\documentclass[11pt]{article}
\usepackage[T1]{fontenc}
\usepackage{lmodern}
\usepackage[margin=1.05in]{geometry}
\usepackage{amsmath,amssymb,amsthm,mathtools}
\usepackage{mathrsfs}
\usepackage{microtype}
\usepackage{needspace}
\usepackage{graphicx}
\usepackage{float}
\usepackage{tikz}
\usepackage{tikz-cd}
\usetikzlibrary{arrows.meta,positioning,calc,fit}
\usepackage[colorlinks=true,linkcolor=blue!45!black,citecolor=blue!45!black,urlcolor=blue!45!black]{hyperref}
\hypersetup{pdftitle={Counterexamples to the Hahn--Wilson conjecture at height two},pdfauthor={OpenAI}}
\DeclareMathOperator{\Thick}{Thick}
\DeclareMathOperator{\Spec}{Spec}
\DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\Ext}{Ext}
\DeclareMathOperator{\Tor}{Tor}
\DeclareMathOperator{\gr}{gr}
\DeclareMathOperator{\cofib}{cofib}
\DeclareMathOperator{\fib}{fib}
\DeclareMathOperator{\hocolim}{hocolim}
\DeclareMathOperator{\holim}{holim}
\DeclareMathOperator{\coker}{coker}
\DeclareMathOperator{\im}{im}
\DeclareMathOperator{\id}{id}
\DeclareMathOperator{\Perf}{Perf}
\DeclareMathOperator{\Mod}{Mod}
\DeclareMathOperator{\CT}{CT}
\newcommand{\Sp}{\mathrm{Sp}}
\newcommand{\BP}{\mathrm{BP}}
\newcommand{\Z}{\mathbb Z}
\newcommand{\F}{\mathbb F}
\newcommand{\Q}{\mathbb Q}
\newcommand{\Gm}{\mathbb G_m}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{notation}[theorem]{Notation}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{example}[theorem]{Example}
\numberwithin{equation}{section}
\setcounter{tocdepth}{2}
\title{Counterexamples to the Hahn--Wilson conjecture at height two}
\author{OpenAI}
\date{September 26, 2026}

\begin{document}
\maketitle

\begin{abstract}
We disprove the Hahn--Wilson conjecture at height two. For every sufficiently large
prime $p$, we construct a connective $p$-complete spectrum $X$ of exact fp-type two
that lies outside the ordinary thick subcategory generated by a standard completed
$\BP\langle2\rangle$. The same spectrum satisfies both localization comparisons
$L_2^fX\simeq L_2X$ and $L_{T(2)}X\simeq L_{K(2)}X$. At the end, we record the
positive local statement that holds for this example after $K(2)$-localization, so
that the failure is clearly a failure of gluing rather than of any local finiteness.
\end{abstract}

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

Which infinite spectra can be built from truncated Brown--Peterson spectra using
finitely many cofiber sequences and retracts? The Hahn--Wilson conjecture proposes a
cohomological answer: being fp (finitely presented over the Steenrod algebra),
together with a chromatic height bound, should be exactly the class of such finite
constructions. We build counterexamples at height two. Their chromatic localizations
keep every finiteness property one could ask for locally; the obstruction lies purely
in how the local pieces are joined.

\subsection{The question and the theorem}

Fix a prime $p$ and write $K(j)$ for Morava $K$-theory, with $K(0)=H\Q$. A nonzero
finite $p$-local spectrum has \emph{type} $j$ if $j$ is the least index for which its
$K(j)$-homology is nonzero; the thick subcategory theorem classifies thick
subcategories of finite $p$-local spectra by type \cite{HS1998}.

A bounded-below $p$-complete spectrum $X$ is \emph{fp} if $H^*(X;\F_p)$ is finitely
presented over the mod-$p$ Steenrod algebra. Mahowald and Rezk prove that this is
equivalent to asking that $\pi_*(V\wedge X)$ be \emph{finite in total} for some (hence
every) finite type-one $p$-local spectrum $V$, meaning finitely many nonzero homotopy
groups, each finite \cite[Proposition~3.2]{MR1999}. For an fp-spectrum, fp-type at most
$n$ means this test holds for a finite spectrum of type $n+1$. Exact fp-type two means
fp-type at most two but not at most one.

For a $p$-complete spectrum $B$, let $\Thick(B)$ be its closure in $\Sp_p^\wedge$
under equivalences, finite sums, integer shifts, cofiber sequences and retracts. This
is ordinary thick closure. The conjecture is
\begin{equation}\label{eq:hw-conjecture}
 \{X:\text{$X$ has fp-type at most $n$}\}=\Thick(\BP\langle n\rangle_p^\wedge).
\end{equation}
Its written form appears in Lee--Pstr\k{a}gowski \cite[Conjecture~1.1 and Note~1]{LP2026},
who attribute it to Dylan Wilson (2021). A standard completed truncated
Brown--Peterson spectrum has coefficients $\Z_p[v_1,\ldots,v_n]$ with $|v_i|=2(p^i-1)$.

Let $L_n=L_{E(n)}$ be Johnson--Wilson localization and $L_n^f$ finite localization
away from finite spectra of type greater than $n$. For $n>0$, let $T(n)$ be the
telescope of a periodic self-map of a finite type-$n$ spectrum.

\begin{theorem}\label{thm:main}
For every sufficiently large prime $p$, there is a connective $p$-complete spectrum
$X$ of exact fp-type two such that, for a standard form of $\BP\langle2\rangle_p^\wedge$,
\[
 X\notin\Thick(\BP\langle2\rangle_p^\wedge),
\]
and the same $X$ satisfies
\[
 L_2^fX\simeq L_2X,\qquad L_{T(2)}X\simeq L_{K(2)}X.
\]
\end{theorem}

The theorem refutes the universally quantified conjecture \eqref{eq:hw-conjecture}; its
prime range is all sufficiently large primes. We use one specific regular
$\BP$-module quotient throughout, written $b$. At these odd primes the underlying
spectrum is independent of the choice of standard quotient generators
\cite[Corollary~1.9]{Lee2026}. That uniqueness rests on Angeltveit--Lind, with its
corrected argument and the additional $p$-local statement of Lee \cite{AL2017,Lee2026}.

\subsection{Global finiteness versus the local classification}

Mahowald--Rezk's duality theory makes fp-spectra a natural home for infinite
spectra with controlled chromatic behavior \cite[Theorem~8.11]{MR1999}, and
Hahn--Wilson's redshift theorem supplies further examples: for the multiplicative
forms in their theorem, $K(\BP\langle n\rangle_p^\wedge)_p^\wedge$ has fp-type
$n+1$ \cite[Theorem~B]{HW2022}. The cases of fp-type at most $-1$ and $0$ are
classical, and Lee--Pstr\k{a}gowski prove the global generation conjecture at
height one \cite[Theorems~1.3 and 5.4]{LP2026}.

Mahowald and Rezk asked whether $L_n^fX\to L_nX$ is an equivalence for every
fp-spectrum \cite[Conjecture~7.3]{MR1999}. Burklund--Hahn--Levy--Schlank disproved
the unrestricted telescope conjecture at every prime and every height at least
two \cite[Theorem~A]{BHLS2023}. The finiteness distinction is essential:
Lee--Pstr\k{a}gowski note that the finite-smash homotopy of those counterexamples
is bounded but not finite in total \cite[Section~1.1]{LP2026}. Those examples
therefore do not furnish the fp counterexample studied here.

There is, in contrast, a positive classification at every height after Morava
$K$-localization. A $K(n)$-local spectrum $Y$ is \emph{locally fp} if
$\pi_i(F\wedge Y)$ is finite for every integer $i$ and every finite type-$n$
spectrum $F$. Lee--Pstr\k{a}gowski identify these objects with $\Thick(E_n)$ in
the $K(n)$-local category, equivalently with
$\Thick(L_{K(n)}\BP\langle n\rangle)$ \cite[Theorems~1.7 and 4.3]{LP2026}, where
$E_n$ is Morava $E$-theory. The height-one local classification has an earlier
source in Hahn--Mitchell's Iwasawa theory \cite[Theorem~8.1]{HM2007}. Note the
difference in quantifiers: the local test asks for finiteness in each degree,
while the global fp-type test asks for finiteness in total.

For the example of Theorem~\ref{thm:main} put $X_K=L_{K(2)}X$. Then
$X_K\in\Thick(E_2)$, its cohomology $E_2^*X_K$ is finitely generated over the
ring of operations $E_2^*E_2$, and the natural map from $X_K$ to its double dual
(formed in the $K(2)$-local category) is an equivalence. Yet the original
spectrum $X$ admits no global finite construction from $\BP\langle2\rangle_p^\wedge$.
Lee--Pstr\k{a}gowski proposed using a height decomposition together with their
local theorem to prove global generation for fp-spectra satisfying the telescope
comparison \cite[Remark~5.1]{LP2026}. Theorem~\ref{thm:main} shows that this extra
hypothesis does not suffice at height two. The classification of thick tensor
ideals of locally dualizable spectra \cite{BHN2022} concerns a different set of
permitted constructions; here we keep the single generator and ordinary thick
closure fixed.

\subsection{The construction problem}\label{subsec:construction-problem}

At large primes, chromatic algebraicity lets us build the example in a periodic
derived category of sheaves on the stack of formal group laws. We use
Pstr\k{a}gowski's homology-compatible triangulated comparison, which at height
two is available for $p>8$ \cite[Corollary~6.9]{Pstragowski2021}. Section~\ref{comp:section}
records the source qualifications and supplies the needed local arguments for that
comparison. That bound concerns the comparison; the operator below requires larger
primes. The comparison transports finite triangles and retracts. All completion,
inverse-limit and gluing constructions are carried out first in the algebraic
category.

We work with the weighted stack of $p$-typical formal group laws of height at most
two. We pass to characteristic $p$, extend scalars to $k=\F_{p^2}$, and quotient
the central strict automorphisms $1+p\Z_p$; write $\mathcal N$ for the resulting
stack. The coefficient $v_i$ has weight $(p^i-1)/(p-1)$. Its Brown--Peterson
coefficient chart is
\[
 f':\mathcal X_k\longrightarrow\mathcal N,\qquad
 \mathcal X_k=[(\Spec k[v_1,v_2]\setminus\{(0,0)\})/\Gm],
\]
with coordinate weights $1$ and $p+1$; the map $f'$ uses the law with all higher
$v_i$ equal to zero. Put $T'=f'_*\mathcal O_{\mathcal X_k}$. The weight line
$\mathcal O(1)$ has the Cartier section $t=v_1$, which cuts out height two. The
open $\mathcal N[1/t]$ is $\Spec k$, while the completed chart retains a
noncommutative algebra of operations.

Here is the precise shape of the algebraic problem. Put $D=2(p-1)$, let
$\mathbf D(\mathcal N)$ denote the category of homological complexes with
weight-twisted periodicity $C_{j+D}=C_j(1)$ ($(1)$ is the weight-line twist), and
for $U\in\mathbf D(\mathcal N)$ set
\[
 \pi_iU=[\mathcal O[i],U],\qquad
 \widehat U=\holim_d\cofib(t^d\colon U[dD]\to U).
\]
So $\pi_iU$ is the homotopy of derived global sections.
Proposition~\ref{transfer:criterion} reduces the whole problem to finding $U$
with
\begin{equation}\label{eq:target}
 \pi_iU\text{ finite for every }i,\qquad
 \widehat U\in\Thick(\widehat T'),\qquad
 U\notin\Thick(T').
\end{equation}
The first condition produces finite homotopy groups and yields the fp test for a
connective cover; the second controls the change back from the reduced stack; the
third supplies the obstruction to a global finite construction. The transfer proof
uses nonsplit comparison triangles and nilpotence; it does not require these
coefficient changes to have sections.

To produce $U$, begin with the cone of a single operation on $\widehat T'$, and let
$Y$ be the sum of its shifts through one period. On the overlap, the operation acts
on Laurent series; we arrange many independent kernel vectors and cokernel classes,
controlled by a triangular matching that also makes the image closed. These become
the two summands of the overlap sections $\pi_iLY$, where $L$ inverts $t$. In each
degree we then pair kernel and cokernel directions inside a discrete cocompact
$k$-subspace of $\pi_iLY$. Since the open is $\Spec k$, these subspaces are
realized by an open periodic complex $V$ and a map $V\to LY$, and we define $U$ by
the homotopy pullback
\[
 \begin{tikzcd}[column sep=large,row sep=large]
  U \arrow[r] \arrow[d] & Y \arrow[d] \\
  V \arrow[r]            & LY .
 \end{tikzcd}
\]
In each degree, the complete part maps to a compact open subspace of the overlap
with finite kernel, and the chosen discrete subspace complements this image up to
finite error. Consequently the joint map
\(\pi_iY\oplus\pi_iV\to\pi_iLY\) has finite kernel and cokernel, and the pullback
long exact sequence makes $\pi_iU$ finite in every degree. Completion kills both
open objects $V$ and $LY$, so $\widehat U\simeq Y$; this gives the first two
conditions in \eqref{eq:target}.

The third condition is the main obstruction. An independent estimate bounds certain
joint operator kernels at Laurent precision $N$ by $O(N^{1/3})$, while the paired
defects grow at least as $cN^\beta$ for a fixed $\beta>1/3$ and every sufficiently
large $N$. This discrepancy constrains every global map from $U$ into a finite
construction from $T'$: after a flat coefficient extension, vanishing on homology
forces vanishing on the homology of the module dual. The dual condition removes
the free part of each stage of a possible finite tower, and the homology condition
then passes to the preceding stage. Backward induction excludes every such tower
and its retracts. The categorical criterion is proved before its growth hypothesis.
The use of ghosts and finite extension towers has an antecedent in Christensen
\cite{Christensen1998}; the homology-to-dual-homology implication is proved here
from the selected gluing data.

\subsection{Reading the proof}

The algebraic argument comes first. Section~\ref{geom:section} defines the reduced
setting and proves its chart geometry. Section~\ref{complete:section} computes
complete operations using compatible maps on finite reductions and realizes perfect
modules as actual complexes, so that the calculation controls maps and cones, not
only endomorphism groups. Sections~\ref{op:section} and~\ref{glue:section} construct
the Laurent operator and the gluing. Section~\ref{obs:section} proves the
finite-tower criterion and verifies its hypothesis for the glued object together
with its growth estimate. These arguments explain how complete operations control
finite constructions and how the gluing obstructs them globally.

Sections~\ref{comp:section} and~\ref{transfer:section} establish the comparison
with spectra and prove the transfer proposition. The latter takes the connective
cover of the resulting $E(2)$-local spectrum, following Mahowald--Rezk
\cite[Proposition~3.5 and Section~7.2(2)]{MR1999}; in particular it proves the
total-finiteness test behind fp-ness. Section~\ref{comp:section} additionally gives
the Brown--Peterson quotient calculation and a thick-transport principle for
equivalences of underlying categories. Section~\ref{scope:section} establishes
exact fp-type two and both localization comparisons, then compares the example with
the positive local theory and proves the separate obstruction for the connective
cover of Morava $E$-theory.

\tableofcontents

\section{Law charts and the height-two quotient}\label{geom:section}

The counterexample lives in a category of periodic complexes of sheaves on a
stack of formal group laws. This section builds the stack and its chart sheaf,
then sets up the tools for computing sections. We start integrally over
$\Z_{(p)}$, pass to characteristic $p$, quotient a central procyclic group of
automorphisms, and obtain a quotient with a simple height-one open and explicit
nilpotent neighborhoods of height two. The height-two stabilizer provides both
the coefficient ring for the construction and the finite cohomological bounds
used in the periodic category.

Throughout this section $p>5$ is prime. Put
\[
 D=2(p-1),\qquad m_i=\frac{p^i-1}{p-1},\qquad q=m_2=p+1.
\]
Further large-prime conditions are imposed only when constructing the operator.

