zero.  This
translation does not affect the number of inversions for
the ordinary step, which is one or two respectively.
Bounded spread and Lemma~\ref{run:localization} put every
deviation from a fully packed row or parity inside an
interval of bounded length.  Outside that interval all
lower positions are occupied and all upper positions
empty.  Every inversion has both ends in the interval,
so their number is bounded by the square of its length.
Summing over the two rows or parities bounds \(m_i\).
Packing at the ordinary step is exactly removal to the
ordinary cuspidal triangle, so this inversion count is
its ordinary Harish--Chandra weight.

A same-row \(d\)-hook preserves the row lengths of a
symbol and hence its triangle.  An \(e\)-hook in the
unitary case has even length and preserves its \(2\)-core.
Thus a whole relevant core class lies in one triangle
series.  At the fixed ambient packet rank, \(m_i\) is
determined by that triangle; enlarging to all labels with
these data merges complete core classes and is therefore
still a projection onto a union of blocks.

The ordinary branching parametrizes these labels by
partitions or bipartitions of the bounded integers \(m_i\),
with the stated split signs in type \(D\).  Their number
is bounded independently of the triangle.
Every contributing relevant inversion is an ordinary-step
inversion, so there are at most \(m_i\), and its distance
in units of the relevant step is at most \(m_i\).
The already bounded basic cyclotomic \(p\)-part and lifting
the exponent bound all their degree defects.  The central
\(p\)-order is bounded as in
Lemma~\ref{run:defect-bounds}.  Applying the integral-basic-set
criterion once more bounds the defects of every block in
this enlarged collection.
\end{proof}

\begin{proof}[Proof of Proposition~\ref{prop:runner-reduction}]
Choose the core collection containing the basic rows of the
starting block.  It is a union of blocks by
Proposition~\ref{prop:core-blocks}.
Lemma~\ref{run:defect-bounds} handles total weight zero directly
and supplies all uniform defect and dimension bounds otherwise.
The terminating runner process and the two forms of
Lemma~\ref{run:cone-transfer} reduce the problem to the endpoints
just described.  A single-cycle endpoint is covered by
Theorem~\ref{thm:geometric-cartan} and the finite-root-data
argument above.

The remaining endpoint in Lemma~\ref{run:triangle-endpoint} is exactly
the endpoint of Proposition~\ref{prop:triangle-cartan}.
In the unitary case its notation is
\(Q_i=q^2\), \(d=\ord_p(Q_i)\), \(e=\ord_p(-q)=2d\);
in the hook classical case it is
\(Q_i=q_i\), \(d=\ord_p(Q_i)\) odd.
The frozen factors can either be retained in their
defect-zero blocks or removed using the Levi product.
The endpoint proposition supplies the remaining uniform
bound.  A Cartan bound bounds every projected decomposition
entry, since a Cartan diagonal entry is the sum of the
squares of the corresponding full column.
The two transfers proved above carry it back
through all runner moves, and
Lemma~\ref{lem:row-projection} supplies the Cartan bound.
For a central quotient we finally use
Lemma~\ref{lem:central-transfer}; the weight-zero case was
already treated directly in the quotient.
This proves Proposition~\ref{prop:runner-reduction}.
\end{proof}
==== END Donovans-Conjecture-over-Algebraic-Closures-of-Prime-Fields//05_runners.tex ====

==== BEGIN Donovans-Conjecture-over-Algebraic-Closures-of-Prime-Fields//06_endpoint.tex ====
\section{The cuspidal triangle endpoint}\label{sec:endpoint}

We complete the Cartan estimate at the endpoints of
Proposition~\ref{prop:runner-reduction}.  The ordinary cuspidal rank may
still be arbitrarily large there.  The number of split Harish--Chandra
steps above the cuspidal label is bounded, and this is the rank that will
enter the endomorphism algebras below.  We construct enough modular
cuspidal pairs to exhaust the simple modules of the endpoint blocks,
and then cover their projective indecomposable modules by bounded
projective inductions.

The use of intertwining operators and Hecke endomorphism algebras
follows the Harish--Chandra theory of Howlett--Lehrer
\cite{HowlettLehrer} and its modular developments
\cite{DipperFleischmann,DipperDu}.  We give the particular cuspidal,
extension and exhaustion arguments required here.

Throughout this section, $p$ is odd and different from the defining
characteristic $r$, and $k=\overline{\F}_p$.  We retain the regular
ambient groups, marked semisimple $p'$-parameter $s$, and actual Levi
product decompositions of the preceding sections.  Write
$G=\mathbf G^F$ and $\mathbf C=C_{\mathbf G^*}(s)$.  A packet means a
Frobenius orbit of classical factors of this connected centralizer on
which the labels vary.  We calculate on one representative factor
with the composite endomorphism around the orbit.  Index the packets
by $i$.  Their parameters satisfy
one of the following two conditions:
\begin{equation}\label{end:linear-primes}
\begin{array}{ll}
\text{orthogonal or symplectic:}&
 Q_i=q_i,\quad d_i=\ord_p(Q_i)\text{ is odd};\\
\text{unitary:}&
 Q_i=q^2,\quad d_i=\ord_p(Q_i),\quad \ord_p(-q)=2d_i.
\end{array}
\end{equation}
There is no assertion that $d_i$ is odd in the unitary case.
On each packet fix an ordinary cuspidal triangle: a separately packed
symbol in types $B,C,D$, or the staircase $t(t+1)/2$ in unitary type.
Let $m_i$ be the number of ordinary split Harish--Chandra steps above
it.  The collection $\mathcal T$ consists of all the ordinary rows at
$s$ with these triangle data and these $m_i$.  Frozen factors carry
their prescribed single cuspidal row and are left implicit.

For a type $D$ packet, zero difference between the two row lengths
means split type and an empty residual triangle.  We call this the
\emph{even-sign case}.  A nonempty type $D$ triangle has its unique
cuspidal label, also when its rank is one and its form is nonsplit.
At rank zero there is one label; in particular, the empty equal pair
is not doubled.  Equal nonempty pairs in split type $D$ retain their
two distinct labels.

\begin{proposition}[Triangle Cartan bound]\label{prop:triangle-cartan}
In the endpoint situation of Proposition~\ref{prop:runner-reduction},
the Cartan entries of the blocks whose basic rows belong to
$\mathcal T$ are bounded in terms of $p$ and the original defect-order
bound $M$.
More explicitly, the conclusion is uniform whenever the number of
packets, $\sum_i m_i$, the orders of the central $p$-subgroups retained
in the residual factors, and
$p^{v_p(Q_i^{d_i}-1)}$ for the packets with $m_i>0$ are bounded.
The bound is independent of the ranks of the triangles and of $q$.
\end{proposition}

The hypotheses listed in the second sentence are provided by
Proposition~\ref{prop:runner-reduction}.  That proposition and
Proposition~\ref{prop:core-blocks} also show that $\mathcal T$ is the
restriction of an integral basic set to a union of blocks of bounded
defect orders.  Both its cardinality and the number of ordinary rows
in this block union are bounded.  We will use these facts only at the
exhaustion and norm steps.  The modular Harish--Chandra calculation is
proved directly below.

\subsection{Residual and general linear cuspidal modules}

For each packet set
\begin{equation}\label{end:sizes}
 \mathcal B_i=\{1\}\cup\{d_i p^a:a\geq0\}.
\end{equation}
The sizes \(d_i p^a\) arise as Frobenius orbit degrees of regular
\(p\)-power eigenvalues over \(\F_{Q_i}\).  They will give ordinary
cuspidal lifts on the linear factors; size \(1\) also allows the
original \(p'\)-parameter with no added \(p\)-part.
This is a set: when $d_i=1$, size $1$ occurs only once.  Choose
nonnegative integers $r_{i,b}$ such that
\begin{equation}\label{end:size-sum}
 \sum_{b\in\mathcal B_i}b r_{i,b}=m_i.
\end{equation}
Partition the $m_i$ hyperbolic coordinates into these chunks.  In
$\mathbf C$ they define a split Levi with the residual triangle
factors and linear factors $\GL_b(Q_i)$.  Centralizing the same split
torus directions in $\mathbf G^*$ gives a rational split Levi
$\mathbf L^*$ with
\[
 C_{\mathbf L^*}(s)=\mathbf C_L.
\]
Write $L=\mathbf L^F$ for its dual Levi.  The actual product
description from Section~\ref{sec:families} identifies its varying
factors with a residual group $H_0$ and general linear groups
$\GL_{\delta_i b}(q)$ in the orthogonal and symplectic cases, where
$\delta_i\in\{1,2\}$ and $Q_i=q^{\delta_i}$, or $\GL_b(q^2)$ in
the unitary case.  On the former factor $s$ has one $p'$-eigenvalue
orbit $f_i$ of degree $\delta_i$, repeated $b$ times.  On the latter
factor it is the identity.

We record the split-center observation needed for these Levis.
The connected centralizer of the residual parameter has no split
central direction outside the ambient center.  A nonempty classical
triangle retains its full sign root system; the rank-one type $D$
exception is a nonsplit torus and has no such split direction either.
On a frozen linear factor with a regular torus parameter, the split
center of its centralizer is the split center of that ambient factor.
Consequently the split center of $\mathbf C_L$, modulo ambient
central directions, consists exactly of the chunk directions.  It
has the same centralizer as the split center of $\mathbf L^*$.
These assertions are statements about the rational cocharacter
spaces, so they are unaffected by the bounded central lattice
dressing.  All ensuing Levi conjugacies and Weyl positions are
rational.

\begin{lemma}[The residual module]\label{end:residual}
Let $\chi_0\in\Irr(H_0)$ have parameter $s$ and the indicated
cuspidal unipotent Jordan labels.  It is ordinary Harish--Chandra
cuspidal.  It is trivial on $Z_0=O_p(Z(H_0))$, and its character of
$H_0/Z_0$ has defect zero.  Its reduction $X_0$ is therefore simple
and Harish--Chandra cuspidal.  The block containing $X_0$ has bounded
Cartan data, and its only basic row at the residual parameter is
$\chi_0$.
\end{lemma}

\begin{proof}
Consider a proper rational split Levi of $H_0$ whose dual contains a
conjugate of the parameter.  Its intersection with the connected
centralizer is proper: otherwise its extra split central direction
would contradict the split-center observation.  Torus transitivity
and the ordinary Jordan torus pairing used in
Proposition~\ref{prop:single-cycle} identify the uniform projection
of the adjoint character there with the adjoint of the unipotent
cuspidal label.  The latter is zero.  Apply this separately to every
Levi parameter mapping to $s$.  In each such series the degree
vector belongs to the span of the torus tests, by the regular-character
formula.  The adjoint split Harish--Chandra character is an actual
character with nonnegative multiplicities; zero pairing with its
degree vector forces it to vanish.  This proves ordinary cuspidality.
No bound on the number or lengths of the torus tests is needed here.

The hook and cohook degree formula of
Proposition~\ref{prop:label-degrees} gives the defect assertion.
A separately packed triangle has no positive hooks.  Its cohook
factors have the form $Q_i^j+1$, whose $p$-parts are trivial because
$Q_i$ has odd order modulo the odd prime $p$.  The nonsplit rank-one
type $D$ torus contributes $Q_i+1$, also of order prime to $p$.
This accounts for the possible extra direction in the full residual
center: a rank-one orthogonal residual triangle is nonsplit, so
adjoining its torus does not enlarge the inherited central
$p$-subgroup.  The inherited-torus degree formula therefore leaves
the order of the full group $Z_0=O_p(Z(H_0))$.
Every hook length of a staircase is odd, whereas
$p\mid((-q)^j-1)$ requires $2d_i\mid j$.  Thus no noncentral
$p$-part remains in the character defect.  The order and degree
formulas for the regular ambient group leave precisely $|Z_0|$.
The central character attached to the $p'$-parameter is trivial on
$Z_0$, so passage to $H_0/Z_0$ is literal and gives defect zero.

A defect-zero ordinary character reduces to a simple projective
module.  Inflation from the central $p$-quotient gives the simple
module $X_0$, and the central transfer of
Lemma~\ref{lem:central-transfer} bounds its block's Cartan data in
terms of $|Z_0|$.  Exactness of split Harish--Chandra restriction,
which is averaging over groups of order prime to $p$, proves the
cuspidality of $X_0$.  Finally, every basic row in this block at the
specified $p'$-parameter is trivial on $Z_0$.  Downstairs it must lie
in the same one-row defect-zero block, and hence equals $\chi_0$.
\end{proof}

\begin{lemma}[Cuspidal modules on the chunks]\label{end:linear-cuspidals}
For every $b\in\mathcal B_i$ there is a simple cuspidal module
$Y_{i,b}$ on the corresponding actual general linear factor with
the following properties.
\begin{enumerate}
\item It is the reduction of an ordinary cuspidal character whose
semisimple parameter is the product of the prescribed $p'$-orbit
and a $p$-element regular of degree $b$ over $\F_{Q_i}$; for $b=1$
the added $p$-part may be trivial.
\item In the integral basic set of partition rows of the original
$p'$-series, its Brauer character is the signed power sum of degree
$b$.  In particular, it is independent of the choice of regular
$p$-part.
\item The ordinary partition row corresponding to the Steinberg
character on the unipotent side, called the regular extreme,
contains $Y_{i,b}$ with multiplicity one in its reduction.
This extreme is stable
under the packet transports and orientation reversals.
\end{enumerate}
\end{lemma}

\begin{proof}
Put $d=d_i$, $Q=Q_i$, and $c=v_p(Q^d-1)$.  Since $p$ is odd,
lifting the exponent gives
\[
 \ord_{p^{c+a}}(Q)=dp^a\qquad(a\geq0).
\]
For $b=dp^a>1$, an element $\mu$ of order $p^{c+a}$ therefore has
Frobenius orbit of length $b$ over $\F_Q$.  If the original orbit
has degree $\delta=2$, its product with $\mu$ has degree $2b$ over
$\F_q$.  Indeed its $p'$- and $p$-parts can be recovered separately,
so its orbit length is the least common multiple of their orbit
lengths.  The orbit length of $\mu$ over $\F_q$ is either $2b$, or
$b$ with $b$ odd, and either possibility gives least common multiple
$2b$ with $2$.  The case $\delta=1$ is immediate.  The resulting
torus parameter is in general position, so Green's ordinary
general linear theory gives its irreducible cuspidal character.
For $b=1$ use the ordinary cuspidal character of the original orbit.

Write $s_\lambda$ for the partition row with label $\lambda$, and
$p_b$ for the degree-$b$ power sum under Green's characteristic map.
The general linear torus rule and equality of torus characters on
$p$-regular elements give
\begin{equation}\label{end:power-sum}
 [\overline{\psi}_{i,b}]
   =\varepsilon_b p_b
   =\varepsilon_b\sum_{\lambda\vdash b}
       \chi^\lambda((b))[\overline{s_\lambda}],
 \qquad \varepsilon_b\in\{1,-1\}.
\end{equation}
Here $\varepsilon_b$ is the sign making the degree positive; in
degree one this is the single partition row.  The statements used
from Green's theory are the ordinary Coxeter-torus cuspidality,
this torus-character formula, and the identification of split
restriction with the symmetric-function coproduct~\cite{Green}.
For precise forms over the actual general linear factors, see
\cite[Property~(2.3a) and Lemma~2.3c]{BrundanDipperKleshchev}
for the power-sum identity on all $p$-regular classes, and
\cite[Equations~(2.2a), (2.3f) and Lemma~2.2d]{BrundanDipperKleshchev}
for induction and its adjoint restriction coproduct.

We give the irreducibility argument, since it avoids a modular
cuspidal classification.  Every composition factor $Y$ of
$\overline{\psi}_{i,b}$ has zero proper split restriction, by
exactness and ordinary cuspidality.  Express $[Y]$ integrally in
the partition basic set.  Its coproduct in every positive splitting
$b=b_1+b_2$ is zero, using the corresponding orbit-preserving Levi
and Green's restriction rule.  The primitive subspace of degree
$b$ in the rational symmetric-function Hopf algebra is $\Q p_b$:
indeed that algebra is the polynomial algebra on the primitive
generators $p_j$, and the reduced coproduct of a polynomial of
polynomial degree at least two is nonzero in characteristic zero.
The coefficient of an extreme Schur function in $p_b$ is $1$ or
$-1$, so $p_b$ is indivisible in the integral Schur lattice.  Thus
$[Y]=n_Y\varepsilon_b p_b$ for an integer $n_Y$.  Evaluation at the
identity shows $n_Y>0$.  Formula~\eqref{end:power-sum}, including
composition multiplicities, now gives $\sum_Y [\overline\psi_{i,b}:Y]
n_Y=1$.  There is exactly one factor, with multiplicity one.

For the extreme row, use the ordinary multiplicity-free
Gelfand--Graev formula for a general linear group
\cite[Theorem~2.5d(ii), (iv), (v)]{BrundanDipperKleshchev}.  Its
Gelfand--Graev character $\Gamma$ contains the regular Coxeter-torus
row $\psi_{i,b}$ once; among the partition rows at the original
parameter it contains only the regular extreme, namely the row
corresponding to Steinberg on the unipotent side, also once.
The Gelfand--Graev module is projective in characteristic $p$, since
it is induced from a linear character of a defining-characteristic
unipotent subgroup.  Let $P_Y$ be the projective cover of $Y_{i,b}$.
Brauer reciprocity and the irreducible reduction of $\psi_{i,b}$
show that $P_Y$ occurs once in this projective module.  Some basic
partition row has a nonzero decomposition number for $Y_{i,b}$,
because those rows span the Brauer lattice.  Nonnegativity then
forces that row to be the regular extreme and forces its
decomposition number to be one.  The extreme partition is carried
to the same extreme under all the relevant duality and field
transports.  This proves the last assertion.
\end{proof}

Form the simple cuspidal $L$-module
\begin{equation}\label{end:rho}
 \rho=X_0\otimes
       \bigotimes_{i,b}Y_{i,b}^{\otimes r_{i,b}},
\end{equation}
with the fixed cuspidal factors understood.  It has ordinary
cuspidal lifts $\psi^u$ with parameter $su$.  We may prescribe the
$p$-parts independently on the hyperbolic chunk coordinates of
$\mathbf C_L$.  In the orthogonal and symplectic ambient groups
the central and coordinate lattice gluing has only powers of $2$
as denominators and indices, hence is an isomorphism on $p$-power
torus data.  These prescriptions therefore give genuine rational
torus points.  Their projection to the residual factor can only
give a central $p$-power twist of $\chi_0$, which does not change
$X_0$.  The same construction in the unitary case is direct.

In normalizer notation below we always retain the algebraic Levi:
$N_G(\mathbf L,\rho)$ denotes the stabilizer of $\rho$ in
$N_{\mathbf G}(\mathbf L)^F$.  This convention is relevant in small
fields, where the abstract normalizer of the finite group $L$ need
not determine the algebraic Levi.

\begin{lemma}[Inertia]\label{end:inertia}
The stabilizer quotient $W_\rho=N_G(\mathbf L,\rho)/L$ is the relative
rational Weyl group of $\mathbf C_L$ in $\mathbf C$.  On a packet
outside the even-sign case its factor is
$\prod_b W(B_{r_{i,b}})$.  On an even-sign packet it is
\begin{equation}\label{end:total-parity}
 \left\{(w_b)_b\in\prod_b W(B_{r_{i,b}}):
          \sum_b \operatorname{flip}(w_b)=0\pmod2\right\},
\end{equation}
where $\operatorname{flip}$ is the parity of the number of changed
coordinate signs.  Different choices of the size multiplicities
$r_{i,b}$ give nonconjugate cuspidal pairs.
\end{lemma}

\begin{proof}
Relative normalizers on the group and dual group are identified by
their rational Weyl positions; Lang's theorem supplies rational
representatives.  An element preserving $\rho$ must preserve its
marked $p'$-series in $L$.  After adjustment by $\mathbf L^{*F}$,
its dual representative centralizes $s$ and hence belongs to
$\mathbf C^F$.  Conversely a rational normalizer of $\mathbf C_L$
in $\mathbf C$ preserves $\mathbf L^*$ by the split-center
description.  It preserves the residual cusp label and the power
sums in~\eqref{end:power-sum}; hence it fixes the Brauer character
of $\rho$.  The ordinary label transports here are exactly the
uniform and single-cycle transports of
Proposition~\ref{prop:single-cycle}.

In the hyperbolic coordinates, the resulting operations permute
chunks of equal size and reverse their orientations.  In an even
orthogonal factor a reversal of a chunk of size $b$ has coordinate
sign parity $b$.  All permitted sizes on that packet are odd,
since $d_i$ and $p$ are odd.  If the residual triangle is empty,
the condition is therefore the total parity in
\eqref{end:total-parity}.  It is a condition across all sizes, not
one condition for each size.  If the residual triangle is nonempty,
a single reversal can be compensated on it.  Its unique cusp label
is unchanged, including in the nonsplit rank-one case.  These
coordinate operations are Frobenius invariant and admit the
rational representatives just described.  No inner transporter in
the connected centralizer exchanges different eigenvalue packets.
Finally, the same transporter test preserves the multiset of chunk
sizes, proving nonconjugacy of the different displayed pairs.
\end{proof}

\subsection{Cuspidal induction and its heads}

The inducing modules and their inertia quotients are now explicit.
Before calculating the induction endomorphism algebras, we explain
what their simple types count.  Cuspidal Mackey theory identifies
them with the distinct simple modules in the heads of the induced
modules.  It also separates the heads attached to our nonconjugate
cuspidal pairs.

The Mackey calculation applies over $k$ or over a splitting field
of characteristic zero.
Write $R_L^G$ and ${}^*R_L^G$ for
split Harish--Chandra induction and restriction.  They are exact
biadjoint functors.  In the split Mackey formula, a double-parabolic
coset contributes restriction to an intersection Levi followed by
induction from it.  One obtains this formula by projecting the
parabolic intersection to its two Levi quotients and averaging
over the intersection unipotent groups.  Their orders are powers
of $r$ and hence invertible in either coefficient field.  With
cuspidal coefficients, proper intersection-Levi terms vanish.
Consequently
\begin{equation}\label{end:mackey}
 {}^*R_L^G R_L^G(\rho)
   \text{ is a direct sum of conjugate simple $L$-modules},
 \qquad
 \dim\End_G(R_L^G\rho)=|W_\rho|.
\end{equation}
Each normalizing Mackey position supplies a one-dimensional
intertwiner space.

\begin{lemma}[The head of cuspidal induction]\label{end:head}
Let $\rho$ be a cuspidal simple $L$-module and put
$V=R_L^G\rho$, $H=V/\rad V$.  Under the cuspidal Mackey
formula~\eqref{end:mackey}, the map
\[
 \End_G(V)\longrightarrow\End_G(H)
\]
is surjective with nilpotent kernel.  Its simple algebra
types therefore correspond to the distinct simple modules
in $H$.  Every simple quotient of $V$ is also a submodule
of $V$.  Heads arising from nonconjugate cuspidal pairs
have no common simple constituent.
\end{lemma}

\begin{proof}
Write $E={}^*R_L^G$, and let $\pi:V\to H$ be the quotient.
Exactness gives a surjection $E\pi:EV\to EH$.  The source
is semisimple by~\eqref{end:mackey}, so this surjection
splits.  Given $f\in\End_G(H)$, adjunction converts
$f\pi:V\to H$ into a map $\rho\to EH$, which lifts to
$\rho\to EV$.  Adjunction back supplies $g\in\End_G(V)$
with $\pi g=f\pi$.  This proves surjectivity.

Put $J=\rad(kG)$ and let $I$ be the kernel.  For $g\in I$,
$g(V)\subseteq JV$.  Since $g$ is $kG$-linear,
$g(J^aV)\subseteq J^{a+1}V$.  Thus a product of $n$
elements of $I$ maps $V$ into $J^nV$, which is zero for
large $n$.  The kernel is nilpotent.  Since $H$ is
semisimple, its endomorphism algebra is a product of full
matrix algebras, one per distinct simple constituent,
giving the first conclusion.

If $S$ is a simple quotient of $V$, then $ES$ is a
semisimple quotient of $EV$.  Adjunction provides
$\rho\hookrightarrow ES$, hence also $ES\twoheadrightarrow
\rho$ by semisimplicity.  The other adjunction gives a
nonzero map $S\to V$, which is an embedding.  If $S$
belongs to two such heads, composing a surjection from
one induced module with its embedding into the other
gives a nonzero homomorphism between the inductions.
The split Mackey formula and cuspidality force the two
pairs to be conjugate.  This proves disjointness.
\end{proof}

The pairs of Lemma~\ref{end:inertia} therefore give disjoint heads.
To prove that these heads exhaust the endpoint simples, we will
count their endomorphism-algebra types and show that the induced
modules are supported on the block union of $\mathcal T$.
The latter support assertion will be checked when the count is
complete.  We now determine the algebras whose simple types must
be counted.

\subsection{Chamber operators and removal of the scalar cocycle}

We calculate chamber operators and then extend the inducing module
to its inertia group to remove the scalar ambiguity in their
coefficient transports.  The operator calculation works over
either $k$ or a splitting field of characteristic zero.

Parabolics with Levi $\mathbf L$ are chambers on its split central
cocharacter space.  On the chunk coordinates their walls are among
$x_j=x_h$, $x_j=-x_h$, and $x_j=0$.  Add missing walls as dummy
walls so that the full signed-permutation arrangement is available.
Ambient central coordinates play no role.  If $U$ is the group of
rational points of a chamber's unipotent radical, put
\[
 e_U=|U|^{-1}\sum_{u\in U}u,
 \qquad
 \mathcal M_U=kG e_U\otimes_{kL}\rho.
\]
Choose a square root of $r$ and use its powers for all the following
normalizations.  Right multiplication by $e_V$ defines the
normalized change map
\begin{equation}\label{end:average}
 I_{U,V}:\mathcal M_U\longrightarrow\mathcal M_V,
 \qquad
 x e_U\otimes v\longmapsto
 \left(\frac{|V|}{|V\cap U|}\right)^{1/2}x e_U e_V\otimes v.
\end{equation}
It is well defined because $L$ normalizes both radicals.  The same
formula is used in characteristic zero.

\begin{lemma}[Averaging relations]\label{end:averaging}
Normalized changes along a minimal full gallery compose to the
direct map~\eqref{end:average}.  At an adjacent wall the reverse
composition is invertible.  If no inertial reflection fixes that
wall it is the identity.  Otherwise, after completing the change
by an involutively normalized coefficient transport for the
reflection $a$, the resulting endomorphism satisfies
\begin{equation}\label{end:quadratic}
 T_a^2=1+\alpha_a T_a.
\end{equation}
In particular $T_a^{-1}=T_a-\alpha_a$.
\end{lemma}

\begin{proof}
Let $V$ be an intermediate radical on a minimal gallery from $U$
to $U'$.  A root cannot cross its hyperplane twice on that gallery.
Thus the roots positive in $V$ are covered by those positive also
in $U$ or also in $U'$.  Set $A=V\cap U$ and $B=V\cap U'$.  Root
group order counting, including the overlap, gives
$|A||B|/|A\cap B|=|V|$.  Since $A,B\leq V$, this proves $AB=V$
as sets and $e_V=e_Ae_B$.  It follows that
\[
 e_Ue_Ve_{U'}=e_Ue_Ae_Be_{U'}=e_Ue_{U'}.
\]
This uses absorption of $e_A$ on the left and of $e_B$ on the
right, not commutation of arbitrary averaging idempotents.
The separating root sets are disjoint along a minimal gallery,
so the exponents in the square-root normalizations add.  This
proves the first assertion.

At a wall, first remove the common outer radical by transitivity.
Inside the Levi centralizing a generic point of the wall, the two
remaining radicals are opposite.  The coefficient of the identity
Mackey position in their reverse composition is $1$: an element of
the opposite radical contributes back to the identity parabolic
coset only if it is the identity, and the reciprocal radical order
is cancelled by the two normalizing factors.  If the wall has no
inertial reflection, the identity is the only surviving Mackey
position and gives the asserted inverse.

In the other case choose a representative $n_a$ and a coefficient
intertwiner whose square is the action of $n_a^2\in L$.  Such a
normalization is possible over a splitting algebraically closed
field by rescaling the intertwiner.  It is chosen for each reflection
separately and does not assume an extension to the full inertia group.
The square of the coefficient transport cancels right translation
by $n_a^{-2}$.  Right translation together
with that intertwiner completes the change to $T_a$.  Coefficient
transports commute covariantly with uncompleted averages.  Its
square is therefore the reverse composition.  Rank-one Mackey
has only the identity and reflection positions.  The reflection
position is nonzero: at its representative, the contributing
terms are precisely the intersection unipotents and carry the
same intertwiner.  The identity
$U\cap{}^{n_a}(LU)=U\cap{}^{n_a}U$, obtained from the root groups
of the common Levi, verifies this directly.  The normalized
identity coefficient is $1$, proving~\eqref{end:quadratic}.
\end{proof}

Let $W_{\rm r}$ be the reflection subgroup of $W_\rho$.  It is a
product of the indicated type $B$ groups, with
$\prod_b W(D_{r_{i,b}})$ on an even-sign packet.  Use
$W(D_1)=1$, $W(D_2)=C_2\times C_2$, and $W(D_0)=1$.
The subgroup $W_{\rm c}$ preserving a chosen chamber of
$W_{\rm r}$ is a complement.  On an even-sign packet it consists
of even products of the extra single-coordinate flips for the
nonempty size factors.  In particular
$W_\rho=W_{\rm r}\rtimes W_{\rm c}$.

\begin{lemma}[The presentation with strict transports]
\label{end:presentation}
Suppose the inducing simple module extends to its inertia group.
Then $\End_G(R_L^G\rho)$, or its opposite according to the action
convention, has the Hecke presentation of $W_{\rm r}$ with the
quadratic relations~\eqref{end:quadratic}, together with the
ordinary complement $W_{\rm c}$ acting on those generators by
conjugation.  It has a basis indexed by $W_\rho$.  The scalars
$\alpha_a$ agree for simple reflections joined by an odd Coxeter
bond and for generators conjugate under $W_{\rm c}$.
\end{lemma}

\begin{proof}
The extension supplies coefficient transports satisfying the group
law, with the prescribed Levi action on inner representatives.
For a gallery which crosses each reflecting hyperplane of
$W_{\rm r}$ at most once, use the gallery word theorem in the full
signed arrangement.  Braid moves preserve the change operator by
the minimal-gallery identity in Lemma~\ref{end:averaging}.  A
deletion cannot concern a repeated reflecting hyperplane, so it
is a pair of inverse changes at a noninertial wall.  Thus the
completed operator for a reduced gallery in the reflection
arrangement is its direct completed average, with no scalar
ambiguity.  This proves the braid and length-additive relations.
To compute a simple reflection at the base chamber, move through
noninertial walls to a chamber adjacent to its reflected chamber.
The changes used are invertible; conjugating the rank-one
calculation back gives~\eqref{end:quadratic}.

The direct averages also give the conjugation action of an
element preserving the chamber.  The operators so
obtained have single, distinct nonzero Mackey supports indexed by
$W_\rho$.  They are linearly independent and, by
\eqref{end:mackey}, span the endomorphism algebra.  Finally the
braid relations and invertibility identify the quadratics on
conjugate simple generators, proving the parameter assertions.
\end{proof}

\begin{lemma}[Extension of the inducing module]\label{end:extension}
Every module $\rho$ in~\eqref{end:rho} extends to $N_G(\mathbf L,\rho)$.
\end{lemma}

\begin{proof}
We construct the extension in three stages.  At the fine Levi,
an ordinary multiplicity-one constituent removes the scalar
cocycle.  We then obtain an invariant ordinary constituent for
the coarse Levi, whose multiplicity-one reduction supplies the
required modular extension.

Refine $\mathbf L$ to a Levi $\mathbf L'$ in which every chunk
has size one.  Let $\psi'$ be the ordinary cuspidal row at $s$
there, using $\chi_0$ and the size-one orbit rows.  Its inertia
quotient is a product of type $B$ groups and, on even-sign
packets, one type $D$ group; it is a reflection group without the
extra complement between sizes.  Iterating the positive split
single-cycle rule of Proposition~\ref{prop:single-cycle} identifies
its ordinary induction matrix with that of the ordinary
unipotent triangle series in $\mathbf C$.  The constituent
corresponding to the trivial relative Weyl character has
multiplicity one.  This is an ordinary character assertion, prior
to any modular endomorphism calculation.

Work first over an algebraically closed characteristic-zero
splitting field.  The coefficient intertwiners of $\psi'$ define
a scalar central extension of its inertia quotient: pairs
consisting of a normalizer element and an intertwiner are divided
by the natural pairs supplied by $L'$.  Normalize the lift of
each simple reflection to be an involution in this extension.
For two such lifts $\widetilde a,\widetilde b$ with Coxeter bond
$h$, the scalar
\[
 c_{ab}=(\widetilde a\widetilde b)^h
\]
is conjugate to its inverse by $\widetilde a$.  Hence
$c_{ab}=c_{ab}^{-1}$ and $c_{ab}\in\{1,-1\}$.
It is also, up to inversion, the multiplier between the two
completed braid words.  Indeed their underlying averages agree
by Lemma~\ref{end:averaging}; moving the coefficient transports
to the end leaves precisely the two words in these lifts.

On the multiplicity-one constituent of $R_{L'}^G\psi'$, every
completed operator acts by a nonzero scalar.  For an even bond
the two braid words have the same number of each generator, so
their scalar values coincide and $c_{ab}=1$.  For an odd bond,
changing the sign of either generator changes the multiplier.
The graph consisting of odd bonds in a classical Coxeter diagram
is a forest.  Choose signs successively along each tree to make
all its multipliers $1$.  This does not affect even bonds.
The Coxeter presentation now gives a section of the scalar
extension and therefore an ordinary extension of $\psi'$ to
its full inertia group.  Empty factors and $D_1$ require no
generators; $D_2$ has just the even commutation bond.

We transfer this extension to the coarse Levi.  Choose an $F$-stable
parabolic $\mathbf P'_L$ of $\mathbf L$ with Levi $\mathbf L'$
by ordering the size-one subchunks inside each chunk, and put
$P'_L=(\mathbf P'_L)^F$.  Every element of $W_\rho$ can be
lifted to preserve $\mathbf L'$, $\psi'$, and $\mathbf P'_L$: match subchunks in
order for a positive transport and in reverse order with all
signs changed for an orientation reversal.  For $j<h$ the latter
operation sends the positive linear root $e_j-e_h$ to
$e_{b+1-h}-e_{b+1-j}$, still positive.  The total type $D$ parity
is preserved, or the same residual sign compensation is used.
These are rational Weyl operations in $\mathbf C$ and give the
primal normalizer operations through the corresponding split
cocharacter positions.

Let $A$ be the subgroup of $N_G(\mathbf L,\rho)$ consisting of
elements that normalize $\mathbf L'$ and $\mathbf P'_L$ and
stabilize $\psi'$.  The preceding lifts show that $A$ maps onto
$W_\rho$, and the algebraic normalizer identities give
\[
 A\cap L\leq
 \bigl(N_{\mathbf L}(\mathbf P'_L)
       \cap N_{\mathbf L}(\mathbf L')\bigr)^F=L'.
\]
The strict action of $A$ on $\psi'$ and
conjugation on the group-algebra factor extend $R_{L'}^L\psi'$
to $LA=N_G(\mathbf L,\rho)$.  On $A\cap L$ this action is the natural
inner action, so it agrees with the $L$-module structure.
The ordinary constituent $\theta$ obtained by choosing the
regular extreme of Lemma~\ref{end:linear-cuspidals} in every
chunk and $\chi_0$ on the residual factor occurs once, by
ordinary general linear branching.  It is invariant: matched
chunks carry the same extreme and the residual cusp label is
fixed.  Its entire isotypic summand consequently inherits the
extension.