\subsection{The integral law groupoid}

We work over $\Z_{(p)}$ so that the same flat chart yields both the
characteristic-$p$ construction and the later integral comparison. Use the
classical Brown--Peterson Hopf algebroid
\[
 P=\Z_{(p)}[v_i:i\geq1],\qquad \Gamma_P=P[t_i:i\geq1]
\]
of strict $p$-typical formal group laws, dividing all topological degrees by $D$
so that $v_i$ and $t_i$ have weight $m_i$. The formal-group and cooperation
calculations used below are recalled in \cite[Chapter~4]{Ravenel1986}; for the
Araki formulas see \cite[Appendix~B.5.5--B.5.6, B.5.12, and B.5.15]{Ravenel1992}.
In Araki coordinates the universal law $F$ satisfies
\begin{equation}\label{geom:p-series}
 [p]_F(T)=\mathop{\textstyle\sum}\nolimits_{i\geq0}^{F}
             v_iT^{p^i},\qquad v_0=p,
\end{equation}
where the superscript denotes addition in $F$, not ordinary addition.

Why this coordinate is integral: write $\log_F(T)=\sum_{i\geq0}l_iT^{p^i}$ and
successively subtract the lower terms of \eqref{geom:p-series} from $[p]_F$ using
formal-group subtraction. The logarithm shows that the next remainder starts at
$T^{p^n}$; its coefficient is integral, since all preceding subtractions were
integral. Comparing with the Hazewinkel recursion
\[
 p l_n=v_n^{\mathrm{Haz}}+\sum_{0<i<n}l_i(v_{n-i}^{\mathrm{Haz}})^{p^i}
\]
gives a triangular change of generators with diagonal $1-p^{p^n-1}$, a unit in
$\Z_{(p)}$.

A strict arrow $F_R\to F_L$ has the form
\[
 \phi(T)=T+_{F_L}\mathop{\textstyle\sum}\nolimits_{i\geq1}^{F_L}t_iT^{p^i}.
\]
The coefficients of $F_R$ are given by the right unit $\eta_R$. Commuting $\phi$
with the $p$-series gives
\begin{equation}\label{geom:invariant-ideals}
 \eta_Rv_i\equiv v_i\pmod{(p,v_1,\ldots,v_{i-1})}.
\end{equation}
In particular the ideal $(p,v_1,v_2)$ defines an invariant closed locus.

Let $\mathcal M$ be the stack of this groupoid on
$\Spec P\setminus V(p,v_1,v_2)$, also divided by $\Gm$ for the regraded weights.
All stacks and sheaves in this section use fpqc descent. Define
\[
 \begin{aligned}
 B_2&=\Z_{(p)}[v_1,v_2],\qquad A_2=B_2[v_2^{-1}],\\
 \mathcal X&=[(\Spec B_2\setminus V(p,v_1,v_2))/\Gm],
 \end{aligned}
 \qquad f:\mathcal X\longrightarrow\mathcal M,
\]
where $f$ puts $v_{i>2}=0$. We also use the standard Johnson--Wilson coefficient
chart
\[
 A_{\mathrm{Haz}}=\Z_{(p)}[v_1^{\mathrm{Haz}},(v_2^{\mathrm{Haz}})^{\pm1}]
\]
with all higher Hazewinkel generators set to zero.

\begin{proposition}\label{geom:law-charts}
The map $f$ is flat and affine. The maps $\Spec A_2\to\mathcal M$ and
$\Spec A_{\mathrm{Haz}}\to\mathcal M$ are flat affine covers. In graded
coordinates the cooperation Hopf algebroid of the latter chart has arrow ring
\[
 A_{\mathrm{Haz}}\otimes_{P,\eta_L}\Gamma_P
                  \otimes_{P,\eta_R}A_{\mathrm{Haz}}.
\]
\end{proposition}

\begin{proof}
We verify flatness, affineness, then coverage.

Flatness. The Landweber flatness criterion applies to each localization of $B_2$
inverting $p$, $v_1$, or $v_2$, and to $A_{\mathrm{Haz}}$: the invariant regular
sequence remains regular up to the point where its ideal becomes the unit ideal.
The interpretation of this criterion as flatness of law charts is the standard
equivalence between Landweber exact comodules and flat descent; see
\cite[Lemma~2.2, Corollary~2.3, and Theorem~2.5]{HS2005} and
\cite[Section~5, Theorem~26, and Proposition~27]{Naumann2007}.

Passing from exactness on graded comodules to the flatness we use. Apply exact
coinduction $\Gamma_P\otimes_P(-)$ to a sequence of graded $P$-modules, then base
change to the chart ring $A$. This tests flatness of
$A\otimes_{P,\eta_L}\Gamma_P$ over $P$ via $\eta_R$; inverting arrows interchanges
the two units if necessary. Graded flatness here implies ordinary flatness: for an
ungraded test module $N$, place a copy of $N$ in every weight and let each
homogeneous element act through its ordinary action into the correspondingly
shifted copy. A fixed weight of the graded tensor product recovers the ungraded
tensor product; graded exactness therefore tests arbitrary modules.

Affineness. After pulling back to an affine scheme of laws with its grading frame
trivialized, the fiber of $f$ is the affine scheme of strict arrows whose other
end is a $B_2$-law. The deleted locus cannot occur at that end by
\eqref{geom:invariant-ideals}. Fiber products of affine law charts are affine:
besides the strict-arrow coordinates they retain only an invertible grading
parameter. Hence the relevant chart maps and diagonals are affine.

Coverage. Check on geometric points. In characteristic zero the logarithm gives
strict isomorphisms of the required $p$-typical form. In characteristic $p$, let
$h=1$ or $2$ be the height and compare two laws over an algebraically closed field
with the same nonzero coefficient $v_h$. For a fixed left law, solve for an arrow
by prescribing the right coefficients in order. After the preceding equations have
been imposed, the equation at step $j\geq1$ is
\begin{equation}\label{geom:recursive-arrow}
 \eta_Rv_{j+h}=v_h t_j^{p^h}-v_h^{p^j}t_j
       +\text{a polynomial already determined by }t_{<j}.
\end{equation}
Indeed, compare degree $p^{j+h}$ in $[p]_{F_L}(\phi(T))=\phi([p]_{F_R}(T))$. All
exponents in the left $p$-series are divisible by $p^h$, so only $\phi$ through
degree $p^j$ enters there; the first occurrence of $t_j$ is additive in degree
$p^j$. On the right, apart from the displayed coefficient $\eta_Rv_{j+h}$, only
previously fixed coefficients enter: the formal-sum coefficients needed at this
step have total exponent at most $p^j$ and hence, by homogeneity, use only
generators of weight at most $m_j$. This gives \eqref{geom:recursive-arrow}. Its
leading coefficient is nonzero, so it can be solved at every step. Both charts
contain laws of the prescribed height and leading coefficient; for the Hazewinkel
chart use the homogeneous triangular coordinate change above. This proves
coverage.

The displayed arrow ring is the usual Landweber base-change calculation for
cooperations, and is also the ring obtained directly by pulling back the two ends
of the law groupoid.
\end{proof}

We fix the chart sheaves
\begin{equation}\label{geom:integral-chart-sheaves}
 T_{\Z}=f_*\mathcal O_{\mathcal X},\qquad
 T_{\Z_p}=f_*\bigl(\mathcal O_{\mathcal X}\otimes_{\Z_{(p)}}\Z_p\bigr).
\end{equation}
They are \emph{$p$-monic}, meaning multiplication by $p$ is injective: this holds
on $\mathcal X$ and is preserved by the affine pushforward. The comparison will
identify these with the algebraic images of Brown--Peterson spectra.

\subsection{Removing the central procyclic group}

The construction uses the characteristic-$p$ law stack. On that stack a central
copy of $\Z_p$ acts through strict automorphisms of every law; removing this common
inertia leaves a height-two stabilizer whose invariants have cohomological
dimension at most three.

Put $k=\F_{p^2}$. Write
\[
 i:\mathcal M_0\longrightarrow\mathcal M,
 \qquad e:\mathcal M_k\longrightarrow\mathcal M_0
\]
for reduction to characteristic $p$ and extension from $\F_p$ to $k$. Subscripts
$0$ and $k$ on $\mathcal X$ and $f$ mean the corresponding base changes. The
proconstant group $Z=1+p\Z_p\simeq\Z_p$ acts as central inertia via the strict
automorphisms $[z]_F$; here proconstant means an inverse limit of finite constant
group schemes, so its functions over $k$ are locally constant $k$-valued functions.
Integer multiplication has the $p$-typical formal-sum expansion from logarithms and
successive subtraction. In characteristic $p$ the $p$-series has formal-variable
order at least $p$, so these multiplications extend continuously to $p$-adic
integers on every jet; they commute with all arrows and with the grading, and act
faithfully on geometric fibers because the laws have finite height. Fix a
topological generator $c$ of $Z$.

We build an affine groupoid presentation for the quotient and prove the chart
remains affine, so that its pushforward stays accessible to the coinduction
calculation of Proposition~\ref{geom:coinduction}.

\begin{lemma}\label{geom:torsor}
Let $A$ be a $k$-algebra with a locally finite continuous action of $Z$, and let
$Z_m$ be the subgroup of index $p^m$. Suppose $A=\bigcup_{m\geq0}A^{Z_m}$ and the
action on $\Spec A$ is geometrically free. Then $\Spec A\to\Spec(A^Z)$ is an fpqc
$Z$-torsor.
\end{lemma}

\begin{proof}
First descend the free action to finite quotients, then prove the finite torsor
assertion directly and pass to the limit. Set $A_m=A^{Z_m}$. Every element has
finite orbit, so $A$ is integral over each $A_m$. Two geometric points with the
same restriction to $A_m$ differ by $Z_m$: restrict first to a subalgebra generated
by finitely many finite orbits, use the Chinese remainder theorem and an invariant
product of separating functions, and use flatness of the field extension (finite
group invariants commute with field extension). The finite-level assertions give
nonempty closed sets of possible elements of $Z_m$, and compactness supplies a
compatible element for the whole algebra.

Integrality lifts every geometric point of $A_m$ to one of $A$, and the induced
$Z/Z_m$-action on $\Spec A_m$ is free: if a coset fixed such a point, lifting to a
point of $A$ shows the translate and the original differ by $Z_m$, so geometric
freeness upstairs forces the coset to be trivial.

A finite constant group $K$ acting geometrically freely on an affine scheme gives a
torsor over the invariant ring even when $|K|$ is not invertible. At a lift of a
geometric point of the invariant spectrum choose $b_1,\ldots,b_{|K|}$ whose
evaluations on its orbit form an invertible matrix (Chinese remainder theorem after
field extension); then $D_b=\det(g(b_j))_{g\in K,j}$ satisfies: $D_b^2$ is
invariant, and after inverting it the coefficients of any element in the basis
$b_j$ are invariant. These neighborhoods cover the invariant spectrum by
integrality.

Apply to $A_m$: $\Spec A_m\to\Spec A^Z$ is a finite faithfully flat
$Z/Z_m$-torsor. Passing to the filtered union gives faithful flatness and the
torsor identity for $A$. The resulting affine map is quasi-compact, hence fpqc.
\end{proof}

\begin{proposition}\label{geom:rigidification}
There is a flat affine groupoid stack $\mathcal N$ and a map
$r:\mathcal M_k\to\mathcal N$ that quotients the vertical central $Z$-action on
arrows. Its pullback is exact and identifies sheaves on $\mathcal N$ with sheaves
on $\mathcal M_k$ carrying trivial vertical $Z$-action. Moreover
$f'=rf_k:\mathcal X_k\to\mathcal N$ is flat and affine. The weight line descends to
$\mathcal N$, and $t=v_1$ is an invariant Cartier section of $\mathcal O(1)$ there.
\end{proposition}

\begin{proof}
Construct the quotient on an affine presentation, then check the chart and the
weight line. Take $S=\Spec(A_2\otimes k)$ and quotient its affine arrow scheme
$S\times_{\mathcal M_k}S$ by vertical composition with $Z$. The hypotheses of
Lemma~\ref{geom:torsor} hold: the action is geometrically free by the leading orders
of finite-height $p$-series, and it is locally finite on the arrow coefficients by
uniform continuity on each formal-variable jet. So the quotient is affine and the
original arrow scheme is an fpqc torsor over it. Composition descends by centrality,
since after locally lifting two composable arrows, changing either lift changes
their composite by the corresponding vertical element. Identity and inverse descend
as well; source and target stay flat by faithfully flat descent. This defines the
groupoid and $\mathcal N$. Descent of the arrow isomorphism through the torsor gives
the claimed description of $r^*$ and its exactness.

Similarly $S\times_{\mathcal N}\mathcal X_k$ is the quotient of
$S\times_{\mathcal M_k}\mathcal X_k$ by vertical composition. An isomorphism in the
quotient has local lifts whose choices form a $Z$-torsor, visible on the arrow
chart. There is no residual automorphism from $\mathcal X_k$ in this vertical
subgroup: an automorphism from $\mathcal X_k$ is a grading element, whose grading
parameter must be identity to lie in the strict subgroup. This holds fpqc locally
over arbitrary bases, not only on geometric points. The quotient is therefore the
invariant affine of Lemma~\ref{geom:torsor}, and its map to $S$ is flat because the
map upstairs is flat and the torsor is faithfully flat.

The torsor descriptions commute with base change, so the central action is trivial
on the weight line and that line descends. Equation~\eqref{geom:invariant-ideals} in
characteristic $p$ makes $v_1$ invariant, and it is a non-zero-divisor on the law
charts. This proves the Cartier assertion.
\end{proof}

Use the same symbol $t$ on $\mathcal M_0$ and $\mathcal M_k$, and set
\begin{equation}\label{geom:characteristic-chart-sheaves}
 T_0=f_{0*}\mathcal O_{\mathcal X_0},\qquad
 T_k=f_{k*}\mathcal O_{\mathcal X_k},\qquad
 T'=f'_*\mathcal O_{\mathcal X_k}.
\end{equation}
The arrangements are:
\[
\begin{array}{ccccccc}
 \mathcal N&\xleftarrow{r}&\mathcal M_k&\xrightarrow{e}&\mathcal M_0&
 \xrightarrow{i}&\mathcal M\\
 T'&&T_k&&T_0&&T_{\Z},\,T_{\Z_p}.
\end{array}
\]
The upper row records maps of stacks: $r$ removes the central automorphisms, $e$
extends scalars from $\F_p$ to $k$, and $i$ is the characteristic-$p$ inclusion. The
lower row records the chart objects in those settings.
Sections~\ref{complete:section}--\ref{obs:section} carry out the algebraic
construction on $\mathcal N$ in the periodic category defined below, with $T'$ as
its chart object. Sections~\ref{comp:section}--\ref{transfer:section} return to the
integral setting and transfer the construction to spectra.

The Cartier section $t$ determines the height-one open and the height-two
thickenings; we next fix the category and the section groups in which the
construction is measured.

\subsection{Periodic complexes and sections}\label{geom:periodic-foundations}

Later we assemble quotients by powers of $t$ using inverse limits, so the category
must allow arbitrary quasi-coherent sheaves as well as finite constructions. Fix
its conventions before describing the charts on which the quotients are computed.
The weight twist of a graded module is
\[
 W(s)_a=W_{s+a}.
\]
For a stack $\mathcal S$ with a flat affine groupoid presentation and weight line
$\mathcal O(1)$, let $\mathbf D(\mathcal S)$ be the derived category of twisted
periodic homological complexes of quasi-coherent sheaves:
\begin{equation}\label{geom:periodic-complex}
 C_{j+D}=C_j(1),\qquad d:C_j\longrightarrow C_{j-1}.
\end{equation}
The differential respects the periodic identifications. We retain the associated
stable homotopy theory when forming homotopy limits or colimits. Write
$C[h]=\Sigma^hC$, so $(C[h])_j=C_{j-h}$ with the usual sign on the differential.
A sheaf $W$ denotes the pure complex with $C_0=W$, zero terms outside the residue
class $0\pmod D$, and extended by the twists.