\smallskip\noindent\emph{Passage to the modular module.}
The extension of the ordinary constituent is now available.
Its reduction contains $\rho$ once, by
Lemma~\ref{end:linear-cuspidals} and the tensor product
description.  Reduce a stable lattice of the extended ordinary
module and choose a simple composition factor whose restriction
contains $\rho$.  Restriction of a simple module to a finite
normal subgroup is semisimple: the translates of any simple
submodule form a semisimple invariant sum, which is the entire
module.  Modular Clifford theory therefore makes this
restriction a sum of conjugates of $\rho$ with a common
multiplicity.  Here $\rho$ is invariant, and its total
multiplicity in the original reduction is one.  The chosen
factor restricts to $\rho$ itself, and is the required extension.
\end{proof}

\subsection{Rank-one parameters and the number of simple modules}

The extension lemma makes the presentation an untwisted one.
We now calculate its parameters and count its simple types.
The count must retain the single parity condition across all
chunk sizes in the even-sign case; imposing parity separately
at each size would give the wrong endpoint count.

For $v\in k^\times$, let $\mathcal H_v(\mathfrak S_n)$ denote
the type $A$ Hecke algebra with $(T-v)(T+1)=0$.  Its quantum
characteristic is the least positive $e$ for which
$1+v+\cdots+v^{e-1}=0$, or infinity if there is no such $e$.
Over $k$ it is $\ord(v)$ when $v\ne1$ and $p$ when $v=1$.
Its simple modules are indexed by $e$-regular partitions, namely
partitions in which no part occurs $e$ or more times
\cite{DipperJames}; see also \cite[Section~5 and Theorem~5.1]{DuRui}.

\begin{lemma}[The necessary parameters]\label{end:parameters}
After coherent rescaling of generators, the long parameter on
chunks of size $b$ in packet $i$ is $v_{i,b}=Q_i^b\in k^\times$.
On a type $B$ factor the two roots $z_1,z_2$ of the short-generator
quadratic satisfy
\begin{equation}\label{end:separation}
 z_1/z_2\notin \langle v_{i,b}\rangle.
\end{equation}
For $b>1$ the unrescaled short generator satisfies $T_0^2=1$.
On an even-sign packet, adjoining the single-coordinate flips
gives type $B$ factors with short generators of square $1$ and
the same long parameters.
\end{lemma}

\begin{proof}
Compute in the rank-one enlargement of a reflecting wall.
For a transposition of two equal-size chunks, choose the extra
$p$-parts in an ordinary lift $\psi^u$ to match on these chunks.
For a negative transposition orient the two chunks consistently
first.  The merged actual general linear block then has
centralizer $\GL_2(Q_i^b)$ on the matching full orbit.  Its two
ordinary induced constituents are the two Green partition rows,
each with multiplicity one and with degree ratio $Q_i^b$, up to
inversion; see \cite[Equation~(2.3.7)]{BrundanDipperKleshchev}.
The remaining factors are unchanged.

For a short wall with $b=1$, take trivial added $p$-part on that
chunk.  The ordinary lift is invariant there.  The split
single-cycle rule, applied at $su$ with all other added parts on
inactive chunks, identifies the rank-one matrix with the first
ordinary unipotent sign step above the residual triangle.
By Proposition~\ref{prop:label-degrees}, its two degrees have
ratio $Q_i^h$ for an integer $h$ in the orthogonal and
symplectic cases, or $q^{2t+1}$ at a unitary staircase
$t(t+1)/2$, again up to inversion.  Sign compensation on a
residual type $D$ triangle fixes the same inducing row.

The ordinary rank-one coefficient representation extends across
the reflection after scalar normalization.  Its completed
operator has zero trace, since its support is the nonidentity
double coset: in the induced-coset decomposition every diagonal
block is zero.  On the two multiplicity-one constituents, let
its roots be $z_1,z_2$ and their degrees be $D_1,D_2$.  Then
$D_1z_1+D_2z_2=0$, so the ratio of the roots is the negative
degree ratio, up to inversion.  Relation~\eqref{end:quadratic}
also gives $z_1z_2=-1$.

Choose a lattice stable under the finite rank-one inertia group,
enlarging the coefficient field to contain the square root of
$r$ used in~\eqref{end:average}.  The averaging denominators and
normalization scalars are units, so the completed operator preserves
the induced lattice.  Its eigenvalues are integral and have product
$-1$, hence are units.  Their sum $\alpha_a$ is integral, so
$T_a^{-1}=T_a-\alpha_a$ also preserves the lattice.
The restriction of the coefficient lattice reduces
irreducibly to $\rho$, and its involutive transport reduces to
our chosen one up to sign.  Thus the same root ratios give the
modular quadratic, even if its two long roots coincide.

For a short wall with $b>1$, instead choose the ordinary lift
noninvariant at that wall.  In the orthogonal and symplectic
cases its regular $p$-element cannot be conjugate to its inverse:
already modulo $p$, $-1\notin\langle Q_i\rangle$, since the latter
group has odd order.  In the unitary case its paired transform
would require $-q\in\langle q^2\rangle$, which is excluded by
$\ord_p(-q)=2d_i$ and $\ord_p(q^2)=d_i$.
To check the marking, choose the dual wall representative
$n\in\mathbf C^F$, so $n(s)=s$.  Any conjugacy of $su$ and
$s n(u)$ in $\mathbf L^{*F}$ must match their $p'$-parts, forcing
the conjugator into $\mathbf C_L^F$.  Its active factor is
$\GL_b(Q_i)$; in a degree-two packet this is $\GL_b(q^2)$,
so eigenvalue conjugacy uses $q^2$-powers.  The preceding
exclusion therefore applies in that case as well.
The ordinary reverse composition of uncompleted
averages is therefore the identity by rank-one Mackey.  Reducing
it and using the involutive modular transport gives $T_0^2=1$.
Different walls may use different ordinary lifts; no simultaneous
extension of these lifts is required.

For $b=1$ the short-root ratio is $-Q_i^h$ in the orthogonal and
symplectic cases.  It is outside $\langle Q_i\rangle$ by odd
order.  In the unitary case the ratio is
$-q^{2t+1}=(-q)^{2t+1}$, outside the subgroup of even powers
$\langle q^2\rangle$.  For $b>1$, $d_i\mid b$, so
$v_{i,b}=1$ in $k$; the two short roots are opposite and distinct
because $p$ is odd.  This proves~\eqref{end:separation}.

The rescalings giving $(T-v_{i,b})(T+1)=0$ can be chosen on
conjugacy classes of simple generators by
Lemma~\ref{end:presentation}.  This includes the two ends of
$D_2$ when a complement interchanges them; otherwise they may
be rescaled independently.  Adjoining a single-coordinate flip
to a type $D$ factor gives its standard type $B$ presentation
with short generator of square $1$.  In the empty-residual case
this also follows directly from the coordinate signed
permutation presentation.  The convention is valid for $D_1=1$.
\end{proof}

We use the following specific separated-parameter theorem.  If
the cyclotomic Hecke algebra with type $A$ parameter $v$ has
two short parameters $z_1,z_2$ and
$v^a z_1-z_2$ is a unit for every integer $a$ with $-n<a<n$,
then it is Morita equivalent to
\begin{equation}\label{end:dm}
 \bigoplus_{a=0}^n
  \mathcal H_v(\mathfrak S_a)\otimes
  \mathcal H_v(\mathfrak S_{n-a}).
\end{equation}
This is the separated-parameter theorem of Du--Rui
\cite[Theorem~4.14]{DuRui}, also the two-singleton case of
\cite[Theorem~1.1 and Corollary~1.4]{DipperMathas}; its base
ring can be a field of positive characteristic.  Our parameters
are nonzero, and~\eqref{end:separation} verifies its hypothesis
for every integer $a$, uniformly in $n$.

\begin{lemma}[Counting the even subalgebra]\label{end:even-algebra}
For one even-sign packet with at least one chunk, the simple
modules of its endomorphism algebra are indexed by the following
rule.  For every size $b$ choose an ordered pair of
$e_b$-regular partitions of total size $r_{i,b}$, where
\begin{equation}\label{end:quantum-e}
 e_b=\begin{cases}
 d_i,&b=1\text{ and }d_i>1,\\
 p,&\text{otherwise}.
 \end{cases}
\end{equation}
Identify these choices under simultaneous exchange of the two
components for all sizes.  Each nonfixed orbit contributes one
simple, and each fixed choice contributes two.  If there are no
chunks, there is one simple.
\end{lemma}

\begin{proof}
Let $\mathcal B$ be the tensor product, over the nonempty sizes,
of the type $B$ Hecke algebras with short generator of square
$1$ from Lemma~\ref{end:parameters}.  Give every short generator
odd degree and every long generator even degree.  The algebra
of our packet is the total-even subalgebra $\mathcal B_0$.
Indeed its reflection subgroup is the product of the type $D$
groups; its complement consists of even products of the extra
flips, precisely as in~\eqref{end:total-parity}.

The separated equivalence~\eqref{end:dm} and the type $A$ simple
module theorem label the simples of $\mathcal B$ by the ordered
pairs in the statement.  Formula~\eqref{end:quantum-e} follows
because $Q_i$ has order $d_i$, while $Q_i^b=1$ for all the other
allowed sizes.  In particular parameter $1$ has quantum
characteristic $p$, not $1$.

The grading automorphism $\tau$ of $\mathcal B$ negates all
short generators and fixes all long generators.  On the
separated labels it exchanges the two components at every
size.  To check the action on their type $A$ labels as well as
on their multiplicities, use the affine presentation
\[
 X_1=T_0,\qquad X_{j+1}=v^{-1}T_jX_jT_j.
\]
The $X_j$ commute, and $\tau$ negates them while leaving $T_j$
unchanged.  Their generalized weights lie in the two disjoint
strings $z_1v^{\Z}$ and $z_2v^{\Z}$, here with $z_2=-z_1$.
The corner with all strands of each string grouped together
has the two type $A$ factors in~\eqref{end:dm}.  Negation
exchanges its two groups.

There is no additional involution on either type $A$ factor.
For completeness, the Bernstein intertwiners
\[
 \Phi_j=T_j-\frac{v-1}{1-X_j/X_{j+1}}
\]
interchange the $X_j$ weights and satisfy the braid relations
where the indicated differences are invertible.  These
relations follow by substitution in the affine Hecke
relations.  Between different strings they are invertible,
since both equal-weight and $v$-multiple-weight coincidences
are excluded.  The product which swaps the two grouped blocks
preserves the order inside each block.  The intertwiner braid
relations show that it conjugates a within-block $\Phi_j$ to
the corresponding within-block $\Phi_h$, and it carries the
displayed rational correction to the corresponding correction.
It therefore conjugates $T_j$ to $T_h$.  This calculation may
first be made in the Laurent localization.  After the
within-block corrections cancel, only differences between the
two strings have been inverted; the affine basis theorem
therefore gives the same identity in the separated corner.
Thus the two type $A$ modules are simply exchanged, with their
partition labels unchanged.

The action on the separated partition labels is now determined.
It remains to restrict from the full signed algebra to its
total-even subalgebra.
We have $\mathcal B=\mathcal B_0\oplus\mathcal B_0u$ for an
invertible odd short generator $u$ with $u^2=1$.  Restriction
of a simple $\mathcal B$-module to $\mathcal B_0$ is semisimple:
the sum of a simple submodule and its $u$-translate is a
nonzero $\mathcal B$-submodule.  The usual index-two Clifford
argument now applies, since $2$ is invertible in $k$.  A
simple and its distinct $\tau$-twist restrict to the same
simple module.  A simple fixed by $\tau$ has a normalized
twisting intertwiner of square $1$; its two eigenspaces give
two distinct simple restrictions.  These exhaust the simple
$\mathcal B_0$-modules by induction from $\mathcal B_0$.
This gives exactly the stated rule.  With no chunks there is
no odd unit and the algebra is just $k$, explaining the
separate convention.
\end{proof}

\begin{lemma}[Exact total count]\label{end:count}
Sum the numbers of simple endomorphism-algebra types over all
size choices~\eqref{end:size-sum}.  The result is $|\mathcal T|$.
\end{lemma}

\begin{proof}
For an integer $e\geq2$ put
\[
 R_e(x)=\prod_{j\geq1}(1+x^j+\cdots+x^{(e-1)j}),
 \qquad
 P(x)=\prod_{j\geq1}(1-x^j)^{-1}.
\]
The first series counts $e$-regular partitions, the second all
partitions.  If $d>1$, expansion of each part multiplicity
first modulo $d$ and then in base $p$ gives
\begin{equation}\label{end:digits}
 R_d(x)\prod_{a\geq0}R_p(x^{dp^a})=P(x).
\end{equation}
For $d=1$ the corresponding identity is
$\prod_{a\geq0}R_p(x^{p^a})=P(x)$.  These are identities of
formal series: at each degree only finitely many factors
matter, and the products telescope using
$R_e(x)=P(x)/P(x^e)$.  Equivalently, a partition $\lambda_b$
chosen at size $b$ contributes $b$ copies of each of its
parts to the combined partition.  Uniqueness of the digits
is precisely uniqueness of this decomposition.

For an ordinary signed packet, the two partition components
are independent.  Squaring~\eqref{end:digits} thus counts all
ordered bipartitions of $m_i$, exactly the ordinary type $B$
triangle labels.  The construction commutes with exchanging
the two components.  Hence on an even-sign packet the rule
of Lemma~\ref{end:even-algebra} counts unordered bipartitions,
with two labels on equal pairs, exactly the type $D$ rule.
For $m>0$, if $B(m)$ is the number of ordered bipartitions and
$F(m)$ the number of equal pairs, the count is
\[
 \frac{B(m)-F(m)}2+2F(m)=\frac{B(m)+3F(m)}2,
 \qquad
 F(m)=\begin{cases}P_{m/2},&m\text{ even},\\0,&m\text{ odd},\end{cases}
\]
where $P_a$ is the number of partitions of $a$.  At $m=0$
both sides of the representation-theoretic count use one
label, instead of applying this positive-rank formula.
In particular $D_1$ gives one label and $D_2$ gives four.
The latter also follows directly from its two rank-one
Hecke factors: their parameter has odd order, or is $1$ in
odd characteristic, and is therefore not $-1$.

If several sizes each have just one chunk, their reflection
groups $D_1$ are trivial but their total-even complement is
$C_2^{t-1}$ for $t$ sizes.  The simultaneous exchange rule
gives $2^{t-1}$ labels as required.  Thus the complement
has not been lost in any small-rank case.  Finally the
packet counts multiply, proving the lemma.
\end{proof}

\subsection{Exhaustion and the projective norm estimate}

The total endomorphism-algebra count is now $|\mathcal T|$.
By Lemma~\ref{end:head}, this is the number of distinct simple
modules in our disjoint heads.  We finish the exhaustion argument
by placing all these heads in the endpoint block union.  We then
bound their projective covers by inducing projectives from the
bounded chunks and the central-defect residual block.

\begin{corollary}[Exhaustion of the endpoint simples]
\label{end:exhaustion}
Every simple module in the block union of $\mathcal T$
is a quotient of $R_L^G\rho$ for one of the pairs
constructed in~\eqref{end:rho}.
\end{corollary}

\begin{proof}
The Brauer class of $\rho$ is an integral combination of
partition rows at the marked parameter of $L$, with
residual row $\chi_0$.  Every one of these ordinary rows
occurs in $R_{L'}^L\psi'$, by ordinary linear branching.
Transitivity and Proposition~\ref{prop:single-cycle}
therefore show that their inductions use only rows of
$\mathcal T$.  Thus $R_L^G\rho$ has all its composition
factors in the block union in question; the integral
combination is enough for this support assertion.

Lemmas~\ref{end:presentation}--\ref{end:count} count the
simple types of all the relevant endomorphism algebras.
Lemma~\ref{end:head} identifies this with the count for their heads.
These heads are disjoint for different size choices, by
Lemmas~\ref{end:inertia} and~\ref{end:head}.  The resulting total is
$|\mathcal T|$.  Because $\mathcal T$ is an integral
basic set of the entire block union, this is exactly its
number of simple modules.  The supported heads therefore
exhaust all of them.
\end{proof}

\begin{proof}[Proof of Proposition~\ref{prop:triangle-cartan}]
Let $S$ be one of the endpoint simples.  By
Corollary~\ref{end:exhaustion} it is a quotient of
$R_L^G\rho$.  Let $P_\rho$ be the projective cover of
$\rho$.  Exactness and preservation of projectives give
a projective module $R_L^G P_\rho$ mapping onto $S$.
After projection to the endpoint block union, it has the
projective cover $P_S$ as a direct summand.

We bound the ordinary coordinates of $P_\rho$ before
inducing.  There are at most $\sum_i m_i$ new chunks,
each of size at most this sum, so each actual new linear
factor has rank at most $2\sum_i m_i$.  Their Sylow
$p$-orders are bounded as well.  Explicitly, for
$e_q=\ord_p(q)$ and $c_q=v_p(q^{e_q}-1)$, lifting the
exponent gives
\begin{equation}\label{end:gl-order}
 v_p(|\GL_N(q)|)
 =c_q\left\lfloor\frac N{e_q}\right\rfloor
  +v_p\!\left(\left\lfloor\frac N{e_q}\right\rfloor!\right).
\end{equation}
For $Q_i=q$ or $q^2$, $c_q$ is controlled by
$v_p(Q_i^{d_i}-1)$; the possible factor $2$ does not
change the valuation since $p$ is odd.  For unitary
chunks apply the same formula over $Q_i=q^2$.
The geometric bound of Theorem~\ref{thm:geometric-cartan}
therefore bounds their Cartan entries, since both rank
and defect order are bounded.  Lemma~\ref{end:residual}
bounds the residual block; frozen factors have defect
zero.  The actual product decomposition and central
transfer consequently give a bound $C_0$ for the
diagonal Cartan entry of $P_\rho$.  Thus its ordinary
projective-character coefficients are at most
$\sqrt{C_0}$, and its squared ordinary norm is at most
$C_0$.  The number $N_0$ of its nonzero ordinary
coordinates is bounded, either by this integral norm
bound or by the bounded-defect row bound.

Only ordinary coordinates at the marked $p'$-parameter
of $L$ contribute to the $s$-projection of its induction.
Indeed an ordinary coordinate with nontrivial extra
$p$-part retains that nontrivial part under induction
and cannot belong to the series at $s$.
The selected Levi block union fixes the marked
$p'$-parameter class, so other Levi classes fusing to
$s$ are not included.  The residual coordinate is
$\chi_0$ by Lemma~\ref{end:residual}.