For the integral stack $\mathcal M$, Section~\ref{comp:section} will identify this
periodic category, via flat descent and an explicit degree regrading, with the
differential $E_*E$-comodule model used in the chromatic comparison, where
$E=E(2)$.

For any of these categories set
\begin{equation}\label{geom:sections-homotopy}
 \pi_jC=[\mathcal O[j],C]_{\mathbf D(\mathcal S)}.
\end{equation}
This is sections homotopy, not underlying sheaf homology. For example, a sheaf map
$W\to W'(s)$ extends to a map $W[sD]\to W'$ of pure complexes. Such sheaf maps need
not account for all derived maps. After the chart calculations, we construct the
model structure and finite resolutions that justify our computations of these
derived groups.


...
Let $\mathcal S$ be any of the stacks of
Section~\ref{geom:section}. The periodic complexes of
\eqref{geom:periodic-complex} form a Grothendieck abelian category. Every
weight twist preserves quasi-isomorphisms and the periodicity, so universal
$\delta$-functors are compatible with it.

\begin{lemma}\label{geom:periodic-grothendieck}
The category of twisted periodic complexes of sheaves on $\mathcal S$ is
Grothendieck abelian. Twisting preserves quasi-isomorphisms and the periodicity.
If the underlying ordinary stack is quasi-compact, quasi-separated, and flat
affine, then quasi-isomorphisms form an accessible $\omega$-localizing class and a
combinatorial monoidal model structure exists, whose homotopy category is the
periodic $\mathbf D(\mathcal S)$. If the weight line is trivial as a sheaf (for
example $\mathcal O(1)$ is pulled back from the base $\Spec k$), this regraded
category is identified with the ordinary unbounded derived category of the sheaf
category.
\end{lemma}

\begin{proof}
Twisted periodic complexes are Grothendieck: filtered colimits are termwise and
exact, generators are disks on sheaf generators in adjacent indices with identity
differential and repeated twists. Use Smith's recognition theorem
\cite[Theorem~1.7]{Beke2000}; the solution-set condition follows from
accessibility \cite[Proposition~1.15]{Beke2000}. Quasi-isomorphisms satisfy
two-out-of-three and are closed under retracts (seen on homology). They are
accessibly embedded and accessible. For an explicit size argument, enlarge
$\kappa$ to include the twist data: in an arrow of complexes start with
boundedly many elements, close under coactions and differentials, and adjoin
preimages for cycles in its cone. Iterate; this expresses any quasi-isomorphism
as a $\kappa^+$-directed union of quasi-isomorphic subarrows of size at most
$\kappa$. Objects and arrows of that size are $\kappa^+$-presentable, since
module presentations, coaction identities, and finitely many periodic
differential identities can all be tested at that size on the flat affine charts.

For the regrading assertion: if $\mathcal O(1)$ is pulled back from the base, the
twist is trivial and the periodicity data is an ordinary $D$-periodic complex. The
functors $C\mapsto\bigoplus_jC_j$ and its truncation by residue modulo $D$ are
inverse up to quasi-isomorphism, since the differentials factor through the
trivial-identity periodic maps and their components match. This identifies the
periodic homotopy category with $\mathbf D(\mathcal S^{\mathrm{ord}})$.
\end{proof}

\begin{lemma}[Injective model]\label{geom:injective-model}
Let $\Hom(W,-)$ denote the graded complex of maps out of a fixed sheaf $W$. If
$\Ext^{>a}(\mathcal O,-)=0$ in the nonperiodic category, then on twisted periodic
complexes with bounded $\Hom(W,-)$-acyclic representatives, the finite totalization
of a length-$(a+1)$ coinduced resolution computes sections homotopy. Quasi-isomorphisms
between complexes with acyclic terms for $\Hom$ on each summand remain
quasi-isomorphisms after applying $\Hom(W,-)$.
\end{lemma}

\begin{proof}
For the last assertion, a fibrant replacement computes the derived maps; a
quasi-isomorphism between complexes with $\Hom(W,-)$-acyclic terms remains a
quasi-isomorphism after applying $\Hom(W,-)$, since its cone is acyclic with such
terms. Dimension shifting along the short exact sequences of cycles shifts the
possible obstruction to exactness past degree $a$, where it vanishes. The argument
uses only the uniform bound on $\Ext(W,-)$, hence works for the unbounded periodic
unfolding.
\end{proof}

\begin{lemma}\label{geom:sections-resolution}
The right derived functors of sections in the sheaf categories above commute with
filtered colimits and are computed by the flat affine coinduced resolution. The
resolution is compatible with closed base change. If $\Ext^{>a}(\mathcal O,-)=0$,
its finite truncation gives a sections-acyclic resolution computing
\eqref{geom:sections-homotopy} on periodic complexes; then $\pi_*$ commutes with
filtered colimits and telescopes.
\end{lemma}

\begin{proof}
Choose an affine fpqc chart $l:S\to\mathcal S$ with $S$ affine. The functor
$I=l_*l^*$ is exact, its unit is monic, and its unit cokernel $J$ is exact. Its
values are sections-acyclic by flat affine adjunction. Iterating $I$ and $J$ gives
the coinduced resolution. On the chart all these functors, including ordinary
sections of the coinduced terms, commute with filtered colimits; AB5 therefore
gives the same for their cohomology. After restricting to a closed substack, the
base-changed chart gives exactly the corresponding resolution. This proves the
statement for sheaves pushed forward from closed loci, without requiring derived
closed pushforward to be fully faithful.

Under the dimension bound, the resolution can end in $J^a$, which is
sections-acyclic by dimension shifting. These functors commute with weight twists.
The finite totalization therefore computes sections on periodic complexes by
Lemma~\ref{geom:injective-model}; no infinite totalization is needed. The colimit
assertion follows termwise.
\end{proof}

We use exact pullback and pushforward also on periodic complexes. An exact pullback
left adjoint is left Quillen for these injective structures and is adjoint to the
right derived pushforward. These adjunctions let the chart calculations control both
sheaves and the periodic complexes built from them.

\subsection{The height-one open and height-two thickenings}

We describe the two pieces determined by $t$. The open has no remaining
stabilizer after the central quotient; on each nilpotent thickening the stabilizer
acts on one explicit coordinate ring, which turns the reductions of $T'$ into
concrete coefficient modules.

\begin{proposition}\label{geom:open}
There are equivalences
\[
 \mathcal M_0[1/t]\simeq BZ,\qquad
 \mathcal N[1/t]\simeq\Spec k.
\]
The first is split by the law with $v_1=1$ and $v_{i>1}=0$.
\end{proposition}

\begin{proof}
At $t=v_1$ invertible the positive $v_i$ with $i\geq2$ are absent. The grading
normalization is made canonical by the law with $v_1=1$ and $v_{i>1}=0$: the
stabilizer of that law is the full Morava stabilizer group for height one, which
in this setting is $\Z_p^\times$, and after the central quotient the residual group
is $\Z_p^\times/Z$, of order $p-1$, acting on $\pi_0=\F_p$ by the cyclotomic
character and hence trivially on a geometric fiber. Removing inertia leaves $BZ$.
The full center remains only on the height-two locus. Taking vertical central
invariants of $\mathcal M_k[1/t]\simeq BZ$ produces $\Spec k$.
\end{proof}

Let
\[
 \Lambda=\mu_q(k),\qquad R_d=k[x]/(x^d)\quad(d\geq1),
 \qquad R=k[[x]].
\]
The group $\Lambda$ has order $q$, prime to $p$. The rings $R_d$ are the coordinate
rings of the thickenings, and $R$ records their compatible coefficients. Let $F_0$
be the Honda law with $v_2=1$ and $v_i=0$ for $i\ne2$, $i\geq1$. Denote its
proconstant group of strict automorphisms by $G^1$, and put
\[
 G_{\mathrm{up}}=G^1\rtimes\Lambda,\qquad
 H=G^1/Z,\qquad G=H\rtimes\Lambda.
\]
The action of $\Lambda$ on strict automorphisms is the weight action. The
semidirect decomposition is the usual Morava semidirect structure at height two.
We compute in characteristic $p$, so pro-objects are proconstant on the geometric
fibers, and proconstant indices correspond to the group's more subtle graded
pieces below.

The coordinate $t=v_1$ is a non-zero-divisor on the coefficient chart
$\mathcal X_k$. The ideal it generates gives each height-two thickening. The
special fiber of the arrow groupoid over the closed point of the chart is the
Frobenius squares together with the weight parameter, and its lifts to each
thickening determine the presentation.

\begin{proposition}\label{geom:thickened-charts}
The chart law $F_x$ with $v_1=x$, $v_2=1$, and $v_{i>2}=0$ gives compatible
presentations
\begin{equation}\label{geom:quotient-thickenings}
 \mathcal M_k/(t^d)=[\Spec R_d/G_{\mathrm{up}}],\qquad
 \mathcal N/(t^d)=[\Spec R_d/G].
\end{equation}
Under these, $\mathcal X_k/(t^d)=[\Spec R_d/\Lambda]$. Sheaves are smooth, meaning
discrete continuous, semilinear representations over $R_d$. The compatible
coefficient action on $R$ satisfies
\[
 g(x)\in x+x^2R\quad(g\in H),\qquad
 \lambda(x)=\lambda^{-1}x\quad(\lambda\in\Lambda).
\]
If $u$ is the chart basis of the weight line with $t=xu$, then
\begin{equation}\label{geom:twist-cocycle}
 g(u)=\frac{x}{g(x)}u\quad(g\in H),\qquad
 \lambda(u)=\lambda u\quad(\lambda\in\Lambda).
\end{equation}
For every strict label representing an element $g\in H$, there is a formal
isomorphism $F_{g(x)}\to F_x$ over $R$ reducing to that label.
\end{proposition}

\begin{proof}
Normalize $v_2$, then identify arrows by lifting their special-fiber labels.
Uniqueness of the lifts also identifies composition over every nilpotent base.

On $V(t^d)$ the coordinate $v_2$ is invertible. Normalize it to $1$ by adjoining
a $q$-th root of the grading parameter. This gives a flat cover by the stated
chart: the map obtained by adjoining a grading parameter $\lambda$ has coordinates
$(v_1,v_2)=(\lambda x,\lambda^q)$ and is the composite of the original flat cover
with the grading action. On $\mathcal X_k$ the remaining grading group is $\Lambda$,
giving the asserted chart there with ambient $v_2\ne0$ open
$[\Spec k[x]/\Lambda]$.

Identify all arrows and composition, not only geometric fibers. If $\rho$ is the
grading parameter and $t_j$ are strict-arrow parameters with left end $x$, the
right chart point is $y=x/\rho$ since $\eta_Rv_1=x$. The remaining equations are
\begin{equation}\label{geom:nilpotent-arrow-equations}
 \eta_Rv_2=\rho^q,\qquad \eta_Rv_{j+2}=0\quad(j\geq1).
\end{equation}
Modulo $x$ the first right unit is $1$. Modulo the preceding equations the
subsequent ones are $t_j^{p^2}-t_j=0$: compare $\phi(T)^{p^2}$ with $\phi(T^{p^2})$
for the Honda law; its coefficients lie in $\F_p$ and all preceding digits are
fixed by the square of Frobenius. Thus the special-fiber arrow scheme is proconstant
with labels
\[
 (\rho_0,t_1^0,t_2^0,\ldots)\in\Lambda\times k^{\mathbb N_{>0}}.
\]
Labels with $\rho_0=1$ give $G^1$, and grading-only arrows split off $\Lambda$.

For the $R_d$ presentations, adjoin a formal $q$-th root $\underline{t}$ of $t^d$
and take $R_d\otimes_{k[t,t^{-1}]}k[t^{\pm1},\underline{t}]$; the chart map is
faithfully flat. On this cover, an arrow is determined by source and target ends
and a single strict label; upgrading to the thickening means solving the displayed
nilpotent equations by induction on the thickness. At square-zero layer
$(x^b)/(x^{b+1})$ only the special-fiber Jacobian acts; its first diagonal entry
is a unit ($q$ prime to $p$) and the other diagonal entries are $-1$, so the system
is lower-triangular and can be solved recursively. This gives the unique lift on
every chart algebra, and proconstant labels are unchanged by the thickening.
Composition descends through $x$-adic compatibility and then through the grading
to the invariants in \eqref{geom:quotient-thickenings}.

The action formula on $R$ is immediate from the arrow lift: for a strict label
$g$ whose formal isomorphism over $R$ has left end $x$ and right end $g(x)$,
reduction of $v_1=x$ through the right unit gives $g(x)$. The formula
$g(x)\in x+x^2R$ reflects reduction of strict arrows to the identity modulo $x$.
The grading $\Lambda$ has order prime to $p$, and its action is the $q$-th root
change, giving $\lambda(x)=\lambda^{-1}x$.

For the line twist: $t=v_1=xu$, so $u=x^{-1}t$ in the chart. Applying $g$ and
using that the grading normalizes $v_1$ to $x$ gives the stated formula; the same
with $\lambda$ gives the grading formula. Every strict label is realized by a
formal isomorphism over the chart algebra by the recursive comparison in
Proposition~\ref{geom:law-charts}.
\end{proof}

The reductions $T'_d$ are pure sheaves with a coinduced structure, and this
structure is what turns their cohomology into an ordinary coefficient-module
calculation. We record it now.

\begin{lemma}\label{geom:arrow-lifts}
On each nilpotent thickening $R_d$, every strict label reduces to a tuple
$(t_1^0,t_2^0,\ldots)$ with $t_j^0\in\F_{p^2}\subset k$ and only finitely many
nonzero entries. For each $d\geq1$ there is a unique continuous lift of every
such special-fiber label to a formal isomorphism over $R_d$ reducing to the
identity on $v_2,v_3,\ldots$.
\end{lemma}

\begin{proof}
At square-zero layer $x^b/(x^{b+1})$, \eqref{geom:recursive-arrow} has leading
coefficient $v_h\ne0$, hence is solvable; after subtracting the linear correction
determined by previous labels, the next equation is again linear with a unit
diagonal. The recursion is functorial in the thickening under $v_2=1$, so finite
algebraic data multiply compatibly. Reducing to $d=1$ recovers the special-fiber
labels of Proposition~\ref{geom:thickened-charts}.
\end{proof}

\subsection{The digit filtration and completed operations}

The coefficient algebra of operations is a completed skew group ring. We give it
an explicit basis by iterating the groups $H(i)$ and their indexes $N_l$. This
will be what makes the finite-level computation of
Section~\ref{complete:section} possible.

Decompose the restricted stabilizer as
\[
 G^1=\varprojlim_{i}H/H(i),\qquad
 H(i)=g_{i+1}^{p^s}H(i+1)\ (2s=2i\text{ or }2i+1),
\]
with coordinates
\[
 (z_{\mathrm{pr}},z_1,z_2,\ldots),
 \qquad z_{\mathrm{pr}}\in\Z_p,\ z_i\in\Z_p.
\]
We arrange this as follows. On the pro-$p$ group the tangent-to-identity
coordinates are the alternating $\gamma^{p^s}$, $\delta^{p^s}$ of a chosen
generator set. The $p$-power map multiplies the tangent by $p$ in each coordinate,
so the coordinates multiply consistently under composition. The consecutive
quotient indexes are
\[
 N_i=[H:H(i)]=\begin{cases} p^{3s}, & i=2s,\\ p^{3s+1}, & i=2s+1,\end{cases}
\]
so $[H:H(i+1)]/N_i=p^a$ with $a\in\{1,2\}$.

\begin{lemma}[Displacement and index]\label{op:displacement}
Let $H(i)$ be the $i$-th filtration piece of $H$ defined above. If
$g\in H(i)\setminus H(i+1)$ then $\ell(g)\leq r_i:=\sum_{j=1}^i p^{\lfloor j/2\rfloor}$.
The action of $H$ on $R$ is faithful. With $r_0=0$,
\[
 \kappa:=\sum_{l\geq1}\frac{1}{N_l}\leq\frac{p}{p-1}.
\]
Moreover, for every $s\geq1$,
\[
 \Gamma_s\subset H_{p^s}\subset H(2s-1)
 \qquad\text{and}\qquad
 N_{p^s}\geq p^{3(s-1)},
\]
where $H_{p^s}=\{g:g(x)\equiv x\bmod x^{p^s+1}\}$ and $\Gamma_s$ is the subgroup
of $g$ with $g\)-coordinates divisible by $p^s$.
\end{lemma}

\begin{proof}
Bound the displacement of an element by its first nonidentity digit. The
comparison is made on torsion points of the deforming formal group, where the
digit gives an exact valuation and the displacement gives a lower bound for the
same expression.

Choose a strict representative of $g$ whose first nonidentity digit is at index
$i$, with nonzero coefficient $b$. For even $i$, choose $b\notin\F_p$ using the
quotient by the central subgroup. The chart construction gives a formal isomorphism
\[
 \widetilde g:F_y\longrightarrow F_x,\qquad y=g(x),
\]
\[
 [p]_{F_x}(T)=xT^p+_{F_x}T^{p^2}.
\]
Its coefficients lie in $R$: the grading change is effected by a $(p-1)$-st root
congruent to $1$, followed by the compatible strict arrow. The reduction of
$\widetilde g(T)-_{F_x}T$ begins with $bT^{p^i}$.

Fix $i$ and take $d$ sufficiently large. The residue of
\[
 \frac{\widetilde g(\zeta)-_{F_x}\zeta}{\zeta^{p^i}}
\]
is $b$: coefficients of lower powers of $T$ lie in $xR$, and their valuations
exceed $p^i\lambda_d$ for large $d$. The differences $\tau-_{F_x}\zeta$, as
$\tau$ ranges over the level-$d$ torsion points, are the multiples $[a]_{F_x}\zeta$
for $a\bmod p^d$. If $s=\nu_p(a)<d$, their valuation is $p^{2s}\lambda_d$; the
zero multiple has infinite valuation. When $2s=i$, the residue after division by
$\zeta^{p^i}$ is $a/p^s\bmod p\in\F_p^\times$, since $[p^s]_{F_0}(T)=T^{p^{2s}}$.
It cannot cancel $b$. Ordinary and formal differences have the same valuation,
because $T-_{F_x}S$ is $T-S$ times a unit. Consequently
\[
 v(\widetilde g(\zeta)-\tau)=p^{\min(i,2s)}\lambda_d,
\]
with $s=\infty$ for $\tau=\zeta$.

There are $(1-1/p)p^{d-s}$ multiples of exact parameter $s<d$. Counting these
factors with multiplicity $p^d$ in the Weierstrass factorization gives
\begin{align}
 v\bigl([p^d]_{F_x}(\widetilde g(\zeta))\bigr)
 &=p^{2d}\lambda_d
   \Bigl((1-1/p)\sum_{0\le s<\lceil i/2\rceil}p^s
                   +p^{i-\lceil i/2\rceil}\Bigr)\notag\\
 &=\frac{p^{\lceil i/2\rceil}+p^{\lfloor i/2\rfloor+1}-1}{p-1}
   =r_i+\frac p{p-1}.
 \label{op:torsion-valuation}
\end{align}
Substitution gives $F_y\equiv F_x\bmod x^{\ell(g)+1}$. Since $[p^d]_{F_x}(\zeta)=0$,
the value $[p^d]_{F_y}(\zeta)$ has valuation strictly greater than $\ell(g)+1$
when $\ell(g)$ is finite. Its image under $\widetilde g$ has the same valuation and
equals $[p^d]_{F_x}(\widetilde g(\zeta))$. Thus
$\ell(g)<r_i+1/(p-1)$, proving the bound for $\ell(g)$. If $y=x$, the value would
instead be zero, contradicting the displays. Since the digit filtration is
separated, the action is faithful.

For $r_i<l\leq r_{i+1}$, the displacement bound gives
$N_l\geq[H:H(i+1)]=p^{2i-\lfloor i/2\rfloor}$. The interval length is
$p^{\lfloor(i+1)/2\rfloor}$, and its contribution to $\kappa$ is at most
$p^{-i}$. Summing proves the $\kappa$ bound. The inequality $r_{2s-2}<p^s$ gives
the second inclusion, and the digit indices give its index bound. For the first
inclusion, every tangent group operator $g-1$ raises the valuation on $R$ by at
least $1$. In characteristic $p$, $g^{p^s}-1=(g-1)^{p^s}$, so
$g^{p^s}(x)\equiv x\bmod x^{p^s+1}$. Apply this to the ordered coordinates of
$\Gamma_s$.
\end{proof}

For the coordinates of $H$, proceed from $s$ to $s+1$. The $p^s$-th powers of
$g_1,g_2$ correct the two independent digits of $H(2s+1)/H(2s+2)$, and do so also
when inserted in their ordered positions because the resulting commutators lie in
the next filtration step. The $p^s$-th power of $g_3$ then corrects the remaining
digit of $H(2s+2)/H(2s+3)$. These corrections are unique. Completeness and
separatedness of the digit filtration give the coordinates on the inverse limit
and all asserted subgroup descriptions. The weight $m_2=q$ is trivial on $\Lambda$
and the other digits of $\gamma$ vanish, proving its commutation assertion.

Define the completed rings, with coefficients on the left, by
\[
 \Omega=k[[H]],\qquad \Delta=R[[H]].
\]
The second is a skew group ring for the action on $R$: it is the inverse limit of
$R_d\#(H/\Gamma_s)$ for $s$ large enough to detect the action on $R_d$, so
$ga=g(a)g$ for $a\in R$. Put $U_i=g_i-1$. The filtration below is an auxiliary
filtration on these coefficient rings; it is distinct from both $x$-adic order
and the homological grading of periodic complexes.

\begin{lemma}[Associated graded of $\Delta$]\label{geom:graded-operations}
Put $U_i=g_i-1$. Filter $\Delta$ by the weight grading in which $x$ has weight
$8$ and the $U_i$ have weight $2$ for $i=1,2$ and weight $1$ for $i=3$. Then
\[
 \gr\Delta\cong k[x,A,B,C],
\]
a polynomial ring on $A,B,C$ representing $U_1,U_2,U_3$ modulo $x$.
\end{lemma}

\begin{proof}
Initially filter by closed spans of all words of weight at least a given integer.
These spaces tend to zero. At each finite quotient the action preserves $(x)$, so
$x$ is normal there and nilpotent; modulo $x$ the ideal generated by the remaining
letters is the augmentation ideal of a finite $p$-group algebra. Thus the ideal
generated by all letters is nilpotent at each finite quotient. Conversely, the
word-filtration spaces contain deep open neighborhoods by the ordered topological
basis.

There is a strict gain in weight whenever letters are interchanged. For
$U_1,U_2$, the commutator lies in $H(2)$, whose first two coordinates are divisible
by $p$. Characteristic-$p$ binomial expansion therefore gives weight at least $3$
for its difference from $1$, exceeding the original weight $2$. Commuting $U_1$ or
$U_2$ with $U_3$ gives an element of $\Gamma_1$, so the difference has weight at
least $p>4$. Finally
\[
 U_ix-xU_i=(g_i(x)-x)g_i
\]
has weight at least $8$, since $g_i(x)-x\in x^2R$. This is larger than either
$\operatorname{wt}(U_i)+4$.

Reorder every word modulo strictly higher weight, iterating to all weights by
completeness. Every element of a word-filtration stage expands in ordered
monomials of at least that weight. Uniqueness in the topological basis proves the
reverse inclusion and shows there are no further associated-graded relations.

Normality of $x$ follows from $gx=x(g(x)/x)g$. Weighting each group coefficient by
the continuous unit $g(x)/x$, or by its inverse, defines a continuous change of
coefficients even in the completion: at precision $x^d$ choose a group quotient
detecting the action modulo $x^{d+1}$. Thus $(x)$ is two-sided and quotienting
simply removes the $x$ coordinate and its weight. This proves the assertion.
\end{proof}

\subsection{Derived sections and cohomological bounds}

Now the chart calculations become bounds for derived sections. First we justify
the model and resolution tools for the periodic categories. The polynomial
associated graded of $k[[H]]$ yields a compact resolution of $k$ of length three.
Applying continuous $\Hom$ computes derived $H$-invariants; exact
$\Lambda$-invariants and the forgetful functor from semilinear $R_d$-modules
then give the bound on the nilpotent charts. Passing through the open, the central
quotient, and the integral base adds the degrees recorded below. The bounds are
uniform for arbitrary sheaves, hence control unbounded periodic complexes.

\begin{proposition}[Cohomological bounds]\label{geom:cohomological-bounds}
Projective dimensions in the $\Ext$ sense satisfy
\begin{equation}\label{geom:dimension-bounds}
 \operatorname{pd}_{\mathcal N}(\mathcal O)\leq4,\qquad
 \operatorname{pd}_{\mathcal M_k}(\mathcal O),
 \operatorname{pd}_{\mathcal M_0}(\mathcal O)\leq5,\qquad
 \operatorname{pd}_{\mathcal M}(\mathcal O)\leq6.
\end{equation}
For sheaves on $\mathcal M_k$, the right adjoint of $r^*$ has
\begin{equation}\label{geom:central-derived-invariants}
 r_*M=\ker(c-1:M\to M),\qquad
 R^1r_*M=\coker(c-1:M\to M),\qquad R^{>1}r_*M=0,
\end{equation}
where the kernel and cokernel descend to $\mathcal N$.
\end{proposition}

\begin{proof}
Pass from the nilpotent charts to $\mathcal N$, then restore the central group,
and finally pass from characteristic $p$ to the integral stack.

A $t^\infty$-torsion sheaf on $\mathcal N$ is a filtered union of sheaves
annihilated by powers of $t$, with the evident weight twists in the kernel maps.
Lemmas~\ref{geom:sections-resolution} and~\ref{geom:cohomological-bounds} show its
sections cohomology vanishes above degree $3$. The open
of Proposition~\ref{geom:open} has exact sections. For an arbitrary sheaf $M$, the
kernel and cokernel of $M\to M[1/t]$ are $t^\infty$-torsion; the open is affine
over the stack, so localization is exact open restriction followed by affine
pushforward. The two short exact sequences for $M\to M[1/t]$ give the extra degree
and prove the bound $4$.

For the central group, write the torsor
$l:\mathcal M_k\to\mathcal N$ with $l_*$ exact and $l_*l^*$ coinduced on the
chart. Since $l$ is a $\Z_p$-torsor and $\Z_p$ is a compact totally disconnected
group, the continuous cohomology of $\Z_p$ on a discrete module has
cohomological dimension $1$; the filtered colimit over continuous finite quotients
of $\Z_p$ recovers the sections cohomology, and gives $R^{>1}l_*l^*=0$. The
Gysin-type long exact sequence from the closed-open decomposition of
$\mathcal M_k$ along $t$ then adds the extra degree, proving the bound $5$.

The formula in \eqref{geom:central-derived-invariants}: the fiber product computing
$r_*M$ is the fixed set of the central action on the underlying $M$; since the
action is generated by $c\in Z$ with $c-1$ the nilpotent (locally) endomorphism on
the module, the limit of the periodic tower reduces to the kernel and cokernel.
The stated vanishing $R^{>1}=0$ is that $\Z_p$ has cohomological dimension one.

Finally, reduction to characteristic $p$: (i) the sheaf $\mathcal O$ on
$\mathcal M$ restricts to the same sheaf $i^*\mathcal O$ on $\mathcal M_0$; by
flatness of $i$ on the chart, $\Ext^*_{\mathcal M}(\mathcal O,-)\to
\Ext^*_{\mathcal M_0}(\mathcal O,i^*(-))$ is compatible. The bound on
$\mathcal M_0$ is thus inherited from that on $\mathcal M$ by suspension of the
cosimplicial affine resolution; the extra degree is the standard contribution from
the closed immersion into the integral chart.
\end{proof}

\begin{lemma}[Exactness of central pushforward on periodic complexes]
\label{geom:central-exactness}
For $M$ a sheaf pulled back from $\mathcal N$, the two coinduction functors
$I=r_*r^*$, $J$(unit cokernel) are exact on the periodic category $D(\mathcal M_k)$,
and the finite totalization
\[
 0\longrightarrow M\longrightarrow r_*Ir^*M
 \longrightarrow r_*Jr^*M\longrightarrow M\longrightarrow0
\]
computes $Rr_*$ in the periodic setting.
\end{lemma}

\begin{proof}
The composite $rl:S\to\mathcal N$ is an affine chart, so $I$ has
$r_*$-acyclic values by affine adjunction.
Equation~\eqref{geom:central-derived-invariants} and the sequence
$0\to M\to IM\to JM\to0$ show $J$ also has $r_*$-acyclic values. Both $r_*I$ and
$r_*J$ are exact (I and J exact, acyclicity of their values), so the finite
totalization preserves quasi-isomorphisms. On injective-term fibrant replacements
it agrees up to quasi-isomorphism with ordinary $r_*$ by the same two-term acyclic
resolution, so it computes $Rr_*$ on periodic complexes. The twist line is pulled
back from downstairs, so all these functors and identifications respect the
periodicity.

Flat affine adjunction along $f_k$ and $f'$ identifies $Rr_*T_k$ with $T'$.
Derived adjunction for $r^*$, with $r^*\mathcal O=\mathcal O$, gives the equality
of sections homotopy. For a sheaf pulled back from $\mathcal N$ the vertical
kernel and cokernel in \eqref{geom:central-derived-invariants} are both the
original sheaf. Hence the four-term sequence. Applying it termwise to periodic
complexes, the two connecting maps give \textup{(the central four-term sequence)} with the
usual homological signs. The triangle is natural; no splitting of its final
connecting map is asserted.
\end{proof}

At this point the geometric framework is in place. The next section carries out
the algebraic construction in $\mathbf D(\mathcal N)$. The open is $\Spec k$, the
thickenings carry the semilinear $G$-action, and the reductions of $T'$ are
coinduced modules.

\section{Complete operations and finite reductions}\label{complete:section}

We work at a prime $p>5$, as in the geometric construction. Throughout this section
the categories are the characteristic-$p$ categories $\mathbf D(\mathcal M_0)$,
$\mathbf D(\mathcal M_k)$, and $\mathbf D(\mathcal N)$ of
Section~\ref{geom:periodic-foundations}. The line section $t=v_1$ has degree
$D=2(p-1)$. Recall the homological and weight conventions
$C_{j+D}=C_j(1)$, $d:C_j\to C_{j-1}$, and $C[b]_j=C_{j-b}$, with the usual sign on
the differential of a shift. We write $\pi_iC=[\mathcal O[i],C]$ for sections
homotopy, which is distinct from the sheaf homology of $C$. We retain
$R=k[[x]]$, $R_d=k[x]/(x^d)$, $\Lambda=\mu_q(k)$, and $\Delta=R[[H]]$ from the
nilpotent charts. All limits and colimits of objects in the derived categories are
homotopy limits and colimits.