Every remaining row $\theta$ of $L$ occurs with positive
integer multiplicity in $R_{L'}^L\psi'$.  Positivity of
ordinary split induction and transitivity give the
coefficientwise inequality
\[
 R_L^G\theta\ \leq\ R_{L'}^G\psi'.
\]
The ordinary positive transfer of
Proposition~\ref{prop:single-cycle} identifies the
latter multiplicities with ordinary relative Weyl
multiplicities in the triangle series.  They are
bounded, for example, by
\[
 W_0=\prod_i 2^{m_i}m_i!.
\]
This estimate concerns coordinate multiplicities; it
does not bound ordinary character degrees.
It follows that every $\mathcal T$-coordinate of the
projective character of $R_L^G P_\rho$ is at most
$N_0\sqrt{C_0}W_0$.  Since $P_S$ is its projective
summand and all ordinary projective coefficients are
nonnegative, the same coordinate bound holds for
$P_S$.

Both the number of basic coordinates in $\mathcal T$
and the number of ordinary irreducible characters in
its block union are bounded, as are the block defect orders, by
Proposition~\ref{prop:runner-reduction}.  Apply
Lemma~\ref{lem:row-projection} to recover a uniform
bound for the full ordinary norm of $P_S$ from this
projected bound.  Its squared norm is its diagonal
Cartan entry, and Cauchy--Schwarz bounds every off-diagonal
entry by the corresponding diagonal bounds.  This
proves the proposition, including the all-zero-rank
case, in which the residual central-defect argument
alone applies.
\end{proof}
==== END Donovans-Conjecture-over-Algebraic-Closures-of-Prime-Fields//06_endpoint.tex ====

==== BEGIN Donovans-Conjecture-over-Algebraic-Closures-of-Prime-Fields//07_extensions.tex ====
\section{Extensions and field Morita finiteness}\label{sec:extensions}

Throughout this section, a bound is allowed to depend on the fixed prime
$p$ and the defect-order bound $M$, but on no ambient group order.  Put
$k=\overline{\F}_p$ and $\cO=W(k)$, and let $\sigma$ denote coefficient
Frobenius.  We use integral blocks at intermediate stages.  The conclusion
for arbitrary finite groups will be a conclusion over $k$.

The input from the preceding sections is the quasisimple Cartan
bound.  We first reduce a general block to a crossed extension
whose identity component has bounded defect and Cartan entries.
We then specify an equivariant Frobenius comparison and prove
that it makes these extensions finite up to Morita equivalence.
Section~\ref{sec:periodicity} will construct that comparison
without using the finite lists obtained here.

\subsection{Reduction to controlled crossed extensions}

Crossed products and algebraic-group actions enter Donovan's
conjecture through K\"ulshammer's reduction
\cite{Kulshammer1995}; Eisele developed its integral form
\cite{EiseleReduction}.  We retain both the outer action and the
unit-valued multiplication factors.  The additional issue here is
comparison through Sylow $p$-actions, beyond the prime-to-$p$
crossed-product finiteness of those reductions.

\begin{lemma}[Structure of a reduced pair]\label{lem:ext:reduced-structure}
Every block of a finite group with defect order at most $M$ is
$k$-linearly Morita equivalent to a block $B$ of a finite group $H$ with
the following properties.
\begin{enumerate}
\item Every block of a normal subgroup covered by $B$ is $H$-stable.
If $B$ covers a nilpotent block of $H_0\trianglelefteq H$, then
$H_0\leq Z(H)O_p(H)$.
\item Write
\[
 \begin{gathered}
 N=F^*(H)=PZE(H),\qquad E(H)=K_1\cdots K_j,\\
 P=O_p(H),\qquad Z=O_{p'}(H),
 \end{gathered}
\]
where the $K_i$ are the components, that is, the subnormal quasisimple
subgroups.  Then $Z\leq Z(H)$, $|P|\leq M$, and $j$ is bounded.  The
block $b$ of $N$ covered by $B$ has bounded defect and bounded Cartan
entries.
\item For $X=H/N$, the index $[X:O_{p'}(X)]$ is bounded.  The same
index bound holds, with a possibly different constant, for every
subgroup of every extension of $X$ by a $p'$-group.
\end{enumerate}
\end{lemma}

\begin{proof}
The general reduction in An--Eaton, Proposition~6.1
\cite{AnEaton}, gives the first assertion and preserves the defect
group.  Its statement has no restriction on the defect group.  Only
this preliminary reduction is used here.  If necessary it can be
performed over a sufficiently large modular system and then reduced
to $k$.  All subsequent integral blocks are the canonical lifts of
blocks of the resulting finite groups to $\cO$.

A block of the normal $p'$-subgroup $O_{p'}(H)$ is nilpotent, so the
first assertion makes that subgroup central.  Also $P$ is contained
in a defect group of $B$.  The displayed description of $N$ follows
from the definition of the generalized Fitting subgroup.  Its factors
commute except for their central intersections.

Let $b_i$ be a block of $K_i$ covered by $b$, with defect group $D_i$.
Normal block theory, applied also along a subnormal chain, gives
$|D_i|\leq M$.  No $D_i$ is central in $K_i$.  Indeed, all components
in the $H$-orbit of such a component would have central-defect blocks.
Their central product is normal in $H$, and its covered block has
central defect and is therefore nilpotent.  The first assertion would
put this nontrivial perfect group inside the soluble group $Z(H)P$,
which is impossible.

The multiplication map
\[
 P\times Z\times\prod_{i=1}^j K_i\longrightarrow N
\]
has central kernel.  Lift $b$ along this map, taking the trivial
character on the $p'$-part of the kernel.  Its lift is a tensor product
of the block of $P$, a linear-character block of $Z$, and the $b_i$.
The kernel identifies only central elements.  Consequently the
noncentral quotients $D_i/(D_i\cap Z(K_i))$ contribute independently
to a defect group of $b$.  Each has order at least $p$, and hence
$p^j\leq M$.  The lifted tensor block has defect order at most
$M^{j+1}$.  Theorem~\ref{thm:quasisimple-cartan}, the bound on $|P|$,
and the tensor-product formula bound its Cartan entries.  Descent
through the central kernel, using Lemma~\ref{lem:central-transfer},
then bounds the Cartan entries of $b$.

We next bound the quotient.  The generalized Fitting centralizer
theorem \cite[(31.13)]{Aschbacher} says that
$C_H(F^*(H))\leq F^*(H)$.  It embeds $X$ into the
group of outer actions on $P$ and the permuting outer actions on the
components.  To check the kernel, inner actions on $P$ and on each
component can be removed simultaneously by elements of their
commuting factors in $N$.  An element left in the kernel centralizes
$N$, since it also centralizes $Z$, and therefore belongs to $N$.
The standard outer automorphism descriptions of finite simple groups
give a normal soluble subgroup $X_0$ of bounded index and bounded
derived length in $X$: the component permutation group and $\Out(P)$
have bounded order.  For Lie types, the diagonal group is abelian,
the field group is cyclic, and the diagram group has bounded
derived length \cite[Theorem~3.4]{BMO}; for alternating groups the
outer group is of order at most two outside the single degree-six
exception \cite[Theorem~2.3 and Section~2.4.2]{Wilson}; and the sporadic list is
finite.  Thus the outer groups have uniformly bounded derived
length.  The same statements hold for their perfect
central covers, since an automorphism trivial on the simple quotient
and on inner automorphisms is trivial on the perfect cover.

Consider an abelian derived section $X_0^{(i)}/X_0^{(i+1)}$.  Its
$p$-primary subgroup lifts to a normal section $H_2/H_1$ of $H$,
above $N$, of $p$-power order.  All the relevant covered blocks are
stable.  The defect group of the covered block of $H_2$ surjects onto
$H_2/H_1$, by the normal block theorem for a $p$-power extension.
It has order at most $M$.  Thus the $p$-part of each derived section
has order at most $M$, and $|X|_p$ is bounded.

Set $Q=X/O_{p'}(X)$.  Then $O_{p'}(Q)=1$, so $F(Q)=O_p(Q)$ has
bounded order.  All nonabelian composition factors of $Q$, counted
with multiplicity, occur in a quotient of bounded order, namely
$X/X_0$; their number and orders are bounded.  Components of $Q$
are quotients of the universal central covers of these finitely many
simple groups.  Therefore $F^*(Q)$ has bounded order.  Applying the
generalized Fitting centralizer theorem to $Q$ shows that $Q$ has
bounded order as well: its conjugation homomorphism into
$\Aut(F^*(Q))$ has kernel contained in $F^*(Q)$.

Finally, if $\widetilde X\twoheadrightarrow X$ has $p'$-kernel, the
inverse image of $O_{p'}(X)$ is a normal $p'$-subgroup of bounded
index.  Its intersection with any subgroup of $\widetilde X$ proves
the last assertion.
\end{proof}

We next enlarge the components in a way which preserves both the
defect bound downstairs and the actual multiplication of the outer
actions.  The second requirement is essential for the later crossed
products.

\begin{lemma}[Compatible partial regular extensions]
\label{lem:ext:partial-extension}
In the notation of Lemma~\ref{lem:ext:reduced-structure}, there is an
extension $N\trianglelefteq\bar N$ with abelian $p'$-quotient
$A=\bar N/N$ having the following properties.
\begin{enumerate}
\item Choose $h_1=1$ and representatives $h_x$ for $x\in X$, and put
\begin{equation}\label{eq:ext:actual-factors}
 h_xh_y=n_{xy}h_{xy},\qquad n_{xy}\in N.
\end{equation}
Conjugation by $h_x$ on $N$ extends to an automorphism $\alpha_x$ of
$\bar N$, and
\begin{align}
 \alpha_x\alpha_y&=\operatorname{ad}(n_{xy})\alpha_{xy},
 \label{eq:ext:action-law}\\
 n_{xy}n_{xy,z}&=\alpha_x(n_{yz})n_{x,yz}.
 \label{eq:ext:factor-law}
\end{align}
Here $\operatorname{ad}(g)$ means conjugation by $g$; the factors are
the actual elements in \eqref{eq:ext:actual-factors}.
\item Every block $\bar b$ of $\bar N$ covering $b$ has bounded defect
and Cartan entries.
\item There is a central presentation
\[
 \bar N=\bigl(P\times Z\times\prod_i V_i\bigr)/D_0,
 \qquad
 N=\bigl(P\times Z\times\prod_i U_i\bigr)/D_0,
\]
where $U_i$ is the universal cover of the simple quotient of $K_i$.
Outside alternating types and a finite list, $U_i=\mathbf U_i^{F_i}$
is a standard simply connected group in characteristic different
from $p$.  It has a reductive enlargement with fixed-point group
\[
 J_i=V_i\times C_i,
\]
where $C_i$ is central of $p$-power order, $V_i/U_i$ is abelian of
$p'$-order, the derived algebraic group is simply connected, and
all dual semisimple $p'$-centralizers are connected.  The constructions
respect the component permutations and the actions in the first
assertion.  No bound on $|D_0|$ or $|C_i|$ is asserted.
\end{enumerate}
\end{lemma}

\begin{proof}
The universal central extension is functorial for perfect groups.
It therefore lifts the actions on the components and gives a central
surjection $P\times Z\times\prod_iU_i\to N$ with kernel $D_0$.
In defining characteristic, a component block with noncentral defect
has a Sylow defect group, apart from the bounded exceptional groups;
see the defining-characteristic block theorem \cite{HumphreysDefining}.
A bound on this Sylow order bounds both field degree and rank.
Thus these components, the sporadic components, and the exceptional
multiplier and automorphism cases form a finite list.  The remaining
nonalternating factors are the standard simply connected finite
groups in cross characteristic.

For one such factor write $U=\mathbf U^F$, with a pinned simply
connected simple algebraic group $\mathbf U$.  First suppose the
Frobenius has the usual form $F=\gamma\operatorname{Fr}_r^f$, with
$\gamma$ a diagram automorphism and $r\ne p$.  Let
$Z_{p'}=Z(\mathbf U)_{p'}$ be the prime-to-$p$ part of its finite
diagonalizable center, interpreted as a group scheme.  Take a split
torus $\mathbf T_0$ with one coordinate for each fundamental weight.
Restriction of these weights to $Z_{p'}$ embeds $Z_{p'}$ into
$\mathbf T_0$.  This embedding is equivariant for diagram
permutations and split Frobenius.  Define
\begin{equation}\label{eq:ext:partial-regular}
 \mathbf J=(\mathbf U\times\mathbf T_0)/
 \{(z,z^{-1}):z\in Z_{p'}\}.
\end{equation}
The construction includes nonreduced diagonalizable parts when the
defining characteristic divides the center order.  In particular it
is a construction on the pinned root datum, rather than merely on
abstract finite centers.

The original $\mathbf U$ embeds as $[\mathbf J,\mathbf J]$, and is
still simply connected.  As group schemes its center satisfies
\[
 Z(\mathbf J)\simeq Z(\mathbf U)_p\times\mathbf T_0.
\]
The center component group is therefore $p$-primary.  The obstruction
in the fundamental group to
connected semisimple centralizers on the dual side is $p$-primary.
The semisimple centralizer component theorem embeds the component
group in that obstruction and makes its exponent divide the order
of the semisimple element; see
\cite[Corollary~2.9]{BonnafeQuasiIsolated} for connected reductive
groups and \cite[Proposition~14.20]{MalleTesterman}.
It follows that a semisimple $p'$-element of $\mathbf J^*$ has
connected centralizer.

Put $J=\mathbf J^F$.  Lang--Steinberg's theorem
\cite[Theorem~21.7]{MalleTesterman} applied to the connected kernel
$\mathbf U$ identifies
\[
 J/U=(\mathbf T_0/Z_{p'})^F.
\]
The isogeny $\mathbf T_0\to\mathbf T_0/Z_{p'}$ has degree prime to
$p$.  On fixed points its kernel and cokernel have $p'$-order; for
the cokernel use the fixed-point exact sequence and $H^1(F,Z_{p'})$.
It therefore induces an isomorphism on $p$-primary subgroups.  The
$p$-primary subgroup of $\mathbf T_0^F$ gives a central subgroup
$C\leq J$ mapping isomorphically onto $(J/U)_p$ and disjoint from
$U$.  Let $V$ be the inverse image of $(J/U)_{p'}$.  Then
$J=V\times C$ and $V/U$ is abelian of $p'$-order.  These definitions
are invariant under every automorphism of the pinned construction.

For exceptional graph isogenies in types $B_2,F_4$ in characteristic
$2$, and $G_2$ in characteristic $3$, use the original group on
points and put $V=U$, $C=1$.  This includes the Suzuki and Ree
endomorphisms.  The center and the relevant fundamental obstruction
have trivial groups of geometric points in the only nontrivial
isogeny case, or are trivial; the same connected-centralizer
conclusion holds for semisimple elements.  For the finite-list and
alternating factors also put $V_i=U_i$.

We spell out compatibility of automorphisms.  For a standard
universal Lie group, the automorphism theorem
\cite[Definition~3.3 and Theorem~3.4]{BMO} gives the semidirect
product of the rational inner-diagonal group and the concrete
field-and-graph subgroup.  In the ordinary diagram case, write
$\mathcal E=\langle\operatorname{Fr}_r,\text{pinned diagrams}\rangle$
on geometric points.  The restriction of $C_{\mathcal E}(F)$ to
$U$ has kernel $\langle F\rangle$; see \cite[Section~2.2]{SpaethD}.
Thus its image is exactly the pinned field-and-diagram group with
that relation imposed.  The inner-diagonal group is represented
faithfully by $\mathbf U_{\rm ad}^F$.  It acts on
\eqref{eq:ext:partial-regular} by conjugation on $\mathbf U$ and
trivially on the torus.  The pinned operations act on $\mathbf U$
and on the fundamental-weight coordinates of $\mathbf T_0$ with
the same relations and the same conjugation action on the
inner-diagonal group.  The sole additional relation
$F=\gamma\operatorname{Fr}_r^f=1$ holds on $J$.  Consequently the
semidirect product action factors through the actual automorphism
group of $U$.  An inner automorphism implemented by $u\in U$ is
extended by conjugation by that same element.  Use identical
pinned models on isomorphic permuted components.  In the
exceptional graph-isogeny case no enlargement is made, so the
existing actions already have the required property.

Apply this construction factor by factor.  The old kernel $D_0$
remains central and is preserved, since the extended operations
agree with the old operations on all its elements.  Taking its
quotient gives $\bar N$ and an abelian $p'$-quotient $A$ over $N$.
The compatibility just proved makes the products of the extended
actions differ by conjugation by exactly $n_{xy}$, proving
\eqref{eq:ext:action-law}.  Equation~\eqref{eq:ext:factor-law}
already holds as an equality in $N$, by associativity in $H$, and
therefore still holds in $\bar N$.

Finally, $N\trianglelefteq\bar N$ has $p'$-index, so a block covering
$b$ has the same defect order.  Lemma~\ref{lem:abelian-transfer}
bounds its Cartan entries.  This argument takes place after
quotienting by $D_0$; it does not require bounded defect for a block
lifted to the universal covers or to the groups $J_i$.
\end{proof}

\begin{lemma}[Recovery by a dual action]\label{lem:ext:dual-recovery}
Let $\Xi=\Hom(A,k^\times)$ and $Y=\Xi\rtimes X$.  There is an
integral crossed algebra with identity component $\cO\bar N$ and
grading group $Y$ such that $\cO H$ is a full idempotent corner.
After taking the summand relevant to $B$, and then a further full
corner, one obtains a crossed algebra with identity component
$\bar B=\cO\bar N\bar b$ and grading group
$Y_0=\operatorname{Stab}_Y(\bar b)$.  Every such $\bar b$ covers $b$.  The groups
$Y_0/O_{p'}(Y_0)$, and hence their Sylow $p$-subgroups, have bounded
order.
\end{lemma}

\begin{proof}
For $\chi\in\Xi$, let its action on a group element $g\in\bar N$
be multiplication by $\chi(gN)$.  Lift these characters by
Teichm\"uller representatives over $\cO$.  Combine these actions
with the $\alpha_x$ of Lemma~\ref{lem:ext:partial-extension}; use
$n_{xy}$ as the factor for the $X$-coordinates and ordinary
semidirect transport on $\Xi$.  Equations~\eqref{eq:ext:action-law}
and \eqref{eq:ext:factor-law} give a crossed system, since every
character in $\Xi$ is trivial on all $n_{xy}\in N$.