\subsection{The fracture square}

For $d\geq1$ define
\begin{equation}\label{complete:definitions}
 P/t^d=\cofib\bigl(P[dD]\xrightarrow{t^d}P\bigr),
 \qquad
 \widehat P=\holim_d P/t^d,
 \qquad
 LP=P[1/t].
\end{equation}
The multiplication maps, their cones, and the transition maps in these towers are
chosen naturally at the level of twisted complexes. We call $P$ \emph{$t$-invertible}
if $t:P[D]\to P$ is an equivalence, and \emph{complete} if $P\to\widehat P$ is an
equivalence.

\begin{proposition}[Fracture square]\label{complete:fracture}
For every $P\in\mathbf D(\mathcal N)$ the square
\begin{equation}\label{complete:fracture-square}
\begin{tikzcd}
 P\arrow[r]\arrow[d] & \widehat P\arrow[d]\\
 LP\arrow[r] & \widehat{LP}
\end{tikzcd}
\end{equation}
is a homotopy pullback. Its fiber is simultaneously $t$-invertible and complete, so
that fiber vanishes. Completion is exact, and a complete object is characterized by
being right orthogonal to all $t$-invertible objects. Finally,
$P/t^d\simeq\widehat P/t^d$ naturally for every $d\geq1$, and localization commutes
with sections homotopy:
\begin{equation}\label{complete:localization-sections}
 \pi_i(LP)\cong\pi_i(P)[1/t].
\end{equation}
\end{proposition}

\begin{proof}
The defining tower for $\widehat P$ is the telescope
$P\to P[-D]\to P[-2D]\to\cdots$ with transition maps $t$, inverted by the
homotopy limit. Exactness of filtered colimits represents the localization
telescope by the ordinary termwise colimit. On the affine charts this is
localization at the principal open defined by $t$, i.e.\ the asserted open
push-pull functor; flat affine adjunction identifies the open category with its
essential image.

Taking the inverse limit of the multiplication-cofiber triangles shows that the
fiber of $P\to\widehat P$ is the limit of
\[
 \cdots\xrightarrow{t}P[(d+1)D]\xrightarrow{t}P[dD]\xrightarrow{t}\cdots.
\]
Shifting the tower one position does not change its limit, so $t$ is invertible on
this fiber. On the other hand $t^d$ is nullhomotopic on $P/t^d$ by the standard
homotopy on its multiplication cone. If $A$ is $t$-invertible, naturality of $t$
gives $[A[j],P/t^d]=0$ for all $j$: multiplication by $t^d$ is simultaneously
invertible from the source and zero on the target. The same for mapping spectra,
followed by the inverse limit, gives $[A[j],\widehat P]=0$.

Consider the map from the square to its totalization and then to the original
$P$. Both rows and columns give the same fiber, namely the $t$-invertible fiber of
$P\to\widehat P$. That fiber is also complete: compose it with the completion map,
and naturality of the completion map turns the composite into itself. Right
orthogonality of the complete objects against the $t$-invertible ones uses the
displayed vanishing. Idempotence of completion follows from the fact that
$\widehat P$ is already right orthogonal.

The identification $P/t^d\simeq\widehat P/t^d$ follows from exactness of the
multiplication-cofiber: completing commutes with forming $P/t^d$, and the
transition maps in the tower conspire to give an equivalence. Finally, the finite
cohomological bounds allow sections-homotopy to commute with the telescope:
compute derived sections within the bounded range and commute the filtered colimit
through the resulting calculation.
\end{proof}

The fracture square reduces reconstruction to a map on the overlap (the open
piece). The following description of the open categories specifies which maps can
be chosen there and records a vanishing used in the transfer step.

\begin{proposition}[Open theory]\label{complete:open}
Under the equivalences $\mathcal N[1/t]\simeq\Spec k$ and
$\mathcal M_0[1/t]\simeq BZ$ of Proposition~\ref{geom:open}, the open category on
$\mathcal N$ is identified with the derived category of $\F_p$-vector spaces, and
that on $\mathcal M_0$ with the derived category of modules over $\F_p[c^{\pm1}]$
for the central-to-open generator $c$. On a periodic complex, taking
$[1/t]$-localization computes the corresponding ordinary periodic spectrum.

There is a vanishing of sections in the open:
\[
 \pi_i(\mathcal O[1/t])=0\qquad(i\ne 0),\qquad
 \pi_0(\mathcal O[1/t])=k.
\]
Equivalently, the derived-global-sections complex of the pure unit on the open is
concentrated in degree $0$.
\end{proposition}

\begin{proof}
The identification is flat affine adjunction to the residual-affine stack. For
$\mathcal N$, the open is $\Spec k$, so the category of quasi-coherent sheaves
is just vector spaces; the periodic structure becomes an ordinary $\mathbb Z$-graded
complex with no twist. For $\mathcal M_0[1/t]$, the group $BZ$ gives the module
category for the group algebra of $\Z$, and the generator $c$ acts invertibly.
Section vanishing is the standard acyclicity of the periodic unit on
$\Spec k$ together with the explicit cohomology of $BZ$.

We will need a lemma on how periodic complexes with torsion support behave under
filtered colimits and under base change. These are consequences of the
cohomological bounds, which the chart in \eqref{geom:dimension-bounds} provides.
\end{proof}

\subsection{The complete operation algebra}

\begin{proposition}[Coinduction]\label{geom:coinduction}
The chart sheaf $T'=f'_*\mathcal O$ is $t$-monic, so multiplication by $t$ is
injective. Its reductions $T'_d=T'/t^d=f'_*\mathcal O_{\mathcal X_k/(t^d)}$ are
pure sheaves. On the nilpotent chart $T'_d$ is $C_{\mathrm{lc}}(H,R_d)$,
coinduced from $\Lambda$, with diagonal semilinear action
$(gF)(h)=g(F(g^{-1}h))$ in coset coordinates.
\end{proposition}

Recall from Proposition~\ref{geom:coinduction} that $T'=f'_*\mathcal O$ is
$t$-monic and that
\[
 T'_d=T'/t^d=f'_*\mathcal O_{\mathcal X_k/(t^d)}
\]
is a pure sheaf. On the nilpotent chart it is $C_{\mathrm{lc}}(H,R_d)$,
coinduced from $\Lambda$, with the diagonal semilinear action
$(gF)(h)=g(F(g^{-1}h))$ in coset coordinates. The chart basis $u$ of the weight
line satisfies $t=xu$ and $h(u)=x/h(x)\,u$. These formulas identify a line twist
with a fractional coefficient lattice.

We use $\Delta[x^{-1}]$ for the two-sided localization at powers of $x$. This
localization contains $\Delta$. With coefficients on the left the skew relation
gives
\[
 hx=x\,\frac{h(x)}x\,h,
 \qquad
 xh=\frac{x}{h(x)}\,h\,x
 \qquad(h\in H).
\]
Both displayed fractions are units of $R$. As in the proof of
Proposition~\ref{geom:graded-operations}, at coefficient precision $x^d$ we may
refine the group quotient until it detects these units modulo $x^d$, using the
coefficient action modulo $x^{d+1}$. The two inverse continuous changes of
coefficients give $\Delta x=x\Delta$. Left multiplication by $x$ is injective on
the compatible coefficient systems defining $\Delta$, and the unit change gives
right injectivity as well. Thus $x$ is normal and regular, and its powers admit
this two-sided localization. This coefficient element is generally noncentral.

We compute the graded algebra
\[
 \mathscr E_n=[\widehat T'[n],\widehat T'].
\]
The product convention is composition: $ab$ means first apply $b$ and then apply
the appropriately shifted $a$. With our periodicity convention, an ordinary sheaf
map $T'_d\to T'_d(s)$ gives a map $T'_d[sD]\to T'_d$. The calculation must retain
these ordinary maps: their strict compositions make matrices of operations into
complexes.

\begin{proposition}[Complete operations and their representatives]
\label{complete:operations}
There is an identification of graded algebras
\begin{equation}\label{complete:operation-algebra}
 \mathscr E_{sD}=(x^{-s}\Delta)^\Lambda\quad(s\in\Z),
 \qquad
 \mathscr E_n=0\quad(n\notin D\Z).
\end{equation}
Products use left-coefficient skew convolution in $\Delta[x^{-1}]$, with degrees
added. Every element of $\mathscr E_{sD}$ has a unique compatible system of
ordinary sheaf-map representatives $T'_d\to T'_d(s)$, and these representatives
realize multiplication strictly. Under \eqref{complete:operation-algebra} the
operation $t$ of degree $D$ is the coefficient $1\in x^{-1}\Delta$, and it is
central in $\mathscr E$.

Put $A_{\mathrm{op}}=(\Delta[x^{-1}])^\Lambda$. With a formal $t$ recording
degree $D$, localization identifies
\begin{equation}\label{complete:localized-operations}
 \mathscr E[t^{-1}]=A_{\mathrm{op}}[t^{\pm1}].
\end{equation}
\end{proposition}

\begin{proof}
Exclude the extension and inverse-limit terms first, so that all operations have
ordinary compatible representatives. Then normalize the line twists and compute
composition at every finite precision.

The fiber of $T'\to\widehat T'$ is $t$-invertible, so
Proposition~\ref{complete:fracture} identifies maps into $\widehat T'$ out of $T'$
or out of $\widehat T'$. Maps into the limit therefore are computed from the tower
of targets $T'_d$.

Flat affine adjunction along $f'$ computes $[T'[n],T'_d]$ on $\mathcal X_k$ with
target $\mathcal O_{\mathcal X_k/(t^d)}$. This target is pushed forward from the
open $v_2\ne0$, so we may work on $[\Spec k[x]/\Lambda]$. Let $M$ be the
restriction of $(f')^*T'$ to this chart. It is $x$-monic and
\[
 M/x^dM=C_{\mathrm{lc}}(H,R_d).
\]
As an $R_d$-module the latter is the scalar extension of $C_{\mathrm{lc}}(H,k)$,
hence free. Since $|\Lambda|=q$ is invertible in $k$, averaging a lift over
$\Lambda$ proves equivariant projectivity.

Choose an equivariant projective resolution of $M$ of length one over $k[x]$. Start
with a surjection from a free module over the skew group ring $k[x]\#\Lambda$; its
underlying $k[x]$-module is free. Since $k[x]$ is hereditary, its kernel is
projective as a $k[x]$-module; averaging promotes this to equivariant
projectivity. Reduction modulo $x^d$ preserves exactness, since $M$ is $x$-monic.
Applying $\Hom$ into $R_du^s$ factors through this reduced resolution. Its
positive cohomology vanishes by the projectivity of $M/x^dM$ just proved.
Periodize the length-one resolution with the line twists. Its bounded resolution
length and projective terms make it valid for computing maps to twisted periodic
complexes: mapping from it takes acyclics to acyclics. Since $D>2$, the
augmentation degree and the relation degree cannot coincide modulo $D$. Thus the
only possible groups are the ordinary maps in degrees $sD$ and their first
extensions; the latter vanish. In particular,
\begin{equation}\label{complete:finite-level-maps}
 [T'[sD],T'_d]=\Hom\bigl(T',T'_d(s)\bigr),
 \qquad
 [T'[n],T'_d]=0\quad(n\notin D\Z).
\end{equation}

The ordinary Hom groups have surjective transition maps: a map at level $d$ can be
viewed as a functional on $M/x^{d+1}M$ with values in $R_du^s$, and equivariant
projectivity of $M/x^{d+1}M$ lifts it through
$R_{d+1}u^s\twoheadrightarrow R_du^s$. The Milnor exact sequence then has zero
derived-limit term. Each ordinary map in \eqref{complete:finite-level-maps} kills
$t^d$ and factors uniquely through $T'_d$. Compatibility in $d$ gives actual maps of
towers of pure complexes, and composition is their ordinary composition. This
factorization is about sheaf maps; it does not assert full faithfulness of closed
pushforward on derived categories.

Compatibility also fixes the coefficient algebra. The chart $[\Spec k[x]/\Lambda]$
restricted to a level $d$ carries the semilinear action, and the transition maps
identify coefficients compatibly. Passing to the limit yields the completed skew
group ring $\Delta$. The twist normalization from \eqref{geom:twist-cocycle}
identifies the line twists as the fractional lattices $x^{-s}R_d$. Taking
$\Lambda$-invariants and then passing to the limit again produces
$(x^{-s}\Delta)^\Lambda$, which is the displayed algebra. The centrality of $t$
follows because $t$ corresponds to multiplication by $x$ times the twist,
and the twist is precisely the central Cartier section; $x$ is not central in
$\Delta$ but the diagonal twist makes $t$ central.
\end{proof}

\begin{corollary}\label{complete:degree-two}
For each $j\in\Z$ there are no self-maps to the $2$-shift of $\widehat T'[j]$,
$\widehat T_k[j]$, or $\widehat T_0[j]$.
\end{corollary}

\begin{proof}
For $\widehat T'$ this is immediate from \eqref{complete:operation-algebra}. The
finite-level calculation \eqref{complete:finite-level-maps} applies equally to the
full-center chart sheaf $T_k$, replacing the functions on $H$ by the functions for
$G_{\mathrm{up}}$. They are again free over $R_d$ and equivariantly projective for
the tame subgroup $\Lambda$. Hence the same extension and Milnor terms vanish.

For $T_0$, use extension of scalars $e^*:\mathbf D(\mathcal M_0)\to\mathbf
D(\mathcal M_k)$. Since $k/\F_p$ is finite separable, trace duality identifies this
extension functor also as a right adjoint to restriction. It preserves homotopy
limits and therefore commutes with completion. Moreover the unit
$F\to e_*e^*F$ is naturally split as an $\F_p$-linear map, so extension of
scalars detects zero maps. The asserted vanishing follows from that for
$\widehat T_k$.
\end{proof}

\subsection{Finite constructions from the complete generator}

The operations act strictly on the entire quotient tower. This turns into a
description of all finite constructions from $\widehat T'$. The key point is that
relations in a differential matrix hold as equalities of maps at every level,
so cones and sections can be calculated from the same operators.

Give $\mathscr E$ its homological grading and zero differential. A
\emph{finite-cell right dg $\mathscr E$-module} is a finite iterated extension of
shifts of the free right module $\mathscr E$; it can be represented by a finite
graded-free module with a filtration whose successive quotients are shifted free
modules. A perfect right module is a retract of a finite-cell module in the derived
category; their category is $\Perf(\mathscr E)$.

The description of an algebraic thick subcategory by perfect modules over a dg
endomorphism algebra is a general form of dg Morita theory; compare Keller
\cite[Section~3.6 and Theorem~3.8(b)]{Keller2006}. Here we use the computed
graded algebra $\mathscr E$ itself with zero differential. Its strict action on the
finite reductions, proved in Proposition~\ref{complete:operations}, is what permits
this choice.

\begin{proposition}[Strict realization]\label{complete:realization}
There is an exact fully faithful functor
\[
 \mathcal R:\Perf(\mathscr E)\longrightarrow\mathbf D(\mathcal N)
\]
from perfect right dg modules sending $\mathscr E$ to $\widehat T'$ and identifying
its image with $\Thick(\widehat T')$. For a finite-cell module $\mathsf P$, its image
is the homotopy limit of an actual compatible tower $Y_d$ of complexes whose terms
are finite sums of shifts of $T'_d$.
\end{proposition}

\begin{proof}
Let $\mathsf P$ be a finite-cell module. Choose a homogeneous basis compatible with
its cell filtration. Replace a basis vector $e_b$ of degree $b$ by the shifted pure
complex $T'_d[b]$ at every level $d$. A degree-$j$ right-module matrix entry
\[
 e_b\longmapsto e_{b'}a,
 \qquad a\in\mathscr E_{b+j-b'},
\]
is represented in chain degree $m$ by
\begin{equation}\label{complete:matrix-degrees}
 a:(T'_d)_{m-b}\longrightarrow(T'_d)_{m+j-b'}.
\end{equation}
Use the unique representatives of Proposition~\ref{complete:operations} for all
entries, including differential entries $j=-1$. Source and target use the same
line-twist periodicity identifications in every degree and at every level $d$.

Strict composition makes the matrix relation $d^2=0$ an equality of actual maps.
All nonzero degrees of $\mathscr E$ are even, so no coefficient sign enters. For a
degree-$j$ matrix $f$, the Hom differential is
\[
 d_{\mathsf Q}f-(-1)^j f d_{\mathsf P}.
\]
Formula~\eqref{complete:matrix-degrees} respects this differential, so it respects
chain maps, chain homotopies, the signed shift, and ordinary mapping cones. It also
makes all these complexes and maps compatible under the transitions $d+1\to d$:
every matrix entry uses its compatible tower map, so its representative is fixed at
all levels at once.

The order of multiplication agrees with the right-module convention. An
endomorphism of the free right module determined by $1\mapsto a$ sends
$z\mapsto az$, by left multiplication. Composing the endomorphism for $b$ then for
$a$ gives $ab$, the same order as the operators in
Proposition~\ref{complete:operations}. Tensoring a right module with a left module
consequently uses the left action of these same operators.

The comparison holds for finite-cell modules and passes to retracts: a retract of a
tower splits levelwise, and the sums and cones used in the filtration are
respected. Thus the functor is defined, exact, and fully faithful on
$\Perf(\mathscr E)$, and its image is exactly the full subcategory generated by
$\widehat T'$ under shifts, finite sums, cones, and retracts, i.e.\
$\Thick(\widehat T')$.
\end{proof}

For example, if $a\in\mathscr E_0$, the two-cell module
$\cofib(a:\mathscr E\to\mathscr E)$ realizes the cofiber
$\cofib(a:\widehat T'\to\widehat T')$. The module map is left multiplication on the
free right module. We next compute the sections of this cone using the same left
operator on a coefficient lattice; its kernel and cokernel are the sections data of
an actual completed object.

\subsection{Sections and finite reductions}

Let $\mathsf L$ be the graded left $\mathscr E$-module
\begin{equation}\label{complete:section-module}
 \mathsf L_{sD}=(x^{-s}R)^\Lambda\quad(s\in\Z),
 \qquad \mathsf L_n=0\quad(n\notin D\Z),
\end{equation}
with zero differential. Its finite reductions are the graded left modules
\[
 (\mathsf L_d)_{sD}=(x^{-s}R/x^{d-s}R)^\Lambda\quad(s\in\Z),
 \qquad (\mathsf L_d)_n=0\quad(n\notin D\Z),
\]
and $\mathsf L=\varprojlim_d\mathsf L_d$ degree by degree. Set
\[
 \Phi=(k((x)))^\Lambda=k((x^q)).
\]
Here $t$ acts on normalized sections by the inclusion of one fractional lattice in
the next. In particular $\mathsf L[t^{-1}]=\Phi[t^{\pm1}]$, with $t$ recording
degree $D$. For a complete object $Y$ write $W(Y)=\pi_*LY$.

\begin{proposition}[Section formulas]\label{complete:sections}
If $Y=\mathcal R(\mathsf P)$ for a perfect right dg $\mathscr E$-module
$\mathsf P$, then naturally
\begin{equation}\label{complete:tensor-formulas}
 \begin{split}
 \pi_*Y&=H_*\bigl(\mathsf P\otimes^{\mathbf L}_{\mathscr E}\mathsf L\bigr),\\
 W(Y)&=H_*\bigl(\mathsf P\otimes^{\mathbf L}_{\mathscr E}\Phi[t^{\pm1}]\bigr).
 \end{split}
\end{equation}
For a finite-cell model the ordinary cellular tensor computes each derived tensor.
The actions are the left actions of \eqref{complete:operation-algebra}.
\end{proposition}

\begin{proof}
Suppose first that $\mathsf P$ is a finite-cell module, and write $Y_d$ for its
realization tower. Each term of $Y_d$ is a finite sum of twists of $T'_d$. Those
twists are sections-acyclic: flat affine adjunction reduces sections to the
nilpotent chart of $\mathcal X_k$, where taking $\Lambda$-invariants is exact. The
bounded derived-sections calculation from
Proposition~\ref{geom:cohomological-bounds} computes $\pi_*Y_d$ by the termwise
sections complex. By \eqref{complete:operation-algebra} that complex is
$\mathsf P\otimes_{\mathscr E}\mathsf L_d$. This identifies differentials and maps
as well as the underlying groups.

These identifications are compatible as $d$ varies: the transition
$\mathsf L_d\to\mathsf L_{d+1}$ corresponds under
\eqref{complete:finite-level-maps} to restriction along the quotient
$T'_{d+1}\to T'_d$, and the ordinary cellular tensor respects this restriction.
Passing to the homotopy limit gives the first formula.

For the second formula, apply the same construction to $L\widehat T'=T'[1/t]$.
Flat affine adjunction on the open $\mathcal N[1/t]\simeq\Spec k$ reduces the
calculation to the periodic unit, whose sections on the open are just $k$ in
degree $0$. The section module localizes to $\Phi$, the action localizes to the
same skew operators, and the second formula follows.
\end{proof}

\begin{lemma}[Intrinsic finite reductions]\label{complete:reductions}
For $Y=\mathcal R(\mathsf P)$ a finite-cell realization, there is a natural
equivalence $Y/t^d\simeq Y_d$. The tower $\{Y_d\}$ therefore determines the
intrinsic reduction $Y\to\widehat Y$. For a retract $Y'$ of $Y$, the reduction
$Y'/t^d$ is the corresponding retract of $Y_d$.
\end{lemma}

\begin{proof}
Multiplication by $t^d$ is strictly zero on $Y_d$, because it is strictly zero on
every shifted $T'_d$ in its terms. For $e\geq d$, map the ordinary cone of
$t^d:Y_e[dD]\to Y_e$ to $Y_d$ by the tower transition on the unshifted summand and
zero on the shifted summand. Strict annihilation makes this a chain map. These
maps are compatible in $e$ and natural for the matrix maps defining our finite-cell
construction. Exactness of homotopy limits gives a map $Y/t^d\to Y_d$.

On the free generator this is the equivalence
$\widehat T'/t^d\simeq T'/t^d=T'_d$ of Proposition~\ref{complete:fracture},
represented by the ordinary quotient. The comparison respects mapping cones and
shifts, with the usual sign isomorphism for a shifted cone. Induction on the
finite cell filtration proves the equivalence for $Y$ and identifies the tower
projection with its intrinsic reduction map.

Retracts: $\pi_iY_d$ is finite by the section calculation, so
$\pi_iY=\varprojlim_d\pi_iY_d$ is profinite. Naturality sends the derived idempotent
defining $Y'$ to an idempotent $e_d$ on every $Y_d$. On the finite group
$\pi_iY_d$, the map $\pi_i e_d$ is an ordinary idempotent, and its image commutes
with the inverse limit because the image and kernel decompositions are compatible
with the transition maps. This gives the formula for $\pi_iY'$ and its intrinsic
reduction topology.
\end{proof}

\subsection{Compact and discrete parts of the overlap}

The complete objects built from $\widehat T'$ are now described. We study the
topology of the overlap vector spaces. When an object has finite sections homotopy
in every degree before completion, its open image occupies a discrete cocompact
subgroup of the overlap. For completions of objects with this finiteness, the
topology is intrinsic, and every map between them induces continuous maps on the
overlap.

\begin{proposition}[Intrinsic overlap topology]\label{complete:topology}
Every $P\in\Thick(T')$ has finite $\pi_iP$ for every integer $i$. Now let
$P\in\mathbf D(\mathcal N)$ have finite $\pi_iP$ for every integer $i$. For each
$i$ the group $\pi_i\widehat P$ is naturally profinite in the intrinsic
inverse-limit topology from the finite groups $\pi_i(\widehat P/t^d)$. The vector
space $W_i(\widehat P)=\pi_iL\widehat P$ has a locally compact Hausdorff
$k$-vector-space topology obtained as the increasing union of the compact open
images of $\pi_{i+sD}\widehat P$ for $s\geq0$. The image of $\pi_iLP$ is discrete
and cocompact in this space.

If $Q\in\mathbf D(\mathcal N)$ also has finite $\pi_jQ$ for every integer $j$, then
every map $f:\widehat P\to\widehat Q$ induces continuous maps
$\pi_i\widehat P\to\pi_i\widehat Q$ and $W_i(\widehat P)\to W_i(\widehat Q)$ in
the topologies just described. No global map $P\to Q$ inducing $f$ is required. If
$\widehat P\simeq\mathcal R(\mathsf P)$ for a perfect right dg
$\mathscr E$-module $\mathsf P$, then under this identification the natural
equivalences of Proposition~\ref{complete:sections} and
Lemma~\ref{complete:reductions} match the profinite topology on
$\pi_i\widehat P$ with the one obtained from the realization's finite-level
sections and intrinsic reductions. They match the compact images and the
increasing-union topology on $W_i(\widehat P)$.
\end{proposition}

\begin{proof}
For $T'$, flat affine adjunction identifies sections homotopy with the cohomology
of twists on $\mathcal X_k=\mathcal P_k(1,q)$. The two-variable principal-open Cech
complex computes it, also in the twisted periodic category. Weight by weight, its
$H^0$ is spanned by nonnegative monomials of that weight, and its $H^1$ by Laurent
monomials with both powers negative. Both sets are finite in a fixed weight because
the two weights $1$ and $q$ are positive. Thus every $\pi_iT'$ is finite. Degreewise
finiteness is preserved by shifts, finite sums, triangles and retracts, which
proves the first assertion.

Now let $P$ satisfy the degreewise finiteness hypothesis. The cofiber sequence for
$t^d$ makes every $\pi_i(P/t^d)$ finite. The Milnor sequence therefore gives
\begin{equation}\label{complete:compact-homotopy}
 \pi_i\widehat P=\varprojlim_d\pi_i(P/t^d),
\end{equation}
with zero first derived-limit term. Give this group its compact Hausdorff
inverse-limit topology. The equivalence $P/t^d\simeq\widehat P/t^d$ is compatible
with the projections: both maps $\widehat P\to P/t^d$ agree after precomposition by
$P\to\widehat P$, and this determines a map uniquely because its fiber is
$t$-invertible whereas the target is complete. Hence
\eqref{complete:compact-homotopy} describes the intrinsic reduction topology.

The fracture square gives a long exact sequence with the map
\begin{equation}\label{complete:joint-overlap-map}
 \pi_i\widehat P\oplus\pi_iLP\longrightarrow W_i(\widehat P).
\end{equation}
Its kernel is a quotient of $\pi_iP$ and its cokernel embeds in $\pi_{i-1}P$; both
are finite. The same holds in every shifted degree. Write
$C_s=\pi_{i+sD}\widehat P$ and let $K_s$ be its image in $W_i(\widehat P)$ under
the localization identifications. The kernel of $C_s\to K_s$ is finite, by applying
\eqref{complete:joint-overlap-map} in degree $i+sD$. Give $K_s$ the compact
Hausdorff quotient topology.

The transition $s\mapsto s+1$ induces an inclusion $K_s\hookrightarrow K_{s+1}$:
the kernel of that inclusion is a quotient of the finite kernel of $C_s\to K_s$.
Therefore the union of the $K_s$ is an increasing union of compact subspaces that
is $T_1$, hence a locally compact $k$-vector space with those spaces as compact
open subspaces.

For continuity, naturality of the fracture square and of
\eqref{complete:joint-overlap-map} shows that $\pi_iLf\cdot K_s$ lands in a compact
open subspace of $W_i(\widehat Q)$ of the same stage, hence is continuous.

Finally, the identification with the realization's own topology: the finite-level
sections are $\mathsf L_d$ and are exactly the finite-level pieces $\pi_iY_d$ by
Proposition~\ref{complete:sections}, and the intrinsic reduction $Y/t^d\simeq Y_d$
matches the profinite topology. The compact images therefore coincide, and the
increasing-union topology matches as well.
\end{proof}

\section{An operator with persistent defects}\label{op:section}

We now build the concrete element $a\in\mathscr E_0$ whose kernel and cokernel
directions, taken together, are large but not complementary. The construction is
finite at each coefficient level and assembles by inverse limit. Its persistent
defects are what obstructs the map into a finite construction later.

\subsection{The target}

Give $\Phi$ its Laurent-series topology, with standard compact open subspace
$k[[w^{-1}]]$. We seek one operator $a\in\mathscr E_0$ with closed image and two
fixed families: nonzero kernel series in $k[[w^{-1}]]$ with distinct leading
positions, and classes of positive monomials $w^j$. The leading position of a
nonzero series is the greatest exponent of $w$ occurring in it. For fixed
constants $c>0$ and $\beta>1/3$, which may depend on $p$, and every sufficiently
large integer $N$, at least $cN^\beta$ leading positions from the first family
should lie in $[-N,0]$, while at least $cN^\beta$ of the selected classes with
$1\le j\le N$ should be independent in
\[
 \Phi\big/\bigl(a\Phi+k[[w^{-1}]]\bigr).
\]
The first family lies in the compact subspace, while the second remains
independent after quotienting the cokernel by the image of that subspace.
Proposition~\ref{op:operator} supplies matrix conditions, and
Section~\ref{glue:section} proves that they yield these analytic properties.

The reason this is the right target is that these are exactly the two pieces in
the overlap decomposition of the fracture square: kernel directions become a
compact open subspace, and cokernel directions become independent of the
discrete-cocompact part. The growth mismatch of the two families with the same
rank count makes the corresponding defect families large in the geometric piece
and small in the compact piece.

\subsection{Coefficient functions and measures}

Recall the digit coordinates
$W_\ell\subset k[[H]]$ from Lemma~\ref{op:displacement}: $H$ acts on $R$ by
$g(x)=x+x^2\varsigma(g)$ with $\varsigma$ the restricted stabilizer character
valued in $R$. For each label $\alpha\in\Z_p$, define the coefficient function
$f_b:\Z_p\to R$ by
\begin{equation}\label{op:coefficients}
 f_b(\alpha)=\int_{g\in H}h_g^{qb}\cdot
             (a-\nu)(h_g)\ \cdot\ x^{-qb}\,\alpha(g)^{-1},
\end{equation}
where $h_g=1+\text{(label $g$)}$ is a local formal coordinate, and $\nu$ is a
formal measure whose coefficients at levels $b$ encode the columns of the future
matrix. The integral extracts the $b$-th Taylor coefficient at a measurement
point indexed by $\alpha$, so different measurements determine the underlying
measure.

Write $z_i$ for the base-$p$ digits of $z\in\Z_p$. If $X=p^r\le qb<pX$, then
$f_b$ and all smaller-depth coefficient functions depend only on $z\bmod pX$,
since $h_g^{pX}\equiv1\bmod x^{qb+1}$.

\begin{lemma}[Coefficient control]\label{op:coefficient-control}
Suppose \eqref{op:parameters} holds, $X=p^r\le qb<pX$, and $b\ge X/(2M)$. By
adding a measure in $(x^{X+1}\Delta)^\Lambda$ to $\nu$, one may change $f_b(z)$
by any polynomial of degree at most $M$ in $z_r$, independently for every lower
label $z\bmod X$, without changing any coefficient function of smaller depth.
Successive such corrections over increasing depths converge in $\Delta$ and do
not change reduction modulo $x$.
\end{lemma}

\begin{proof}
The measure with values in $(x^{X+1}\Delta)^\Lambda$ annihilates the coefficient
functions of depth smaller than $b$: it acts through a formal series whose
coefficients at those depths lie in $x^{X+1}R_d$, and the pairing
$h_g^{qb}x^{-qb}$ picks up only coefficients of $h_g^{qb}$ at depth up to $qb$.
The remaining action on $z_r$ is by
multiplication by a polynomial of degree at most $\lfloor pX/X\rfloor$: the
new-digit variable $z_r$ runs over $\F_p$, and the change in $f_b$ is the value
of the modified measure paired against $h^{qb}$ at each new digit independently.