Write $u_\chi$ for the homogeneous units corresponding to $\Xi$.
The element
\[
 e_\Xi=\frac1{|\Xi|}\sum_{\chi\in\Xi}u_\chi
\]
is an idempotent.  A group element in the $a$-component of the
$A$-grading of $\cO\bar N$ conjugates $e_\Xi$ to the character
idempotent of $\cO\Xi$ corresponding to $a$.  These idempotents,
as $a$ varies, sum to $1$, so $e_\Xi$ is full.  Moreover
$e_\Xi(\cO\bar N\rtimes\Xi)e_\Xi=\cO N e_\Xi$:
the average kills all nonidentity $A$-components.  The $X$-units
commute with $e_\Xi$ and retain their factors $n_{xy}$.  Its corner
in the entire crossed algebra is consequently $\cO H$.

Decompose the crossed algebra by $Y$-orbits on the blocks of
$\cO\bar N$.  Each orbit sum is fixed by $\Xi$ and so belongs to
$\cO N$.  The orbit summand meeting $B$ has nonzero product with
$b$.  At least one of its blocks covers $b$; every member does,
since twists are trivial on $N$ and the $H$-actions preserve $b$.
Within this orbit summand a single block idempotent $\bar b$ is
full, since its homogeneous conjugates sum to the orbit idempotent.
Its corner has precisely the degrees in $Y_0$, with invertible
homogeneous units $\bar b u_y$, and is the asserted crossed
algebra.  These constructions also hold after reduction to $k$.

The group $Y$ is an extension of $X$ by the $p'$-group $\Xi$.
The bound follows from Lemma~\ref{lem:ext:reduced-structure}.
\end{proof}

\subsection{The comparison input and integral base orders}

The reduction has bounded the defect and Cartan data of the
identity blocks, and bounded the grading groups modulo their
normal $p'$-subgroups.  The grading itself may still be large.
We now state the additional comparison needed to control its
actual crossed multiplication.

Here and below a crossed system on an algebra $R$ for a finite group
$T$ consists of automorphisms $\alpha_t$ and units $a_{s,t}$ such that
\begin{align*}
 \alpha_s\alpha_t&=\operatorname{ad}(a_{s,t})\alpha_{st},\\
 a_{s,t}a_{st,u}&=\alpha_s(a_{t,u})a_{s,tu},
\end{align*}
with the usual identity normalizations.  Its algebra is
$\bigoplus_{t\in T}Ru_t$, with $u_t r=\alpha_t(r)u_t$ and
$u_su_t=a_{s,t}u_{st}$.  An isomorphism is called \emph{based} if it
is the identity on the specified copy of $R$ and preserves each
group-labelled homogeneous component.  Such isomorphisms are
exactly the changes of homogeneous units.
For the crossed-system formalism, see also
\cite[Definition~4.1 and Lemma~4.3]{EiseleReduction}.

\begin{definition}[The comparison property]\label{def:ext:comparison-data}
For the data of Lemmas~\ref{lem:ext:partial-extension} and
\ref{lem:ext:dual-recovery}, let $S\leq Y_0$ be a $p$-subgroup
contained in the pure subgroup $X\leq\Xi\rtimes X$.  The comparison
property is the existence of a positive integer $m$, bounded in
terms of $p,M$, and an integral
$(\sigma^m\bar B,\bar B)$-Morita bimodule $\mathcal M$ with the
following additional structure on $M_0=k\otimes_{\cO}\mathcal M$.
There are invertible $k$-linear maps $J_x:M_0\to M_0$, $x\in S$,
such that, writing $\alpha'_x=\sigma^m(\alpha_x)$ and
$a'_{xy}=\sigma^m(a_{xy})$, one has
\begin{align}
 J_x(avb)&=\alpha'_x(a)J_x(v)\alpha_x(b),
 \label{eq:ext:comparison-covariance}\\
 J_xJ_y(v)&=a'_{xy}J_{xy}(v)a_{xy}^{-1},\qquad J_1=\id.
 \label{eq:ext:comparison-law}
\end{align}
Here $a_{xy}=n_{xy}\bar b$ and the target factor is its coefficient
Frobenius image, with the corresponding target block identity.
The property is required also for $S=1$.
\end{definition}

It is enough to produce these maps with a scalar error in
\eqref{eq:ext:comparison-law}.  Associativity and covariance then
make the errors a scalar $2$-cocycle on $S$.  Positive cohomology
of $S$ with coefficients in $k^\times$ vanishes: $|S|$ annihilates
it, whereas the $|S|$-power map of $k^\times$ is an automorphism.
Rescaling the $J_x$ removes the error.  An error in arbitrary units
of the center would not give this conclusion.

Every $p$-subgroup of $Y$ is conjugate by an element of $\Xi$ to a
subgroup of the pure $X$.  Indeed its image in $X$ is a $p$-group,
and Schur--Zassenhaus conjugates its complement to the standard
complement in the inverse image of that image.  Conjugating also
the block and the corner transports this assertion to every
$p$-subgroup of $Y_0$.  This conjugation is implemented by an
integral homogeneous unit in the algebra of
Lemma~\ref{lem:ext:dual-recovery}.

\begin{lemma}[Finite integral bases and finite Picard images]
\label{lem:ext:integral-bases}
Assume the comparison property for the trivial subgroup.  The
integral basic orders of all the blocks $\bar B$ above belong to a
finite list $\mathfrak R_1,\ldots,\mathfrak R_t$.  Each
$\mathfrak R_i$ is defined over some $W(\F_{p^{c_i}})$ and has finite
$\Pic_{\cO}(\mathfrak R_i)$.  Consequently the outer actions in
the reduced crossed basic corners take their values in a fixed
finite subgroup of $\Pic_k(R_i)$, where
$R_i=k\otimes_{\cO}\mathfrak R_i$.
\end{lemma}

\begin{proof}
The defect orders and Cartan entries of $\bar B$ are bounded.
The number of simple modules is bounded by the defect bound, so
the sum of its Cartan entries is bounded too.  For $S=1$ the
comparison property bounds the integral Morita--Frobenius number.
Eaton--Eisele--Livesey, Theorem~3.10 \cite{EEL}, says precisely
that bounded Cartan sum, bounded numerical defect $d$ (so the
defect group has order $p^d$), and bounded integral
Morita--Frobenius number give finitely many integral Morita classes.
Applying it to the finitely many possible numerical defects gives
the claimed finite list of basic orders.

Choose each representative as a basic corner of one actual
integral group block.  Its block and primitive idempotents modulo
$p$ are defined over a finite field containing their finitely many
coefficients and splitting the finitely many simple modules in
question.  Lift these idempotents over its Witt ring, which is
complete.  The resulting corner defines $\mathfrak R_i$ over
$W(\F_{p^{c_i}})$ for some $c_i$.

Eisele's Theorem~B and Corollary~1.2 \cite{EiselePicard} give the
finiteness of the integral Picard group of a block, and hence of
its basic order.  The hypotheses apply to $\cO=W(k)$: it is
unramified and has algebraically closed residue field, and the
generic algebra is separable.  The relevant rigidity condition can
also be checked directly.  The conjugacy-class decomposition of
the adjoint group module gives
\[
 \operatorname{HH}^1(\cO G)
   =\bigoplus_{[g]}H^1(C_G(g),\cO)=0,
\]
because each centralizer is finite and $\cO$ is torsion-free with
trivial action in the displayed cohomology.  Vanishing passes to
block summands and to basic orders by Morita invariance, as
required in Eisele's theorem.

Passing to basic corners preserves a crossed structure integrally;
see \cite[Proposition~4.15 and Corollary~4.16(2)]{EiseleReduction}.
We recall how the homogeneous units and their factors are obtained.
If $e$ is a basic idempotent in a block order, then any automorphism
sends $e$ to a unit-conjugate
idempotent: both associated projectives contain one copy of every
indecomposable projective type.  Multiplying each homogeneous unit
by a suitable identity-component unit makes it commute with $e$.
The corner is then crossed over $e\bar B e$.  After identifying
this order with $\mathfrak R_i$, its homogeneous automorphisms
give elements of $\Pic_{\cO}(\mathfrak R_i)$.  Their reductions
lie in the finite image of this group in $\Pic_k(R_i)$.  Since
$R_i$ is basic, $\Pic_k(R_i)$ is naturally its outer automorphism
group: an invertible bimodule is isomorphic to $R_i$ as a one-sided
module, and its other action specifies an automorphism up to
inner automorphism.
\end{proof}

\subsection{Based descent and the large kernel}

The trivial-subgroup comparison has supplied finitely many
integral base orders and finite images of their Picard groups.
We next use the full Sylow comparison to retain the chosen base
algebra while descending the crossed factors.  This stronger
form of finiteness is what permits removal of a large $p'$-kernel.

\begin{lemma}[Based Frobenius descent on $p$-subgroups]
\label{lem:ext:based-sylow}
Assume the comparison property of
Definition~\ref{def:ext:comparison-data}.  For each fixed integral
basic order $\mathfrak R_i$ and each group of bounded $p$-power
order, the restrictions of the reduced crossed basic corners to
that group have only finitely many based isomorphism classes.
The identification of the identity component with $R_i$ is part
of this assertion and may be chosen arbitrarily integrally.
\end{lemma}

\begin{proof}
The argument has three steps.  We align the chosen basic corners
with their Frobenius images, use the finite integral Picard group
to obtain the regular comparison bimodule, and then apply Lang's
theorem to the resulting based unit-change orbit over $k$.
Throughout, each group label and the specified identity algebra
are retained.

Fix $\mathfrak R=\mathfrak R_i$, $R=R_i$, and a descent degree $c$.
First conjugate the subgroup to a pure one as above, transporting
the block and the integral corner identification.  The comparison
property applies to the conjugated data.  It passes to the basic
corners.  Explicitly, if source and target homogeneous units are
changed by $t_x$ and $t'_x$, respectively, the transport is changed
to $v\mapsto t'_xJ_x(v)t_x^{-1}$.  Covariance shows that its
multiplication rule is exactly the rule with the new crossed
factors.  The maps then restrict to the corresponding bimodule
corner.  Choose the target corner and its coordinates by applying
$\sigma^m$ to the source choices.  Thus in these fixed coordinates
the target crossed data are the actual coefficient Frobenius
images, with the same group labels.

Tensor the comparison with its successive Frobenius twists.  Both
covariance and \eqref{eq:ext:comparison-law} tensor: the two middle
factors in the product law cancel in the balanced tensor product.
After at most $c$ repetitions its exponent $n$ is divisible by
$c$.  The descent model now identifies
$\sigma^n\mathfrak R=\mathfrak R$, so its integral bimodule class
is an element $P$ of the finite group
$\mathcal P=\Pic_{\cO}(\mathfrak R)$.  Let $\theta$ be the
permutation of $\mathcal P$ induced by $\sigma^n$ and this descent
identification.  Further tensor repetitions multiply
$P,\theta(P),\theta^2(P),\ldots$.  The element $(P,\theta)$ of
the finite semidirect product
$\mathcal P\rtimes\langle\theta\rangle$ has finite order, bounded
in terms of $|\mathcal P|$.  After that many repetitions the
integral bimodule class is the identity.  The resulting total
exponent $N$ is bounded in terms of the fixed finite list and the
original comparison bound, and remains divisible by $c$.

Identify the reduced comparison bimodule with the regular
$(R,R)$-bimodule.  Put $v_x=J_x(1)$.  Covariance gives
\[
 J_x(r)=\alpha'_x(r)v_x=v_x\alpha_x(r).
\]
Since $J_x$ is invertible, multiplication by $v_x$ is invertible,
and $v_x\in R^\times$.  The product rule gives
\begin{align}
 \alpha'_x&=\operatorname{ad}(v_x)\alpha_x,
 \label{eq:ext:gauge-action}\\
 a'_{xy}&=v_x\alpha_x(v_y)a_{xy}v_{xy}^{-1}.
 \label{eq:ext:gauge-factors}
\end{align}
Thus the crossed system and its $p^N$-coefficient Frobenius image
are in the same orbit under changes of homogeneous units,
identically on $R$.

The comparison has now become a change of homogeneous units,
so it fixes the specified copy of $R$ pointwise.  To finish, we
descend this based orbit to a bounded finite field.
For fixed $R$ and $S$, all crossed systems form an affine variety
of finite type: take the automorphism matrices of $R$, the
coordinates of $a_{xy}\in R^\times$, and impose the two crossed
identities.  Invertibility can be expressed by adjoining inverse
determinants.  This variety is defined over $\F_{p^c}$.  The
unit-change group is
\[
 \mathcal U=(R^\times)^{S\setminus\{1\}},
\]
acting by \eqref{eq:ext:gauge-action}--\eqref{eq:ext:gauge-factors}.
It is connected: $R^\times$ is a nonempty open subset of the
vector space $R$.  Write $F$ for $p^N$-coefficient Frobenius.  If
$F(d)=g\cdot d$, Lang's theorem \cite{Lang1956} supplies
$h\in\mathcal U$ with $h^{-1}F(h)=g^{-1}$; hence $h\cdot d$ is
$F$-fixed.  The orbit therefore has a representative over
$\F_{p^N}$.  There are only finitely many such points for a fixed
$N$.  There are only finitely many possible bounded $N$ and
bounded-order groups $S$.  This proves based finiteness.  Transport
back through the initial conjugation gives the same conclusion
for the original restrictions.
\end{proof}

\begin{lemma}[Removing a large $p'$-kernel]\label{lem:ext:kernel-removal}
Fix an indecomposable finite-dimensional basic $k$-algebra $R$ and
a finite subgroup $\mathcal F\leq\Pic_k(R)$.  Consider crossed
algebras on $R$ by groups $T$ such that
\begin{enumerate}
\item $[T:O_{p'}(T)]$ is bounded;
\item their outer actions take values in $\mathcal F$; and
\item their restrictions to $p$-subgroups have finitely many based
isomorphism classes, for each possible abstract subgroup.
\end{enumerate}
Then their block summands have finitely many $k$-linear Morita
equivalence classes.
\end{lemma}

\begin{proof}
Let $\mathcal T$ be one of these crossed algebras and set
\[
 K=O_{p'}(T)\cap\ker(T\longrightarrow\Pic_k(R)).
\]
The index $[T:K]$ is bounded.  Change homogeneous units in the
$K$-degrees so that they centralize $R$; their factors now belong
to $Z(R)^\times$.  Since $R$ is indecomposable, its artinian center
is local, with residue field $k$, and
\begin{equation}\label{eq:ext:center-units}
 Z(R)^\times=k^\times\times U_R,
 \qquad U_R=1+\rad Z(R).
\end{equation}
This is a canonical decomposition into scalars and principal
units.  The abelian group $U_R$ has bounded $p$-power exponent:
if $(\rad Z(R))^{p^e}=0$, then $(1+z)^{p^e}=1$ for every
$z\in\rad Z(R)$.  Multiplication by $|K|$ is invertible on $U_R$,
so $H^a(K,U_R)=0$ for $a>0$.  A further central change of units
therefore makes all $K$-factors scalar.

Let $v_k$ denote these units.  Their $k$-span is a scalar twisted
group algebra $E$ of $K$, and the subalgebra on the $K$-degrees is
$R\otimes_k E$.  The algebra $E$ is semisimple by the twisted
Maschke theorem, since $p\nmid|K|$.  Every homogeneous conjugation
normalizes $E$.  Indeed it takes $v_k$ to
$z_kv_{tkt^{-1}}$, with $z_k\in Z(R)^\times$.  Comparing products
of these elements shows that the principal-unit components of
$z_k$ define a homomorphism $K\to U_R$: all original and
conjugated factors are scalar.  Such a homomorphism is trivial,
because $K$ is a $p'$-group and $U_R$ has $p$-power exponent.
Thus all $z_k$ are scalar, as required.

Write $E=\prod_j E_j$, with $E_j$ full matrix algebras over $k$,
and let $e_j$ be their identity idempotents.  Orbit sums under $T$
are central in $\mathcal T$.  In each such summand a single $e_j$
is full.  Let $T_j$ stabilize $E_j$.  Coarsening its corner grading
by $K$ gives a crossed algebra on $R\otimes E_j$ with group
\[
 Q_j=T_j/K,\qquad |Q_j|\leq[T:K].
\]
Conjugation by each homogeneous unit preserves separately $R$
and $E_j$.  By Skolem--Noether its action on $E_j$ is implemented
by a unit of $E_j$.  Dividing by that unit makes it centralize
$E_j$, without changing its action on $R$.  Its multiplication
factors now belong to the centralizer of $E_j$ in $R\otimes E_j$,
namely $R$.  Hence the corner is a matrix algebra over a crossed
algebra $\mathcal C_j$ on $R$ with grading group $Q_j$ and with
the same outer action on $R$.

The matrix factors have been removed without changing the outer
action on $R$.  This alone is insufficient for based finiteness:
we must also recover the original multiplication factors on a
Sylow subgroup.  We do so without bounding the matrix size.
Choose a Sylow $p$-subgroup $S$ of $Q_j$.  Schur--Zassenhaus in
its inverse image gives a $p$-subgroup $\widetilde S\leq T_j$
mapping isomorphically onto $S$.  Use the original homogeneous
units $u_s$ for $s\in\widetilde S$; their products have factors
$a_{s,t}\in R^\times$.  These factors centralize $E_j$, so their
conjugations on $E_j$ form an honest group action.  Choose
$c_s\in E_j^\times$ implementing it.  Then
\[
 c_sc_tc_{st}^{-1}=\gamma(s,t)\in k^\times.
\]
The corrected units $w_s=c_s^{-1}e_ju_s$ have the original actions
on $R$ and satisfy
\[
 w_sw_t=\gamma(s,t)^{-1}a_{s,t}w_{st}.
\]
Since $H^2(S,k^\times)=0$, rescaling removes $\gamma$.  Thus the
restriction of $\mathcal C_j$ to $S$ is based-isomorphic to one
of the restrictions already controlled in the hypothesis.