The measure of a polynomial of degree $M$ at points $\{0,\ldots,p-1\}$ spans a
space of dimension $M+1$, and the constraint is that its values at each
pre-chosen point are free. Concretely, choose a $p$-adic interpolation
polynomial of degree $M$ matching the desired values on any $M+1$ chosen points
and prescribed elsewhere: such a polynomial exists by linear independence of the
evaluation functionals at distinct points (divided differences); when $p>M$ and
the points are distinct modulo $p$, the Vandermonde determinant is invertible.
The convergence of successive corrections: at depth $X+1$, corrections live in
$x^{X+1}R_d$, and the $x$-adic filtration is complete, so the sum over $X$ is a
well-defined measure. Reduction modulo $x$ is unchanged because the corrections
each vanish to depth at least $X+1\ge 1$.
\end{proof}

\subsection{Triangular matrices and rank matching}

Let $\mathsf A=(\mathsf A_{l,j})_{l,j\in\Z}$ be a lower-going triangular matrix,
so $\mathsf A_{l,j}=0$ for $l>j$. For $l\le j$ write $\mathsf R(l,j)$ for the rank
of its finite block on row and column indices from $l$ to $j$, and put
$\mathsf R(l,j)=0$ when $l>j$. This is the rank on rows at least $l$ and columns
at most $j$: triangularity makes all remaining entries in that range zero.

\begin{lemma}[Rank matching]\label{op:matching}
For such a matrix, the numbers
\begin{equation}\label{op:rank-matching}
 \mathsf R(l,j)-\mathsf R(l,j-1)-\mathsf R(l+1,j)+\mathsf R(l+1,j-1)
\end{equation}
are $0$ or $1$ and define a partial matching of rows with columns. The rank
$\mathsf R(l,j)$ counts the matched pairs in its block. When entries are fixed in
order of increasing depth $b=j-l$, a new match at $(l,j)$ is possible exactly
when both its row and column are still unmatched and the corner entry avoids one
specified scalar. That scalar is determined by smaller-depth entries. Decisions
at a fixed depth are independent of one another.
\end{lemma}

\begin{proof}
For fixed $j$, put $d_j(l)=\mathsf R(l,j)-\mathsf R(l,j-1)$. Adding one column
changes the rank by $0$ or $1$. If column $j$ is independent of the earlier
columns on rows at least $l+1$, it remains independent after row $l$ is added: the
newly admitted earlier column $l$ is zero on the old rows. Thus
$d_j(l)\ge d_j(l+1)$. The difference $d_j(l)-d_j(l+1)$ is therefore $0$ or $1$,
with at most one nonzero value in each column. The analogous argument for row
increments gives at most one in each row. Telescoping over the finite block proves
that $\mathsf R(l,j)$ counts its matched pairs.

Both corner indices are still unmatched precisely when
\[
 \mathsf R(l,j-1)=\mathsf R(l+1,j-1)=\mathsf R(l+1,j).
\]
To see the exceptional corner value, permute the corner column to the first
position and write the block as
\[
 \begin{pmatrix}c&r\\ u&B\end{pmatrix}.
\]
The displayed rank equalities say $r=\rho B$ and $u=B\eta$ for some row vector
$\rho$ and column vector $\eta$. Row and column operations reduce the block to $B$
together with the single entry $c-\rho B\eta$. The value $\rho B\eta$ is
independent of the choices: changing $\rho$ by a vector annihilating $B$, or
changing $\eta$ by a vector annihilated by $B$, leaves it fixed. Exactly the value
$c=\rho B\eta$ keeps the old rank; every other value creates a new match. All
entries of $r,u,B$ have smaller depth, which also proves independence of decisions
at the current depth.
\end{proof}

At a depth $b$ with $X=p^r\le qb<pX$, every coefficient of the matrix in
\eqref{op:coefficients} used in the current rank tests depends only on the column
label modulo $pX$. Translating both indices by $pX$ therefore preserves each finite
block used in a test. All eligibility tests and their exceptional corner values
are periodic in $j$ with that period. Fixing a lower label $(\alpha+j)\bmod X$
leaves $p$ replicas, indexed by the next digit $z_r$. At any allowed depth,
Lemma~\ref{op:coefficient-control} permits the following rule:
\begin{quote}
Prescribe no pivot at any chosen set of at most $M$ eligible next digits, and
prescribe a pivot at every other eligible next digit.
\end{quote}
Choose a polynomial correction of degree at most $M$ whose value at each chosen
digit is the exceptional corner value minus the current coefficient. Because at
most $M$ values are prescribed, the solution space has positive dimension. At any
other eligible digit, equality of the corrected entry with its exceptional corner
value cuts out a proper affine hyperplane in this space: a polynomial vanishing at
the prescribed digits need not vanish at the additional one. There are at most $p$
such hyperplanes, each containing at most a fraction $1/p^2$ of the space, so their
union cannot cover it. The rule is applied independently at each lower label; it
does not distinguish replicas at still higher digits.

Define center digits and a central set of their indices by
\begin{align}
 d_i&=\left\lfloor\frac{(i+1/2)p}{M}\right\rfloor\quad(0\le i<M),\notag\\
 s_0&=(M-1)/2,\qquad d_{s_0}=\delta,\qquad R_0=\lfloor M/8\rfloor,\notag\\
 I&=\{s_0-R_0+1,\ldots,s_0+R_0\},\qquad B_0=|I|=2R_0,\notag\\
 \beta&=\log_p B_0>1/3.
 \label{op:centers}
\end{align}
The cyclic positive gaps between the $d_i$ are $p/M\pm1$. The set $I$ meets
neither endpoint $0,M-1$. For an index set $S$, write $d(S)=\{d_i:i\in S\}$; in
particular $I-1=\{i-1:i\in I\}$ is again a set of center indices.

The central digits serve two related purposes. Their repeated choices give $B_0$
surviving possibilities at each new digit, producing the exponent $\beta$. At
sufficiently large stages, the digit $\delta$ is retained for every possible pivot
crossing a fixed pole bound. The next proposition proves these assertions for the
resulting matrix, including the uniform conclusion for every large cutoff.

The whole construction depends on a small set of parameters. Write
\begin{equation}\label{op:parameters}
 p>16M^2,\qquad M\ge 81,\qquad M\equiv 1\pmod 8,\qquad \beta=\log_p(\lfloor M/4\rfloor).
\end{equation}
Then $\beta>1/3$, and $p$ is at least linear in $M^2$. The condition $M\ge 81$
ensures $B_0\ge 20$, hence a sizeable growth exponent. The congruence is needed so
that $s_0=(M-1)/2$ is integral and $I$ sits strictly inside the digit range.

\begin{proposition}[Operator with persistent defects]\label{op:operator}
Under \eqref{op:parameters}, there is $a\in\mathscr E_0$ satisfying
\textup{(with initial symbol $\sigma(\bar a)=C$)} whose triangular matrix on $\Phi$ has the following
properties.
\begin{enumerate}
\item For every fixed $J\in\Z$, only finitely many matched pairs $(l,j)$ satisfy
$l\le J<j$.
\item There are constants $c>0$ and $N_0$ such that for every integer $N\ge N_0$,
at least $cN^\beta$ columns in $[-N,0]$ and at least $cN^\beta$ rows in $(0,N]$
are forever unmatched.
\end{enumerate}
Here the matching is that of Lemma~\ref{op:matching}; the constants may depend on
$p$.
\end{proposition}

\begin{proof}
Begin at one initial depth and proceed by stages. At each subsequent stage,
adjacent clusters of lower labels are paired, and only specified central clusters
retain next-digit replicas. We record the surviving labels exactly. Once the
stages have defined the operator, that record simultaneously excludes late
crossings of a fixed bound and counts the unmatched labels at every large scale.

\emph{Initialization.} Give every row and column index $j$ the label $z=\alpha+j$.
Begin with $d=\lfloor p/M\rfloor$. All entries of smaller depth are zero by
initialization, since $qd<p^2$. This depth belongs to the $X=p$ stage and satisfies
$d\ge p/(2M)$. Match every column except those with
\[
 z_0=\delta,\qquad z_1\in\{d_0,\ldots,d_{M-1}\}.
\]
The pivot rule permits this: at the one specified lower label protect $M$ digits,
and at all other lower labels protect none. The corresponding surviving rows have
\[
 z_0=\delta-d,\qquad z_1\in\{d_0,\ldots,d_{M-1}\},
\]
without a borrow in subtraction. These statuses persist until the end of
$qb<p^2$. Surviving lower labels could meet again only at a depth congruent to $d$
modulo $p$, and no second such depth occurs in this range.

\emph{Induction.} Take $X=p^r$ with $r\ge2$. Suppose the surviving labels modulo
$X$ are exactly
\begin{equation}\label{op:induction}
\begin{array}{c|ccc}
 &z_0&z_j\ (1\le j\le r-2)&z_{r-1}\\ \hline
 \text{columns}&\delta&d(I)&d_i\ (0\le i<M)\\
 \text{rows}&\delta-d&d(I-1)&d_i\ (0\le i<M).
\end{array}
\end{equation}
All displayed choices are independent, and each has all $p$ values of the new
digit $z_r$. Group these lower labels into clusters indexed by $i$, according to
their last digit $d_i$.

Process this first band by increasing depth. Call a lower label \emph{full} if all
its $p$ next-digit replicas remain. When an edge has two full endpoints, match them
at every next digit, except that for column cluster $i\in I$ retain the $M$ digits
\[
 z_r\in\{d_0,\ldots,d_{M-1}\}.
 \label{op:retained-digits}
\]
In this case there is no cyclic wrap to the row cluster: the depth is the positive
difference of the ordinary lower representatives. The retained next digits on the
two sides therefore agree. Mark both endpoints no longer full. Whenever either
endpoint is not full, there are at most $M$ eligible next digits, since the
next-digit replicas of the two endpoints correspond bijectively at a fixed depth.
Require no pivot at all of them; these are precisely the choices allowed by the
pivot rule. At one depth, different column lower labels have different row lower
labels, so the prescriptions are compatible.

After the band, drop the last digit and merge endpoints that now agree. The
surviving labels modulo $X$ are concentrated on the central clusters $I-1$ for
rows and $I$ for columns, following \eqref{op:induction}. Retained clusters use the
$M$ surviving digits from \eqref{op:retained-digits} to seed the next band; the
digit $\delta$ is preserved for the columns because no central drop occurs at
digit $0$. This establishes the induction on $X=p^r$.

\emph{Late crossings and counting.} A match at $(l,j)$ with $j-l$ large would
require a surviving row at $l$ and column at $j$ whose clusters overlap in a high
digit. By the induction, rows and columns live in disjoint center classes at each
stage, so a match is impossible once the depth exceeds the base scale. This gives
the finiteness in the proposition.

Counting: the surviving set of columns modulo $X=p^r$ has size
$B_0^{\,\lfloor r/2\rfloor}\cdot p^{\lceil r/2\rceil}$ before reduction; in the
band of $[X, pX)$ the number of surviving new columns is
$B_0^{r/2}p^{r/2}$. For a cutoff $N=p^r$, this gives $N^{\beta}$ survivors in
$[-N,0]$, and the same count holds for rows in $(0,N]$ by symmetry of the matching
record. The constant is bounded below by $B_0^{-1}$.
\end{proof}

Attached to the operator is the matrix over $\Phi$ discussed above. Because the
matrix arises from a genuine element $a\in\mathscr E_0$, its coefficients are
compatible with the periodicity, with the $\Lambda$-action, and with the skew
relations of $\Delta$. The operator $a$ has initial symbol $C=U_3$, so its kernel
and cokernel exist in a controlled amount.

\section{From pivots to a gluing lattice}\label{glue:section}

We now use the operator $a$ to produce an object $U$ whose finite-level
homotopy is finite, but whose completion carries structured kernel and
cokernel directions. This is the algebraic object $U$ of the target
conditions \eqref{eq:target}.

The two kinds of defect of the matrix of $a$ on $\Phi$ become a compact
open and a discrete cocompact subspace of the overlap. A gluing of one
inside the other produces the object $U$. Its two pieces survive
independently.

\subsection{Pivots and the pole topology}

Lemma~\ref{op:matching} associates a partial matching to any matrix
representing an element of $\mathscr E_0$ on the Laurent field
$\Phi$. Read this matching in terms of the kernel and the cokernel of
the map on $\Phi$.

\begin{lemma}[Pivots and the pole topology]\label{glue:pivot-topology}
An integer $j$ is an unmatched column of $a$ if and only if there is a
vector in $\mathcal K=\ker(a:K_0\to K_0)$ with leading term $w^j$. An
integer $l$ is a matched row if and only if it is the leading position
of a nonzero vector in $a\Phi$. Moreover, for every $J\in\Z$,
\begin{equation}\label{glue:crossing-quotient}
 \dim_k\frac{a\Phi\cap F_J\Phi}{a(F_J\Phi)}
   =\#\{(l,j)\text{ matched}:l\le J<j\}<\infty.
\end{equation}
The image $a\Phi$ is closed. Thus $\mathcal Q=\Phi/a\Phi$ and
$\mathcal K$ are Hausdorff locally compact $k$-vector spaces, with
compact open subspaces given respectively by the images of $F_J\Phi$
and by $\mathcal K\cap F_J\Phi$.
\end{lemma}

\begin{proof}
Suppose column $j$ is unmatched. For every lower cutoff $L\le j$, its
rank increment in the block $[L,j]$ is zero. Hence
\[
 S_L=\{f\in w^j+F_{j-1}\Phi:a(f)\in F_{L-1}\Phi\}
\]
is nonempty: a solution of the finite coefficient equations extends by
a zero tail. Continuity of $a$ makes $S_L$ closed in the compact space
$w^j+F_{j-1}\Phi$. Lowering $L$ adds equations, so these sets are
nested. Compactness gives a point in their intersection, and its image
lies in $\bigcap_LF_{L-1}\Phi=0$. This is an actual kernel vector with
leading term $w^j$. Conversely, such a vector makes column $j$
dependent in every finite block, so the column is unmatched. In
particular,
\begin{equation}\label{glue:image-increment}
 \dim_k\frac{a(F_j\Phi)}{a(F_{j-1}\Phi)}
 =
 \begin{cases}
 1,&\text{column $j$ matched},\\
 0,&\text{column $j$ unmatched}.
 \end{cases}
\end{equation}

For the second assertion, a matched row $l$ means by definition that
$l$ is the leading position of a nonzero vector in $a\Phi$, which is
$a(w^j)$ for the matched column $j$ paired with $l$. Conversely, if $l$
is the leading position of $a(f)\in a\Phi$ with $f\ne0$, take $j$ the
leading position of $f$; the pairing $(l,j)$ is matched, because the
image increment in degree $j$ is nonzero.

For the dimension formula, the numerator counts $a(w^j)$ for matched
$(l,j)$ with $l\le J<j$, and the denominator counts the same for
$j\le J$. The difference is exactly the pairs $(l,j)$ matched with
$l\le J<j$.

The image $a(F_J\Phi)$ is compact by continuity. The quotient in
\eqref{glue:crossing-quotient} is finite-dimensional over the finite
field $k$, hence a finite set. Thus $a\Phi\cap F_J\Phi$ is a finite
union of translates of $a(F_J\Phi)$, and is compact and closed. The
$F_J\Phi$ form an open cover of $\Phi$, so $a\Phi$ is closed in $\Phi$.
The quotient is Hausdorff, and the quotient map is open, proving the
asserted compact open subspaces of $\mathcal Q$. The kernel is closed
by continuity, and its intersections with the compact open stages give
the assertion for $\mathcal K$.
\end{proof}

\subsection{The complete cone and its overlap}

In the category of complete objects, or equivalently in the perfect
$\mathscr E$-module calculus of
Proposition~\ref{complete:realization}, define
\begin{equation}\label{glue:complete-object}
 m=\cofib(a:\widehat T'\longrightarrow\widehat T'),
 \qquad Y=\bigoplus_{r=0}^{D-1}m[r].
\end{equation}
The finite sum over all residues ensures that both kinds of defect
occur in every degree.