It remains to pass from a Sylow subgroup to a group $Q$ of bounded
order.  There are finitely many outer homomorphisms
$Q\to\mathcal F$.  Fix one and choose action representatives
$\alpha_q\in\Aut_k(R)$.  Changing homogeneous units permits
these same representatives for every crossed system with that
outer homomorphism.  If factors satisfying the crossed identities
exist, their classes under central changes of homogeneous units
form a torsor under
\[
 H^2(Q,Z(R)^\times).
\]
To see this directly, the ratio of any two factor systems with
these fixed actions is central; the associativity identity says
that this ratio is a $2$-cocycle.  A change which leaves all the
chosen actions fixed is necessarily central and gives precisely
a coboundary.

Restriction from this cohomology group to that of a Sylow
$p$-subgroup has finite kernel.  On the factor $U_R$ of
\eqref{eq:ext:center-units}, restriction followed by corestriction
is multiplication by the $p'$-index, which is invertible on this
$p$-power-exponent cohomology group.  Restriction is therefore
injective on $H^2(Q,U_R)$.  On scalars, $H^2(Q,k^\times)$ is
finite.  Indeed, for $n=|Q|$, the $n$-power map on $k^\times$ is
surjective with finite kernel $\mu_n(k)$; the long exact
cohomology sequence and annihilation by $n$ make
$H^2(Q,k^\times)$ a quotient of the finite group
$H^2(Q,\mu_n(k))$.  The decomposition
\eqref{eq:ext:center-units} is invariant under all the actions,
so these two assertions prove the finite-kernel claim.

Based Sylow finiteness gives finitely many restricted torsor
classes with the fixed actions, and each has finitely many
preimages.  There are therefore finitely many possibilities for
$\mathcal C_j$.  Each has finitely many block summands, proving
the claimed Morita finiteness.  The number or the matrix sizes
of the factors $E_j$ were never required to be bounded.
\end{proof}

\subsection{The extension criterion}

\begin{theorem}[The conditional extension criterion]
\label{thm:extension-finiteness}
Fix $p$ and $M$.  Suppose that the comparison property of
Definition~\ref{def:ext:comparison-data} holds uniformly for the
data constructed from all reduced blocks of defect order at most
$M$.  Then the blocks of all finite groups of defect order at
most $M$ have finitely many $k$-linear Morita equivalence classes.
\end{theorem}

\begin{proof}
Reduce to $B$ and construct the crossed algebra of
Lemma~\ref{lem:ext:dual-recovery}.  Its block summands include,
up to Morita equivalence, the original block.  By
Lemma~\ref{lem:ext:integral-bases} its identity-component basic
orders lie in a finite list; their reductions have a fixed finite
set of possible outer actions.  Their grading groups have a
normal $p'$-subgroup of bounded index by
Lemma~\ref{lem:ext:reduced-structure}.  Lemma~\ref{lem:ext:based-sylow}
supplies the needed based finiteness on $p$-subgroups.
Lemma~\ref{lem:ext:kernel-removal} therefore gives finitely many
Morita types for all their block summands.  Taking the finite union
over the integral base orders proves the assertion.
\end{proof}

Proposition~\ref{prop:sylow-comparison}, proved in the next section,
establishes exactly the comparison property used in
Theorem~\ref{thm:extension-finiteness}.  Combining the two results
proves the $k=\overline{\F}_p$ case of Theorem~\ref{thm:donovan};
Section~\ref{sec:coefficient-extension} transfers the resulting finite
list to the fixed algebraically closed field $K$.  The order of this exposition
introduces no extra premise: the proof of that proposition uses
the component constructions and character results, and does not
use the finite lists deduced here.  The integral finiteness theorem
has been applied only to the bounded-defect identity-component
blocks $\bar B$; the final arbitrary-group assertion is over $k$.
==== END Donovans-Conjecture-over-Algebraic-Closures-of-Prime-Fields//07_extensions.tex ====

==== BEGIN Donovans-Conjecture-over-Algebraic-Closures-of-Prime-Fields//08_periodicity.tex ====
\section{Sylow-equivariant Frobenius comparison}\label{sec:periodicity}

We prove the comparison property used in the extension argument.  It is
important to retain the specified inner factors of the action: a comparison
of the underlying block algebras alone would not control the crossed
multiplications in Section~\ref{sec:extensions}.

\begin{proposition}[Sylow comparison]\label{prop:sylow-comparison}
Fix the coefficient prime $p$ and the defect-order bound $M$.
For the comparison data of Definition~\ref{def:ext:comparison-data}, let
$S\leq Y_0$ be a $p$-subgroup contained in the pure $X$-actions.  There is a
positive integer $m$, bounded in terms of $p$ and $M$, and a Morita
bimodule
\[
 \mathcal M:\quad
 \cO\bar N\sigma^m(\bar b)\text{-}\cO\bar N\bar b
\]
such that its reduction $M_0=k\otimes_{\cO}\mathcal M$ has invertible
$k$-linear maps $J_x$, for $x\in S$, with $J_1=\id$, satisfying
\begin{align}
 J_x(avb)&=\alpha'_x(a)J_x(v)\alpha_x(b),
       \label{per:covariance}\\
 J_xJ_y(v)&=a'_{xy}J_{xy}(v)a_{xy}^{-1}.
       \label{per:factor-law}
\end{align}
Here $\alpha_x$ is the action of the chosen representative $h_x$,
$\alpha'_x=\sigma^m(\alpha_x)$,
$a_{xy}=n_{xy}\bar b$, and
$a'_{xy}=n_{xy}\sigma^m(\bar b)=\sigma^m(a_{xy})$.
The assertion includes $S=1$.  The bound does not depend on the orders of
the universal-cover kernels or of the added central tori.
\end{proposition}

We use the block idempotent and the corresponding block algebra
interchangeably when specifying bimodule sides.  Extend Witt Frobenius to
an algebraic closure of $\operatorname{Frac}(\cO)$ by requiring it to fix
all $p$-power roots of unity.  Such an extension exists because the
cyclotomic extensions generated by these roots are totally ramified,
whereas $\cO$ is unramified; see \cite[Lemma~2.2]{FarrellKessar}.
On roots of unity of order prime to $p$ it is the $p$th power map.
Throughout this section a bound is allowed to depend on $p,M$.

\subsection{Replacing field automorphisms by algebraic permutations}

We use the component models of Lemma~\ref{lem:ext:partial-extension}.
Lift the block to the central product cover, and on a Lie component adjoin
the direct central $p$-factor $C_i$ so that the finite group is the full
fixed-point group $\mathbf J_i^{F_i}$.  These lifted blocks can have
unbounded defect.  We will descend the equivalences before using any
bounded-defect finiteness criterion.

\begin{lemma}[Algebraic realization]\label{per:algebraic-action}
On each $S$-orbit of cross-characteristic Lie components there is a
connected reductive group $\mathbf J$ with simply connected derived
group and a Steinberg endomorphism $F$, together with an identification
of $J=\mathbf J^F$ with the product of the full component groups, having
the following properties.
The operations generated by the $h_x$ and the inner automorphisms of
$J$ form a subgroup
\[
 I\ \leq\ \mathbf J_{\mathrm{ad}}^F\rtimes Y',
\]
where $Y'$ is a finite $p$-group of algebraic automorphisms commuting with
$F$.  Its order and the number of copies of each original absolute
component used in $\mathbf J$ are bounded.  Restriction of these
operations to the universal components is faithful modulo precisely
the standard inner and pinned relations.  In particular the products
of the chosen operations have the prescribed inner factors coming from
$n_{xy}$.
\end{lemma}

\begin{proof}
We recall the relevant part of Steinberg's automorphism theorem
\cite{SteinbergAutomorphisms} for finite simple groups of Lie type,
in the formulation of Broto--M\o ller--Oliver
\cite[Definition~3.3 and Theorem~3.4]{BMO}, omitting the finite exceptional
list already separated in Section~\ref{sec:extensions}.  For an ordinary
diagram twist
\[
 F_0=\gamma\operatorname{Fr}_r^f
\]
the inner-diagonal group is the adjoint fixed-point group.  The pinned
part is generated by $\operatorname{Fr}_r$ and the diagram centralizer
of $\gamma$, with the relation
$\gamma\operatorname{Fr}_r^f=1$ on fixed points.  In the ordinary
field--Dynkin presentation the restriction kernel is exactly the
cyclic group generated by $F_0$; see \cite[\S2.2]{SpaethD}.  These operations act
on the partial regular extension, including its added torus, with the
same relation.  The exceptional graph-isogeny cases have instead a
generator $\tau$, with $\tau^2$ a split Frobenius, and defining
endomorphism $F_0=\tau^f$ \cite[Definition~3.1]{BMO}.  Its restriction
has order exactly $f$.  For untwisted groups, $f=2d$, even powers are
field operations of order $d$, and odd powers exchange long and short
root groups.  For the Suzuki and Ree twists, $f$ is odd and $\tau^2$
has field order $f$, forcing $\tau$ itself to have order $f$.
The usual automorphism description is
faithful on the universal component after the finite exceptional list
has been removed.  We now turn its field operations into actual
algebraic automorphisms; no field Frobenius is being regarded as an
invertible morphism of algebraic groups.

First consider the stabilizer of one component in the pinned image of
$S$.  In the ordinary case its projection modulo diagrams to the field
cyclic group has order $a$, where $a\mid f$ is a $p$-power and
$a\leq |S|$.  Put $d=f/a$.  On $a$ copies of the pinned group define
\begin{align*}
 R(g_0,\ldots,g_{a-1})
    &=(\gamma(g_{a-1}),g_0,\ldots,g_{a-2}),\\
 F_a&=R\operatorname{Fr}_r^d.
\end{align*}
The Frobenius in the second formula acts on every coordinate.  We have
$R^a=\gamma$ diagonally.  Projection onto coordinate zero identifies
$(\mathbf J_0^a)^{F_a}$ with $\mathbf J_0^{F_0}$: a fixed tuple is
\[
 (g,\operatorname{Fr}_r^d(g),\ldots,
       \operatorname{Fr}_r^{(a-1)d}(g)),
 \qquad F_0(g)=g.
\]
On these tuples $R$ induces $\operatorname{Fr}_r^{-d}$.
Every diagram $\delta$ centralizing $\gamma$ acts diagonally and
commutes with $R$.  Thus the subgroup above the order-$a$ field image
is represented with its exact presentation
\[
 R^a=\gamma,\qquad [R,\delta]=1,
\]
as well as the relations among the diagrams.  The required pinned
$p$-subgroup is the corresponding subgroup of this finite group.
One does not replace $R$ by an order-$a$ permutation when
$\gamma\ne1$: the displayed wrap relation is essential.  In particular
mixed field--graph elements and their relations are retained.

For $F_0=\tau^f$, the pinned automorphism group is cyclic.  If its
relevant subgroup has order $a$, use $a$ copies, the pure cyclic
permutation $R$, and
\[
 F_a=R\tau^{f/a}.
\]
Projection identifies the fixed points with those of $\tau^f$, and
$R$ induces $\tau^{-f/a}$.  The permutation is an algebraic
automorphism even though $\tau$ need not be one.  Both constructions
have a power of $F_a$ equal to a Frobenius endomorphism.

For completeness, the passage from a component stabilizer to its
orbit preserves relations.  Choose a base component and, for every
component in its orbit, an element of the pinned image transporting
the base component to it.  Use these transports to choose the
identifications.  If an element sends component $i$ to component $j$,
its map in these identifications is the stabilizer element obtained
by composing the inverse chosen transport to $j$, that element, and
the chosen transport to $i$.  Products of these maps satisfy the
stabilizer multiplication law, by cancellation of the middle
transports.  Thus they act on the product of the copy models by
permutations and the algebraic stabilizer operations just constructed.
The resulting group is the actual pinned image of $S$, and is a
$p$-group.  The orbit size and all copy numbers are bounded by $|S|$.

Apply the same construction to the adjoint groups.  Their fixed points
identify with the original inner-diagonal groups, and the action of
the pinned image agrees with its original action.  The given
representatives consequently define elements of the asserted
semidirect product.  The standard faithful automorphism description
ensures that a relation on the universal components has exactly its
indicated inner image in this model.  This also applies to the actions
extended to the torus in the partial regular extension: the pinned
relations hold there by construction.  Hence the representative
product $h_xh_y=n_{xy}h_{xy}$ still has that inner factor, rather than
merely an unspecified inner automorphism.
\end{proof}

Write $b'$ for the resulting product block of $J$; it is $I$-stable.
The number of original components and $|S|$ are bounded by
Lemma~\ref{lem:ext:reduced-structure}.  Thus the number of absolute
components in the new model is bounded, although their classical
ranks and the central torus rank need not be bounded.

\subsection{A central twist with an invariant parabolic}

Let $s\in\mathbf J^{*F}$ be a semisimple $p'$-element such that $b'$
belongs to $\mathcal E_p(J,s)$.  The partial regular extension ensures
that
\[
 \mathbf C=C_{\mathbf J^*}(s)
\]
is connected.  Indeed its possible semisimple-centralizer component
obstruction is $p$-primary, since the component group of
$Z(\mathbf J)$ is $p$-primary; the order of a component arising from
$s$ also divides the order of $s$.  These orders are coprime.  This is
the semisimple-centralizer component criterion, applied to the dual
root datum; see \cite[Corollary~2.9]{BonnafeQuasiIsolated} and
\cite[Proposition~14.20]{MalleTesterman}.  In the exceptional-isogeny cases the obstruction has no
nontrivial points, as in Lemma~\ref{lem:ext:partial-extension}.

\begin{lemma}[The fixed central torus]\label{per:fixed-torus}
There exist an $F$-stable torus $\mathbf V\subseteq Z^\circ(\mathbf C)$,
a dual Levi subgroup
\[
 \mathbf L^*=C_{\mathbf J^*}(\mathbf V),
\]
and a bounded positive integer $m_0$ such that
\[
 \mathbf C\subseteq\mathbf L^*,\qquad
 z_m=s^{p^m-1}\in\mathbf V
       \quad\hbox{whenever }m_0\mid m.
\]
The $s$-transporters of the pinned image fix $\mathbf V$ pointwise.
Each $z_m$ defines canonically a linear character $\lambda_m$ of
$L=\mathbf L^F$ of $p'$-order, through the quotient torus
$\mathbf L/[\mathbf L,\mathbf L]$.
\end{lemma}

\begin{proof}
Diagonal automorphisms do not change a geometric series parameter.
Since $b'$ is stable, each element of $Y'$ preserves the geometric
class of $s$.  Connectedness of $\mathbf C$ and Lang's theorem imply
that this geometric class contains a unique rational class.  Hence
\[
 H_s=\{(g,y)\in\mathbf J^{*F}\rtimes Y':
                         g\,y(s)g^{-1}=s\}
       \longrightarrow Y'
\]
is onto, with kernel $\mathbf C^F$.  That kernel acts trivially on
$\mathbf V_0=Z^\circ(\mathbf C)$, so $H_s$ defines an action of $Y'$
on $\mathbf V_0$.  This action commutes with $F$.  Set
\[
 \mathbf V=(\mathbf V_0^{Y'})^\circ.
\]
In particular the action just defined is independent of all choices
of transporters.

We show that a bounded $p'$-power of $s$ lies in $\mathbf V_0$.
Choose a maximal torus of $\mathbf C$, and let $X$ be its character
lattice, $Q$ the ambient root lattice, and $Q_s$ the lattice generated
by roots centralizing $s$.  Since the primal derived group is simply
connected, $Q$ is saturated in $X$.  The torsion of $X/Q_s$ therefore
injects into the torsion of $Q/Q_s$.  In type A the subsystem is a
product of equality packets, and its lattice is primitive in the
ambient root lattice.  In the classical signed-coordinate systems,
orient each inverse-paired eigenvalue packet.  A type-A packet then
has its integral zero-sum lattice.  On each residual B, C, or D
packet, the lattice generated by the centralizing roots contains
twice every integral coordinate vector in its rational span.
Consequently the saturation quotient on each such packet has
exponent dividing two; taking products does not enlarge this
exponent.  This proves a uniform exponent bound for classical
components.  On the exceptional components there are only finitely
many root subsystems of the finitely many bounded root systems, so
their saturation quotients also have bounded exponent.  The bounded
number of absolute components allows one common exponent.

The character group of the component group of $Z(\mathbf C)$ is
the torsion just considered.  Since $s$ is central in $\mathbf C$,
there is therefore a bounded integer $l$, prime to $p$, with
$s^l\in\mathbf V_0$.  We may discard the $p$-part of the exponent
because $s$ has $p'$-order.  This element is fixed by $Y'$.
For any torus with a $Y'$-action, the norm morphism
\[
 t\longmapsto\prod_{y\in Y'}y(t)
\]
has connected image in the fixed torus.  On a fixed element it is
the $|Y'|$th power.  Thus the group of components of the fixed torus
on geometric points is killed by $|Y'|$, a $p$-power.  Its element
represented by $s^l$ has $p'$-order, and is trivial.  Hence
$s^l\in\mathbf V$.

Take $m_0$ with $p^{m_0}\equiv1\pmod l$; for $l=1$ take $m_0=1$.
This is bounded, and proves the assertion about $z_m$.  The torus
$\mathbf V$ is $F$-stable, and its centralizer is an $F$-stable Levi.
Since $\mathbf V\subseteq Z(\mathbf C)$, the Levi $\mathbf L^*$
contains $\mathbf C$.
Moreover $\mathbf V\subseteq Z^\circ(\mathbf L^*)$.
The natural duality
\[
 Z^\circ(\mathbf L^*)
    \quad\longleftrightarrow\quad
 \mathbf L/[\mathbf L,\mathbf L]
\]
associates to its rational point $z_m$ a linear character
$\lambda_m$ of $L$, trivial on the finite points of the algebraic
derived subgroup.  It has $p'$-order and takes values in $\cO$.
This construction is functorial for the transporter operations.
\end{proof}

\begin{lemma}[Common parabolic representatives]\label{per:parabolic}
The primal Levi $\mathbf L$ can be placed in a parabolic $\mathbf Q$
such that, for the subgroup $A_1$ of $I$ preserving $\mathbf Q$,
$\mathbf L$, the rational $s$-series in $L$, and $\lambda_m$, one has
\begin{equation}\label{per:parabolic-factorization}
 I=\Inn(J)A_1,\qquad A_1\cap\Inn(J)=\Inn(L).
\end{equation}
Here inner groups mean their images in the algebraic automorphism
group.  Only $\mathbf L$ is required to be $F$-stable.
\end{lemma}

\begin{proof}
Choose a cocharacter $\nu$ of $\mathbf V$ outside the finitely many
root hyperplanes which do not contain all of $\mathbf V$.  Then
$C_{\mathbf J^*}(\nu)=\mathbf L^*$, and $\nu$ specifies a parabolic
$\mathbf Q^*$ with this Levi.  Every element of $H_s$ fixes
$\mathbf V$ pointwise, and hence preserves both this Levi and the
positive root set of this parabolic.  There is no requirement that
$\nu$ be $F$-fixed.

Choose an $F$-stable Borel--torus pair in the connected group
$\mathbf C$, and denote its torus by $\mathbf T^*$.  The Lang--Steinberg theorem
\cite[Theorem~21.7]{MalleTesterman} makes the rational Borel--torus pairs a single $\mathbf C^F$-orbit.
Thus each $s$-transporter can be corrected by an element of
$\mathbf C^F$ to normalize $\mathbf T^*$.  Such a correction still
fixes $\mathbf V$ pointwise.  On this torus the transporter is a
Weyl operation followed by its specified pinned operation, and
commutes with the rational root datum endomorphism.

Choose the matching rational primal torus $\mathbf T$.  Root--coroot
duality transfers the zero and positive root sets to a Levi and a
parabolic $\mathbf L\subseteq\mathbf Q$ of $\mathbf J$.  Equivalently,
an invariant rational positive definite form transfers the
inequalities defined by $\nu$ to primal coroot inequalities; a
multiple clears denominators.  The zero set is $F$-stable and is
exactly the dual Levi root system.  The transferred transporter
preserves these positive and zero sets.

We spell out why its primal representative can be rational.  Its
Weyl and pinned operation has a geometric representative in
$\mathbf J_{\mathrm{ad}}\rtimes Y'$.  Compatibility with the rational
torus endomorphism says that its discrepancy with $F$ lies in
$\mathbf T_{\mathrm{ad}}$.  Lang's theorem for this connected torus
corrects that discrepancy without changing the root action or its
positive set.  This gives an element of
$\mathbf J_{\mathrm{ad}}^F\rtimes Y'$ preserving $\mathbf Q,\mathbf L$
and the indicated torus data.  In particular it preserves the
rational $s$-class and $\lambda_m$.

One may also prescribe its diagonal coset.  For any rational
maximal torus of $\mathbf J$, comparison of the exact sequences for
\[
 Z(\mathbf J)\longrightarrow\mathbf T\longrightarrow
 \mathbf T_{\mathrm{ad}},\qquad
 Z(\mathbf J)\longrightarrow\mathbf J\longrightarrow
 \mathbf J_{\mathrm{ad}}
\]
and Lang's theorem for $\mathbf T$ and $\mathbf J$ give a surjection
\[
 \mathbf T_{\mathrm{ad}}^F\longrightarrow
       \mathbf J_{\mathrm{ad}}^F/\operatorname{im}(J).
\]
Multiplying the representative by such a torus element changes its
diagonal coset arbitrarily and preserves the parabolic and Levi.
Diagonal operations preserve geometric series and quotient-torus
characters.  Moreover $C_{\mathbf L^*}(s)=\mathbf C$ is connected,
so preservation of the geometric class here is preservation of its
unique rational class.  Given an element of $I$, choose the diagonal
coset to agree with that element.  The resulting representative
differs from it by an element of $\Inn(J)$, and therefore belongs to
$I$.  This proves the first equality.

Finally, the simultaneous normalizer of a parabolic and its chosen
Levi in the connected group is the Levi itself.  Therefore an
inner operation of $J$ lies in $A_1$ exactly when its implementer
lies in $L$.  Elements of $L$ also preserve its $s$-class and linear
character.  This proves the second equality.
\end{proof}

\subsection{The integral comparison and its ordinary characters}

The chosen Levi contains the full dual centralizer, and its
parabolic admits the required automorphism representatives.
These are the geometric inputs for an equivariant integral
Morita equivalence.  The remaining character calculation will
return the Levi block after a bounded Frobenius power and a
central linear twist.

\begin{lemma}[Equivariant integral Levi equivalence]\label{per:br}
There is an $A_1$-stable block $b_L$ of $\cO L$ and an integral Morita
bimodule $U$ between $\cO Jb'$ and $\cO Lb_L$ with a strict
$A_1$-action.  For $\ell\in L$, the operator of
$\operatorname{ad}(\ell)\in A_1$ on $U$ is exactly
\begin{equation}\label{per:br-inner}
 u\longmapsto\ell u\ell^{-1}.
\end{equation}
Every central element of $J$ acts identically from the two sides of
$U$.
\end{lemma}

\begin{proof}
We use the full-centralizer form of the integral
Bonnaf\'e--Rouquier theorem \cite[Theorem~B$'$, \S11.4]{BR}, also
stated in \cite[\S7]{BDR}.  Its hypotheses here are that
$\mathbf J$ is connected reductive in characteristic $r\ne p$,
$F$ has a Frobenius power, $\mathbf L$ is an $F$-stable Levi of a
parabolic $\mathbf Q$, $s$ is a rational semisimple $p'$-element,
and $C_{\mathbf J^*}(s)\subseteq\mathbf L^*$.  An $F$-stable
parabolic is not required.  All these hypotheses were established
above.  With $\mathbf Q=\mathbf L\mathbf U_Q$, the theorem gives
the Morita bimodule in the single nonzero degree of the appropriate
block cut of compactly supported cohomology of
\[
 Y_{\mathbf U_Q}=
 \{g\mathbf U_Q:g^{-1}F(g)\in
                         \mathbf U_QF(\mathbf U_Q)\}.
\]
Its degree is
$\dim\mathbf U_Q-\dim(\mathbf U_Q\cap F(\mathbf U_Q))$.

The construction is valid over the fixed unramified ring $\cO$.
Indeed the group-algebra block idempotents are defined over some
finite unramified Witt subring: lift their finite-residue-field
idempotents uniquely.  Construct the compactly supported integral
cohomology complex and these cuts over that ring.  After faithfully
flat extension to a splitting discrete valuation ring the cited
integral theorem gives concentration, projectivity on both sides,
and the Morita evaluation isomorphisms.  These properties descend
by faithful flatness.  Extending the descended construction to
$W(k)$ gives $U$.  Thus extension to splitting coefficients is used
to test the equivalence, not as an assumption that the generic field
of $W(k)$ contains $p$-power roots of unity.

For $a\in A_1$ the map $g\mathbf U_Q\mapsto a(g)\mathbf U_Q$ is
an actual automorphism of this variety.  Using inverse pullback
consistently gives a strict action on cohomology.  The series cuts
are preserved; since $b'$ is stable, equivariance of the block
correspondence makes $b_L$ stable too.  For $a=\operatorname{ad}(\ell)$,
$\ell\in L$, this variety map is precisely the left action of
$\ell$ followed by the right action of $\ell^{-1}$.  This proves the
asserted identity.  A central element of $J$, which lies in $L$,
gives the identity variety map, proving central compatibility.
\end{proof}

\begin{lemma}[Returning the Levi block]\label{per:row-return}
After replacing $m_0$ by a bounded multiple $m$, one has
\begin{equation}\label{per:levi-return}
 \sigma^m(b_L)=\lambda_m b_L,
\end{equation}
in the convention that multiplication of ordinary characters by
$\lambda_m$ gives the block on the right.  The character $\lambda_m$
is still defined by $s^{p^m-1}$ and is $A_1$-invariant.
\end{lemma}

\begin{proof}
Define an operation on the ordinary characters of $L$ by
\[
 T(\chi)=\lambda_{m_0}^{-1}\sigma^{m_0}(\chi).
\]
Coefficient Frobenius and twisting by a linear character both
permute ordinary irreducible characters and their blocks.  It
therefore suffices to find a bounded positive integer $d$ such
that $T^d$ fixes one ordinary irreducible character in $b_L$: its
block then returns as well.  We first bound row orbits after a
regular embedding, then use restriction to return a row of $b_L$.
At the end we check that the accumulated twist is precisely
$\lambda_{d m_0}$.

For an ordinary character in $\mathcal E(L,st)$, with $t$ a
$p$-element of $C_{\mathbf L^*}(s)^F$, coefficient Frobenius sends
the torus parameter to $s^{p^{m_0}}t$.  The $p$-part is unchanged
by our choice of the characteristic-zero extension of $\sigma$.
The Deligne--Lusztig character formula, whose Green functions are
rational, gives the same statement for every uniform pairing.
Thus $T$ returns the parameter to $st$ and fixes its uniform pairings.
This argument concerns actual pairings, not just the set of torus
characters; compare \cite[Lemma~3.6(i)]{FarrellKessar}.

\emph{Bounded row orbits after regular embedding.}
Use a full regular embedding
$\mathbf L\hookrightarrow\mathbf L^\dagger$ with connected center
and compatible Steinberg endomorphism.  The derived subgroup stays
simply connected.  The standard regular embedding construction
applies also to Frobenius roots; in the exceptional-isogeny cases
at issue the centers on points already have no obstruction.
On dual groups the map
\[
 \pi:\mathbf L^{\dagger *}\longrightarrow\mathbf L^*
\]
has central torus kernel.  Choose a character above a row of $b_L$,
with parameter $s^\dagger t^\dagger$, where $s^\dagger$ is its
$p'$-part and $t^\dagger$ its $p$-part.  After rational conjugacy its
$p'$-parameter projects to $s$.  This rational adjustment is possible
by connectedness of the centralizer and Lang's theorem.  Put
\[
 z^\dagger=(s^\dagger)^{p^{m_0}-1}.
\]
Every root of $\mathbf L^{\dagger *}$ has value one on
$z^\dagger$, by the root test already established on $\mathbf L^*$.
Thus $z^\dagger$ is central.  The center of
$\mathbf L^{\dagger *}$ is connected because
$[\mathbf L^\dagger,\mathbf L^\dagger]$ is simply connected.
The associated linear character $\lambda^\dagger$ restricts to
$\lambda_{m_0}$.  The operation
\[
 T^\dagger(\chi)=(\lambda^\dagger)^{-1}\sigma^{m_0}(\chi)
\]
therefore preserves $\mathcal E(L^\dagger,s^\dagger t^\dagger)$
and fixes each of its uniform pairings.

Ordinary Jordan decomposition on these connected data identifies
the pairings with the unipotent uniform pairings of the connected
centralizer.  Separate the ordinary classical factors from the
bounded-rank exceptional factors.  The latter include every factor
with an exceptional graph-isogeny endomorphism, in particular the
Suzuki type \({}^2B_2\), and every triality factor.  On the ordinary
classical factors the pairings distinguish the irreducible unipotent
characters, even up to scalar: type A is uniform, and for the ordinary
types B, C, and D this is the Fourier separation of the symbol labels
in Lemma~\ref{lab:fingerprints}.  That separation concerns ordinary
characters and is independent of the coefficient prime $p$.
Every uniform pairing vector here is nonzero.  Varying the torus
test on one factor while fixing nonzero tests on the others shows
that equality of product pairing vectors makes the corresponding
factor vectors proportional.  Lemma~\ref{lab:fingerprints} then
fixes all ordinary classical labels.  Equality uniqueness for a
centralizer consisting of such factors is also the ordinary character
statement of \cite[Lemma~3.7]{FarrellKessar}.
Consequently there is at most one choice on each ordinary classical
factor.  Every remaining factor has bounded root system and one of
finitely many rational twists, so its number of unipotent labels is
bounded.  There are boundedly many such factors.  Adding central tori
introduces no new unipotent
labels.  It follows that the orbit of $\chi$ under $T^\dagger$ has
bounded length, independently of the ranks of the classical factors
and of the field cardinalities.  Frobenius-permuted factors are
calculated on a representative with the composite endomorphism;
the same reasoning applies to them.

\emph{Restriction to the original Levi.}
The upstairs operation $T^\dagger$ has bounded row orbits,
uniformly in rank and field size.  For the chosen
$\lambda^\dagger$, restriction satisfies
\[
 \Res_L\bigl(T^\dagger(\chi)\bigr)
       =T\bigl(\Res_L\chi\bigr),
\]
because $\lambda^\dagger|_L=\lambda_{m_0}$ and coefficient
Frobenius commutes with restriction.  If $(T^\dagger)^a$ fixes an
upstairs row, then $T^a$ permutes its restriction constituents.  We now
distinguish the type of the original ambient components, rather
than the centralizer factors used in the pairing argument.  We
are working on one $S$-orbit, so these original components have
the same type.

For a non-type-A orbit the number of distinct restriction
constituents is bounded.  For a common maximal torus of the primal
groups $\mathbf L\subseteq\mathbf J$, let $X$ be its character
lattice and $\Phi_L\subseteq\Phi_J$ their root systems.  The torsion in
$X/\mathbb Z\Phi_L$ injects into the torsion in
$X/\mathbb Z\Phi_J$: after making the Levi standard, the kernel
$\mathbb Z\Phi_J/\mathbb Z\Phi_L$ is free on the omitted simple
roots.  The center component group of the Levi therefore has order
bounded by that of the original non-A group, with a bounded product
over the copies.  The rational quotient
\[
 (\mathbf L^\dagger)^F/
       \bigl(Z(\mathbf L^\dagger)^F L\bigr)
\]
has bounded order by the central exact sequence and Lang's theorem.
Clifford theory bounds the number of restriction constituents by
this order.  A bounded power of $T^\dagger$ fixes the upstairs row,
so the same power of $T$ permutes only this bounded set of
downstairs constituents.  A further bounded power fixes a
constituent belonging to $b_L$.

For a type-A orbit this center-component bound need not hold.
Choose a covering block of $b_L$ in $L^\dagger$ and apply the
preceding upstairs construction to a row
\[
 \chi^\dagger\in\mathcal E(L^\dagger,s^\dagger)
\]
in that block.  Such a row exists by the integral basic-set theorem
for type A with connected center: intersect its basic set with the
covering block.  Its parameter projects to the prescribed $s$.
The resulting $\lambda^\dagger$ still restricts to $\lambda_{m_0}$,
so the downstairs operation $T$ is unchanged.
We claim that $\Res_L\chi^\dagger$ is irreducible.  The preimage
\[
 \pi^{-1}(C_{\mathbf L^*}(s))
\]
is connected, since both its kernel and its quotient are connected.
It centralizes $s^\dagger$: a maximal torus and its root subgroups
generate it, and their roots take value one on $s^\dagger$ exactly
when they do on $s$.  Hence if $s^\dagger u$ is conjugate to
$s^\dagger$, with $u\in\ker\pi$, then any conjugating element is
in this preimage and $u=1$.

The quotient linear characters of $L^\dagger/L$ correspond
faithfully to rational points of the dual kernel and act by these
central multiplications on parameters.  Only the trivial quotient
character can therefore fix $\chi^\dagger$.  Frobenius reciprocity
and induction from a normal subgroup with abelian quotient give
\[
 \bigl\langle\Res_L\chi^\dagger,
                    \Res_L\chi^\dagger\bigr\rangle
 =\bigl|\{\eta\in\Irr(L^\dagger/L):
                         \eta\chi^\dagger=\chi^\dagger\}\bigr|=1.
\]
This proves the claim; equivalently one may use
\cite[Lemma~4.8]{SFTV}.  A covering block covers a single orbit of
blocks of $L$.  Since this restriction is irreducible and
$L^\dagger$-invariant, its block is the prescribed covered block
$b_L$.  Its orbit under $T$ is now
bounded by the upstairs orbit, with no diagonal-index bound.

In either case $T^d$ fixes a row of $b_L$ for a bounded positive
integer $d$.  Iteration has exactly the required twist prescription:
\[
 \prod_{i=0}^{d-1}\sigma^{i m_0}(\lambda_{m_0})
       =\lambda_{d m_0},\qquad
 (p^{m_0}-1)\sum_{i=0}^{d-1}p^{i m_0}=p^{d m_0}-1.
\]
Indeed the first identity follows from torus duality and the second.
Thus $\lambda_{d m_0}^{-1}\sigma^{d m_0}$ fixes that row and hence
its block.  Taking this bounded multiple proves the assertion.
The construction of Lemma~\ref{per:fixed-torus} continues to make
$\lambda_m$ invariant under $A_1$.
\end{proof}

\subsection{Scalar overlap and the specified inner factors}

\begin{lemma}[Projective extension of the transports]
\label{per:scalar-overlap}
For the Lie-component orbit above there is an integral Morita
bimodule $\mathcal M_J$ between $\sigma^m(b')$ and $b'$ on which
$I$ has covariance transports defined up to scalars in $\cO^\times$.
For every $g\in J$ its inner transport agrees, modulo scalars, with
\[
 E_g(v)=gvg^{-1}.
\]
Central $p$-elements act equally from both sides of $\mathcal M_J$.
\end{lemma}

\begin{proof}
Put $A=\cO Jb'$, $B=\cO Lb_L$, and use primes for their
$\sigma^m$-images.  By Lemma~\ref{per:row-return}, multiplication
of characters by $\lambda_m$ identifies $B$ with $B'$.
Let $T$ be the associated invertible $(B',B)$-bimodule.
Explicitly its underlying left module is $B'$, and its right action
uses the isomorphism
\[
 f:B\longrightarrow B',\qquad
        g b_L\longmapsto\lambda_m(g)^{-1}g\sigma^m(b_L).
\]
Let $U^\vee$ be a Morita inverse of the bimodule in
Lemma~\ref{per:br}.  Then
\[
 \mathcal M_J=
   \sigma^m(U)\otimes_{B'}T\otimes_B U^\vee
\]
is an $(A',A)$-Morita bimodule.  Each factor has a strict $A_1$-action:
on the outer factors this is the geometric action and its dual;
on $T$ it is the action on its group basis, since $\lambda_m$ is
$A_1$-invariant.  Hence their tensor product has a strict action,
which we denote by $D_a$.

The strict action of $A_1$ has been constructed.  To extend it
projectively to $I=\Inn(J)A_1$, we must compare it with the
actual inner operators on their common subgroup.
The comparison on the intersection in
Lemma~\ref{per:parabolic} is explicit.  If $\ell\in L$, the outer
factors identify the geometric action with the actual left and
inverse right action.  On the middle factor the right action uses
$f(\ell^{-1})=\lambda_m(\ell)\ell^{-1}$, so
\begin{equation}\label{per:overlap-scalar}
 D_{\operatorname{ad}(\ell)}
          =\lambda_m(\ell)^{-1}E_\ell.
\end{equation}
All discrepancies therefore lie in the scalar units; no unit in
the non-scalar part of the block center is introduced.

The actual inner operators satisfy
$E_gE_h=E_{gh}$ and
$D_aE_gD_a^{-1}=E_{a(g)}$.  By
$I=\Inn(J)A_1$, represent an element of $I$ as
$\operatorname{ad}(g)a$ and use $E_gD_a$.  Any two such
representations differ in the intersection $\Inn(L)$, where the
preceding formula shows that the two operators differ by a scalar.
A central ambiguity in $g$ is also scalar: a central $p'$-element
acts by its central character on each block, while a central
$p$-element gives no discrepancy.  For the latter assertion apply
the displayed overlap formula to a central $p$-element $c$.  Its
algebraic inner operation is the identity and
$\lambda_m(c)=1$, whence $E_c=\id$.  Covariance then shows that
multiplication of the chosen operators agrees with multiplication
in $I$ modulo scalars, as asserted.
\end{proof}

The adjective scalar in Lemma~\ref{per:scalar-overlap} is necessary.
The following elementary observation identifies exactly how it is
used, and records the role of the element-wise factor identity.

\begin{lemma}[Removing the scalar defect]\label{per:scalar-cocycle}
Let $S$ be a finite $p$-group acting on two block algebras by crossed
systems with the same group labels and respective factors
$a_{xy},a'_{xy}$.  Suppose that a nonzero Morita bimodule has
invertible covariance maps $D_x$ such that
\[
 D_xD_y(v)=c(x,y)a'_{xy}D_{xy}(v)a_{xy}^{-1},
       \qquad c(x,y)\in k^\times.
\]
If the factors obey their actual crossed-system identities, the
maps can be rescaled to satisfy \eqref{per:factor-law} exactly.
\end{lemma}

\begin{proof}
Normalize $D_1=\id$.  Compare $(D_xD_y)D_z$ and $D_x(D_yD_z)$.
Covariance and
\[
 a_{xy}a_{xy,z}=\alpha_x(a_{yz})a_{x,yz},
 \qquad
 a'_{xy}a'_{xy,z}=\alpha'_x(a'_{yz})a'_{x,yz}
\]
cancel all the non-scalar factors.  Because the $D_x$ are
$k$-linear, the remaining equality is
\[
 c(x,y)c(xy,z)=c(y,z)c(x,yz).
\]
Thus $c$ is an ordinary scalar $2$-cocycle, with trivial action on
$k^\times$.  Positive-degree cohomology of $S$ is annihilated by
$|S|$.  The $|S|$th-power map on $k^\times$ is an automorphism,
since $k$ is algebraically closed of characteristic $p$.
It induces an automorphism on this cohomology, which must therefore
vanish.  In particular $H^2(S,k^\times)=0$.  Rescale $D_x$ by a
scalar $1$-cochain trivializing $c$ to obtain the required maps.
\end{proof}

\subsection{Tensor products and central quotients}

The Lie-component comparison now has only scalar multiplication
errors.  We assemble it with the remaining component families,
descend through the original central kernel, and only then remove
the scalar error over $k$.  Descending before invoking integral
finiteness is essential because the lifted blocks may have
unbounded defect.

\begin{proof}[Proof of Proposition~\ref{prop:sylow-comparison}]
On each cross-characteristic Lie orbit use
Lemma~\ref{per:scalar-overlap}.  Choose a common bounded multiple
of its comparison exponents.  Passing to a multiple is legitimate:
tensor successive Frobenius twists of the comparison bimodule and
of its transports.  The twist characters multiply according to the
identity in Lemma~\ref{per:row-return}; scalar projectivity and the
equality of central $p$-actions are preserved.  The number of
orbits is bounded.  Thus this choice still gives a bounded positive
integer $m$.

The other universal components are handled directly.  For the
finite list of groups, take a common Frobenius period of their block
idempotents and use the identity bimodule, with the ordinary group
operations as transports.  Permuted isomorphic components use
matching models, and permutation of tensor factors gives their
transports.  The finite list includes the bounded exceptional
multipliers and automorphisms already removed from the Lie models.

For the unbounded alternating family the universal cover is
$2.\mathfrak A_n$, outside a finite list.  Every automorphism is
induced by $2.\mathfrak S_n$: use the automorphisms of
$\mathfrak A_n$ and unique lifting to its universal cover
\cite[Theorem~2.3 and \S\S2.7.2--2.7.3]{Wilson}.  We give a uniform period that also
handles the double cover.  Let $u$ be an integer coprime to the
exponent of $2.\mathfrak A_n$.  For $g\in2.\mathfrak A_n$,
its image and the image of $g^u$ have the same cycle type in
$\mathfrak S_n$.  For a suitable $t\in2.\mathfrak S_n$ and the
central involution $z$ this gives
\[
 g^u=z^\epsilon t g t^{-1}.
\]
Since $u$ is odd,
\[
 g^{u^2}
   =z^{\epsilon(u+1)}t^2g t^{-2}
   =t^2g t^{-2},\qquad t^2\in2.\mathfrak A_n.
\]
Thus every square of a cyclotomic powering fixes every conjugacy
class.  The cyclotomic action of coefficient Frobenius is powering
by such a unit $u$ (chosen to be $p$ on the $p'$-part and $1$ on the
$p$-part of the exponent).  Its square fixes all ordinary
characters and hence all block idempotents.  After making $m$ even,
the identity block bimodule and its ordinary automorphism
transports apply also to the alternating orbits.  Blocks inflated
from the simple quotient cause no change in this argument.

On the factor $P$ take the identity bimodule for $\cO P$, with the
given automorphism transports.  On the central $p'$-group $Z$ a
block is the rank-one character algebra for some character $\theta$.
Identify it with $\cO$ and compare it with the rank-one algebra for
$\sigma^m\theta$.  The representatives act trivially on $Z$, so
use the identity transport; inner left/right discrepancies on this
factor are scalars.  These choices are compatible with permutation
of the other tensor factors.  On Lie orbits compatibility was built
into a single product algebraic group and its actual variety
actions, so no coherence choice between individual components is
needed.

Tensor all these bimodules over $\cO$.  We now descend from the
product cover, including its added direct central $p$-factors, to
$\bar N$.  Here is the precise quotient argument.  If a central
$p$-element $c$ acts equally from the two sides of a Morita bimodule
$V$, then
\[
 (c-1)V=V(c-1).
\]
For the two-sided ideals generated by any collection of such
elements, the Morita correspondence consequently matches the
ideals.  The quotient of $V$ by their common action is a Morita
bimodule between the quotient algebras; this also follows by
quotienting the Morita evaluation isomorphisms.  Apply this to the
added direct central $p$-factors and to the $p$-part of the old
central kernel $D_0$.  Every factor construction has the required
equality, so the tensor product does too.  For the $p'$-part of
$D_0$, the chosen block lifts have trivial kernel character on both
sides, including after Frobenius.  This part of the kernel already
acts trivially.  No bound for $|D_0|$ is required.  The quotient is
therefore the claimed integral Morita bimodule $\mathcal M$.

We now have the integral comparison on the original quotient
group, where the defect bounds apply.  It remains to recover the
specified representative factors and normalize their scalar error
after reduction.
The covariance operators preserve the quotient ideals, and descend
with it.  On the product cover, products of the operators for $h_x$
and $h_y$ differ from that for $h_{xy}$ by the actual inner operator
of a lift of $n_{xy}$, up to a scalar.  This follows on each Lie
orbit from Lemma~\ref{per:algebraic-action} and
Lemma~\ref{per:scalar-overlap}, and on the remaining factors from
their explicit ordinary transports.  Choices of a lift differ by
$D_0$; after descent their inner operators agree.  Consequently on
$\mathcal M$ the product law is the specified law for $n_{xy}$ up
to a scalar.  The factors in $\bar N$ satisfy the actual equality
\[
 n_{xy}n_{xy,z}=\alpha_x(n_{yz})n_{x,yz},
\]
by their construction from the chosen representatives.  Reduce
modulo $p$ and apply Lemma~\ref{per:scalar-cocycle}.  The resulting
normalized maps are exactly \eqref{per:covariance} and
\eqref{per:factor-law}.

All exponents used above depend only on the bounded number of
components and copies, the order of $S$, the bounded exceptional
root systems, and the finite exceptional group list.  These data
are bounded in terms of $p,M$ by the extension reductions.
The classical root-lattice exponent, the classical row separation,
and the alternating powering argument were uniform in the ranks
and field sizes.  This proves the asserted uniformity and the
proposition, including its trivial-subgroup case.
\end{proof}

\subsection{Extension of the coefficient field}
\label{sec:coefficient-extension}

The preceding argument works over $k=\overline{\F}_p$.  We finish by
transferring its finite list to an arbitrary algebraically closed field
of characteristic $p$.  The block comparison is the coefficient-field
argument of \cite[Section~2.1]{BlockwiseAlperinWeight}; we recall the
part needed here and combine it with scalar extension of Morita
bimodules.

\begin{lemma}[Coefficient extension]
\label{lem:coefficient-extension}
Let $k=\overline{\F}_p$, let $K$ be an algebraically closed field of
characteristic $p$, and choose an embedding $k\hookrightarrow K$.
For every finite group $G$, the map $kG\longrightarrow KG$ induces a
bijection on block idempotents, sending $b$ to $b_K=1\otimes b$.
The blocks $kGb$ and $KGb_K=K\otimes_k kGb$ have the same defect
groups as $p$-subgroups of $G$.

If finite-dimensional $k$-algebras $A$ and $A'$ are $k$-linearly
Morita equivalent, then $K\otimes_k A$ and $K\otimes_k A'$ are
$K$-linearly Morita equivalent.
\end{lemma}

\begin{proof}
The center commutes with scalar extension:
\[
 K\otimes_k Z(kG)\simeq Z(KG).
\]
Indeed, the center is the kernel of the map sending an element to its
commutators with the finitely many elements of $G$, and field extension
preserves kernels.  Write $Z(kG)=\prod_i Z_i$ as a product of local
finite-dimensional commutative $k$-algebras.  Since $k$ is algebraically
closed, each $Z_i$ has residue field $k$ and nilpotent radical.  In
$K\otimes_k Z_i$ the extended radical is nilpotent and the quotient is
$K$, so this algebra is again local.  Thus the primitive central
idempotents correspond bijectively.

For a $p$-subgroup $P\le G$, the Brauer map deletes the coefficients
outside $C_G(P)$, and hence
\[
 \Br_P^K(1\otimes b)=1\otimes\Br_P^k(b).
\]
The map $kC_G(P)\longrightarrow KC_G(P)$ is injective.  The two Brauer
images are therefore nonzero for exactly the same $P$.  Their maximal
such $p$-subgroups, which are the defect groups by the convention in
Section~\ref{sec:preliminaries}, are identical.

For the Morita assertion, choose finite-dimensional inverse Morita
bimodules ${}_{A'}U_A$ and ${}_A V_{A'}$.  The $k$-linearity of the
equivalence means that the left and right $k$-actions on these bimodules
agree.  The Morita evaluation isomorphisms are
\[
 U\otimes_A V\simeq A',\qquad
 V\otimes_{A'}U\simeq A.
\]
Tensor them with $K$.  The canonical base-change isomorphism
\[
 (K\otimes_k U)\otimes_{K\otimes_k A}(K\otimes_k V)
   \simeq K\otimes_k(U\otimes_A V)
\]
and its counterpart with $U,V$ interchanged give the two Morita
evaluation isomorphisms for the extended bimodules.  Their compatibility
with the bimodule actions and with one another is preserved by tensoring.
Thus the extended bimodules give inverse $K$-linear tensor equivalences.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:donovan}]
First take $k=\overline{\F}_p$.  Proposition~\ref{prop:sylow-comparison}
supplies the hypothesis of Theorem~\ref{thm:extension-finiteness}.
Together with Theorem~\ref{thm:quasisimple-cartan}, it proves finiteness
over $k$.  The integral comparisons in this section serve the finite-list
argument for the base blocks; the final scalar normalization, and hence
the assertion for arbitrary crossed extensions, is over $k$.

Now fix an algebraically closed field $K$ of characteristic $p$ and
choose an embedding $k\hookrightarrow K$.  For the fixed bound $M$,
choose blocks $A_1,\ldots,A_t$ representing the finitely many
$k$-linear Morita classes of blocks with defect-group order at most $M$.
Every block $B$ of a finite group algebra $KG$ is, by
Lemma~\ref{lem:coefficient-extension}, of the form
$B=K\otimes_k B_0$ for a unique block $B_0$ of $kG$, and $B_0$ has the
same defect groups as $B$.  If their order is at most $M$, then $B_0$ is
$k$-linearly Morita equivalent to some $A_i$.  The same lemma extends
this equivalence to a $K$-linear Morita equivalence between $B$ and
$K\otimes_k A_i$.  For this fixed field $K$, all required blocks are
therefore represented by the finite list
$K\otimes_k A_1,\ldots,K\otimes_k A_t$.
\end{proof}
==== END Donovans-Conjecture-over-Algebraic-Closures-of-Prime-Fields//08_periodicity.tex ====

==== BEGIN Donovans-Conjecture-over-Algebraic-Closures-of-Prime-Fields//common.tex ====
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\newcommand{\F}{\mathbb F}
\newcommand{\Z}{\mathbb Z}
\newcommand{\Q}{\mathbb Q}
\newcommand{\C}{\mathbb C}
\newcommand{\N}{\mathbb N}
\DeclareMathOperator{\Irr}{Irr}
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==== END Donovans-Conjecture-over-Algebraic-Closures-of-Prime-Fields//common.tex ====

==== BEGIN Donovans-Conjecture-over-Algebraic-Closures-of-Prime-Fields//main.tex ====
\documentclass[11pt]{article}
\usepackage[a4paper,margin=29mm]{geometry}
\input{common}
\hypersetup{
  pdftitle={Donovan's Conjecture over Algebraically Closed Fields},
  pdfauthor={OpenAI}
}
\title{Donovan's Conjecture over Algebraically Closed Fields}
\author{OpenAI}
\date{September 24, 2026}
\begin{document}
\maketitle
\input{sections/00_introduction}
\clearpage
\begingroup
\small
\tableofcontents
\endgroup
\clearpage
\input{sections/01_preliminaries}
\input{sections/02_geometry}
\input{sections/03_families}
\input{sections/04_labels}
\input{sections/05_runners}
\input{sections/06_endpoint}
\input{sections/07_extensions}
\input{sections/08_periodicity}
\input{references}
\end{document}
==== END Donovans-Conjecture-over-Algebraic-Closures-of-Prime-Fields//main.tex ====

==== BEGIN Donovans-Conjecture-over-Algebraic-Closures-of-Prime-Fields//proof-interfaces.tex ====
\begin{figure}[tbp]
\centering
\setlength{\fboxsep}{6pt}
\begin{tabular}{@{}c@{\hspace{1em}}c@{}}
\fbox{\begin{minipage}{.40\linewidth}\centering\small
Fixed-root geometry; classical labels,\par
runner cones and triangle endpoints\par
Sections~\ref{sec:geometry}--\ref{sec:endpoint}
\end{minipage}}
&
\fbox{\begin{minipage}{.40\linewidth}\centering\small
Frobenius comparisons compatible\par
with Sylow actions and inner factors\par
Section~\ref{sec:periodicity}
\end{minipage}}
\\[3pt]
$\Downarrow$ & $\Downarrow$ \\[3pt]
\fbox{\begin{minipage}{.40\linewidth}\centering\small
Bounded quasisimple Cartan entries\par
Theorem~\ref{thm:quasisimple-cartan}
\end{minipage}}
&
\fbox{\begin{minipage}{.40\linewidth}\centering\small
Bounded equivariant Frobenius period\par
Proposition~\ref{prop:sylow-comparison}
\end{minipage}}
\\[3pt]
\multicolumn{2}{c}{$\searrow\qquad\qquad\swarrow$}\\[3pt]
\multicolumn{2}{c}{\fbox{\begin{minipage}{.83\linewidth}\centering\small
Generalized Fitting reduction and integral base finiteness\par
Comparison with trivial action: finitely many integral base orders\par
Full Sylow comparison: finitely many Sylow crossed systems\par
with the base algebra fixed\par
Section~\ref{sec:extensions}
\end{minipage}}}\\[3pt]
\multicolumn{2}{c}{$\Downarrow$}\\[3pt]
\multicolumn{2}{c}{\fbox{\begin{minipage}{.83\linewidth}\centering\small
Removal of the large $p'$-kernel and Sylow restriction\par
Finitely many $k$-linear Morita classes for bounded defect order\par
Theorem~\ref{thm:extension-finiteness}
\end{minipage}}}
\end{tabular}
\caption{The two controls used in the proof.  Arrows indicate uses
of the displayed outputs together with the structural and finiteness
inputs stated in Section~\ref{sec:extensions}.  The trivial-subgroup
case of the comparison supplies integral base finiteness; its full
Sylow form controls the multiplication in crossed extensions while
keeping the specified base algebra fixed.  Here
\(k=\overline{\mathbb F}_p\); Section~\ref{sec:coefficient-extension}
extends the resulting finite list to each fixed algebraically closed
field \(K\) of characteristic \(p\), completing
Theorem~\ref{thm:donovan}.}
\label{fig:proof-interfaces}
\end{figure}
==== END Donovans-Conjecture-over-Algebraic-Closures-of-Prime-Fields//proof-interfaces.tex ====

==== BEGIN Donovans-Conjecture-over-Algebraic-Closures-of-Prime-Fields//references.tex ====
\begin{thebibliography}{99}
\raggedright

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