Write $W_i(Y)=\pi_iLY$ for the normalized overlap of $Y$. Explicitly,
$W_i(Y)$ is the $i$-th sections-homotopy of $LY$, which is naturally
identified with the sum, over the $D$-periodic residue classes, of
pieces on which the line twist acts by an explicit fractional lattice.
Each piece reduces to the Laurent field $\Phi$ together with a sign and
shift.

\begin{lemma}[Overlap decomposition]\label{glue:overlap}
For each $i\in\Z$ there is a natural identification
\[
 W_i(Y)=\mathcal Q_i\oplus\mathcal K_i,
\]
where
\[
 \mathcal Q_i=\bigoplus_{r=0}^{D-1}\mathcal Q_{i-r}
 \qquad
 \mathcal K_i=\bigoplus_{r=0}^{D-1}\mathcal K_{i-r},
\]
and, in each residue class $r$, the factors
$\mathcal Q_{i-r}, \mathcal K_{i-r}$ are respectively the cokernel and
kernel of the operator $a$ on $\Phi$ in the corresponding grading.
Moreover, $\mathcal Q_i$ is discrete, $\mathcal K_i$ is compact, and
both depend naturally on $i$.
\end{lemma}

\begin{proof}
Section-homotopy on $Ly$ is computed by the section formula of
Proposition~\ref{complete:sections}, applied to $Y$ and then localized.
For the cone $m=\cofib(a)$, the section formula gives the mapping-cone
complex of $a:\Phi\to\Phi$ on the overlap. Its homology is
exactly $\ker a\oplus\coker a$ in each shift, so the sum over residues
gives the stated decomposition.
\end{proof}

For each forever unmatched positive row $j$, let $q_j\in\mathcal Q$ be
the class of $w^j$. For each forever unmatched nonpositive column $l$,
choose $k_l\in\mathcal K$ with leading term $w^l$, as supplied by
Lemma~\ref{glue:pivot-topology}. In particular
$k_l\in\mathcal K\cap F_0\Phi\subset R$. Pair these lists in order of
increasing $j$ and increasing $|l|$; write $(j,l)$ for a resulting
pair of an unmatched row and an unmatched column. Since
$(0,k_l)\in C_0^{\mathrm{lat}}$ for the compact open
$C_0^{\mathrm{lat}}=\mathcal K\cap R$ in each residue, the pair
$(q_j,k_l)$ and the vector $(q_j,0)$ have the same image in the
quotient. The next lemma verifies independence of these quotient
classes and records simultaneous bounds on their leading positions.

\begin{lemma}[Simultaneous defect pairs]\label{glue:pairs}
The vectors $(q_j,k_l)\in\mathcal Q\oplus\mathcal K$, indexed by the paired
lists, are linearly independent modulo $C_0^{\mathrm{lat}}$. For every
sufficiently large integer $N$, at least $cN^\beta$ pairs, for a fixed
$c>0$, simultaneously satisfy $j\le N$ and $|l|\le N$. For these pairs,
their kernel coordinates are linearly independent modulo $x^{qN+1}R$.
\end{lemma}

\begin{proof}
The $q_j$ are independent modulo $\mathrm{im}_{\mathcal Q}(F_0\Phi)$. Otherwise
a finite nontrivial relation, after subtracting a vector in $F_0\Phi$, would be an
actual image of $a$ whose leading position is an unmatched positive row, which
contradicts Lemma~\ref{glue:pivot-topology}. It also proves independence of the
pairs modulo $C_0^{\mathrm{lat}}$.

Let $n_+(N)$ count the positive unmatched rows at most $N$, and let $n_-(N)$
count the nonpositive unmatched columns of absolute value at most $N$. In the
ordered pairing, the first condition holds precisely for the first $n_+(N)$
pairs and the second for the first $n_-(N)$ pairs. So exactly
$\min\{n_+(N),n_-(N)\}$ pairs satisfy both conditions.
Proposition~\ref{op:operator} bounds each count below by a constant times
$N^\beta$ for every sufficiently large $N$, giving the simultaneous bound.

The leading terms of the kernel vectors are the distinct powers
$w^l=x^{q|l|}$, with $q|l|\le qN$. A nonzero combination has one of these as its
lowest $x$-power, so it cannot lie in $x^{qN+1}R$. This proves independence modulo
$x^{qN+1}R$.
\end{proof}

In every period residue, extend the independent images of the pairs to an
algebraic basis of $(\mathcal Q\oplus\mathcal K)/C_0^{\mathrm{lat}}$. Choose
lifts retaining the specified pairs, and let $\mathcal L_i$ be their span. The
projection of this algebraic span to the quotient is an isomorphism. It
therefore meets $C_0^{\mathrm{lat}}$ trivially and, together with
$C_0^{\mathrm{lat}}$, fills the whole space. Since $C_0^{\mathrm{lat}}$ is
open, the span is discrete. Each open coset of $C_0^{\mathrm{lat}}$ contains
exactly one point of the span, so the span is also closed. Finally, the compact
space $C_0^{\mathrm{lat}}$ maps onto the quotient by the span, proving
cocompactness. Make the same choice along each $t$-period.

\begin{lemma}[Gluing lattice and its compact quotients]\label{glue:lattice}
Let $\mathcal L_i\subset W_i(Y)$ be the discrete cocompact lattice just
constructed. Then $\mathcal L_i$ is spanned by vectors with distinct leading
positions of the form $w^j$ or $w^l$, and the compact quotient
$W_i(Y)/\mathcal L_i$ is naturally isomorphic as a topological $k$-vector space
to the compact subspace $\mathcal K_i\cap F_0\Phi_i$ modulo the lattice of
kernel coordinates.
\end{lemma}

\begin{proof}
The quotient identification is inspection: each summand of the algebraic span
fits with a compact piece in a coset. Topological compactness follows by
combining cocompactness of the lattice in the sum and compactness of
$C_0^{\mathrm{lat}}=\mathcal K\cap R$: every point of $W_i(Y)$ lies within a
compact translate of $\mathcal L_i$, so the quotient is compact.
\end{proof}

\subsection{The glued object}

Recall the equivalence $\mathcal N[1/t]\simeq\Spec k$
(Proposition~\ref{geom:open}). The open category is the derived category of
$k$-vector spaces, and its objects are sums of finite-dimensional spaces in
finitely many degrees. The localization map $Y\to LY$ is $t$-inversion; on the
overlap it induces the identification $W_i(Y)=\mathcal Q_i\oplus\mathcal K_i$,
and the lattice $\mathcal L_i$ is a {$k$}-basis for the discrete-cocompact piece.

We now construct $U$ as a homotopy pullback in $\mathbf D(\mathcal N)$. See the
diagram below.

\begin{proposition}[The glued object]\label{glue:object}
There is an object $U\in\mathbf D(\mathcal N)$ fitting into a homotopy pullback
\[
\begin{tikzcd}[column sep=large,row sep=large]
 U \arrow[r]\arrow[d] & Y \arrow[d] \\
 V \arrow[r] & LY
\end{tikzcd}
\]
where
\begin{itemize}
\item $Y=\bigoplus_{r=0}^{D-1}m[r]$ is the object of
\eqref{glue:complete-object};
\item $V$ is a finite sum of shifts of the pure open unit $\mathcal O[1/t]$
chosen so that, in every degree $i$, the image of $V$ in $W_i(Y)$ is exactly
the discrete lattice $\mathcal L_i$;
\item the map $V\to LY$ realizes this identification, and its cofiber is
identical to the cone on the sum of the specified lattice basis vectors.
\end{itemize}
Then:
\begin{enumerate}
\item $\pi_iU$ is a finite group for every $i\in\Z$.
\item $\widehat U\simeq Y$; in particular
$\widehat U\in\Thick(\widehat T')$.
\item The lattice pairs of Lemma~\ref{glue:pairs} survive as concrete
subspaces of the overlap map $\pi_iY\oplus\pi_iV\to W_i(Y)$. The joint map has
finite kernel and cokernel in every degree.
\end{enumerate}
\end{proposition}

\begin{proof}
The homotopy pullback of sheaves is a homotopy pullback of complexes; its
existence uses the derived category model structure on
$\mathbf D(\mathcal N)$.

For (1): the fracture square of $Y$ and the analogous square for $V$
(long exact in $\pi_*$) fit into a commutative diagram with the long exact
sequence of the pullback. Each of $\pi_iY$, $\pi_iV$, $\pi_iLY$ is finite or is
torsion: $\pi_iY$ is finite by \eqref{glue:complete-object} (the cone of
$T'$ over an operation preserves finiteness); $\pi_iV$ is finite because the
open unit is the unit of $\F_p$-vector spaces and $V$ is finite; and
$\pi_iLY$ is a compact piece plus a discrete piece, and both have finite
projection in every degree by the construction of $\mathcal L_i$. The long
exact sequence of the pullback gives that $\pi_iU$ is an extension of a
subquotient of a finite group by another finite group, hence finite.

For (2): completion is exact and kills the open unit. Since $V$ is a finite sum
of shifts of $\mathcal O[1/t]$ and $\mathcal O[1/t]$ is $t$-invertible, $V$ is
killed by completion: $\widehat V\simeq 0$. Applying completion to the
pullback square gives $\widehat U\simeq Y$, as claimed. In particular
$\widehat U\in\Thick(\widehat T')$, because $Y$ is a finite sum of shifts of
$m=\cofib(a:\widehat T'\to\widehat T')$.

For (3): the joint map in degree $i$ is the identity of $Y$'s overlap plus
the chosen lattice embedding on $V$'s overlap. Its kernel is the finite set of
pairs whose image falls into $Y$'s compact piece; this uses the simultaneous
independence of Lemma~\ref{glue:pairs}. Its cokernel is what remains once the
lattice pairs are removed from $W_i(Y)$; this is the compact quotient computed
in Lemma~\ref{glue:lattice}, which is finite by compactness and discreteness.
\end{proof}

\section{The obstruction to finite construction}\label{obs:section}

We now isolate the finite-tower argument. Its input is an exact functor to
perfect modules over a periodic algebra over a domain with principal left and
right ideals. The criterion uses ordinary homology and the homology of module
duals to pass backward through any finite tower: the dual test removes the free
part of a tower tail, and the ordinary test then uses torsion to force vanishing.
We next construct a flat extension of the operation coefficients for which $Y$
has nonzero torsion homology. The final part verifies the relation between the
two tests for maps that come from global objects.

\subsection{A criterion for arbitrary finite constructions}

We state the two tests abstractly before constructing the coefficient extension
that makes them apply to the operator $a$ and the gluing lattice. Torsion
homology by itself cannot be the obstruction: the cone of $2:\Z\to\Z$ is a
two-cell object with homology $\Z/2$. The additional information will restrict
the maps that exist in the original category.

Let $\mathcal C$ be an idempotent-complete triangulated category, let
$T_0,U_0\in\mathcal C$, and let
\[
 \Psi:\mathcal C\longrightarrow\Perf(\mathscr B_0)
\]
be an exact functor with $\Psi(T_0)\simeq\mathscr B_0$.

\begin{theorem}[A finite-tower criterion]\label{rew:obs:tower-criterion}
Assume $H(\Psi(U_0))$ is torsion as a right $S_0$-module in every degree and is
nonzero in at least one degree. Assume also that, for every
$P\in\Thick_{\mathcal C}(T_0)$ and every map $g:U_0\to P$ in $\mathcal C$,
\begin{equation}\label{rew:obs:two-tests}
 H(\Psi(g))=0 \quad\Longrightarrow\quad H\bigl((\Psi(g))^*\bigr)=0.
\end{equation}
Then $U_0\notin\Thick_{\mathcal C}(T_0)$.
\end{theorem}

Here $\Thick_{\mathcal C}(T_0)$ is the ordinary thick closure, using
isomorphisms, finite sums, all integer shifts, triangles, and retracts. The
quantifier in \eqref{rew:obs:two-tests} concerns maps in $\mathcal C$; in our
application, their global origin is precisely what imposes a constraint on the
gluing lattice.

The first algebraic fact separates free homology from torsion homology over
the coefficient ring.

\begin{lemma}[Periodic splitting]\label{obs:periodic-splitting}
Let $S_0$ be a domain whose left and right ideals are principal, and let $D_0$ be
a positive even integer. Every finitely generated right $S_0$-module is a direct
sum of a torsion module and a finite free module and has a finite free resolution
of length at most one. Every perfect right dg module over $S_0[t^{\pm1}]$, with
$|t|=D_0$ and zero differential, splits in its derived category into the
$D_0$-periodizations of its shifted homology modules. In particular it splits as
a finite sum of free shifts and a perfect module with torsion homology.
\end{lemma}

\begin{proof}
$S_0$ is noetherian on both sides, since every one-sided ideal is principal.
Nonzero right ideals $dS_0$ and $eS_0$ must intersect; otherwise the right
ideals $e^idS_0$ for $i\geq0$ would form an infinite internal direct sum in
$S_0$: in a finite relation, its term of least $e$-power belongs both to $dS_0$
and to $eS_0$ after canceling that power, hence vanishes; continue inductively.
This contradicts noetherianity. The same argument on the left gives the
two-sided Ore division ring of fractions $Q_0$. Clearing common denominators
gives exact localization of modules. Consequently, for a finitely generated
right module $M$, the kernel
\[
 M_{\mathrm{tor}}=\ker\bigl(M\longrightarrow M\otimes_{S_0}Q_0\bigr)
\]
consists exactly of the elements right-annihilated by a nonzero element of
$S_0$.

The quotient $M/M_{\mathrm{tor}}$ embeds in the finite-dimensional right
$Q_0$-vector space $M\otimes_{S_0}Q_0$. Choose coordinates in that space and
finitely many generators of the embedded module. In each coordinate clear their
common \emph{left} denominators; left multiplication in a coordinate is right
$S_0$-linear, so this produces an injection
\[
 M/M_{\mathrm{tor}}\longrightarrow S_0^r
\]
for some finite $r$. The choice of left denominators is the one compatible with
right modules.

Every submodule of $S_0^r$ is free of finite rank: induct on $r$ and project to
the first coordinate. The image is a right ideal. If zero, the induction
hypothesis applies inside $S_0^{r-1}$. If nonzero, its principal generator is a
nonzero element of the domain, so the image is free of rank one. The projection
then splits, and its kernel lies in $S_0^{r-1}$. Hence $M/M_{\mathrm{tor}}$ is
finite free, and the sequence with this quotient splits. The kernel of a finite
free presentation of $M$ is also finite free by the same argument. This gives a
finite free resolution of length at most one.

Periodic case: a perfect right dg module over $S_0[t^{\pm1}]$ (with $|t|=D_0$,
zero differential) has finitely generated homology in each residue and is
determined by it. Split each $H_j$ as torsion plus free by the module-theoretic
part. Because $D_0$ is even, the shifts by multiples of $D_0$ preserve
$\Z$-grading, and the $D_0$-periodizations of the shifted free modules and the
torsion modules realize the direct sum. This provides the asserted splitting.
\end{proof}

The second fact puts every finite construction into a tower of free layers,
with all retracts taken at the end.

\begin{lemma}[Finite towers and retracts]\label{rew:obs:tower-normal-form}
For an idempotent-complete triangulated category $\mathcal C$ and an object
$T_0$, every $P\in\Thick_{\mathcal C}(T_0)$ is a retract of an object $P_0$
admitting a finite tower of triangles
\[
 F_j\longrightarrow P_j\longrightarrow P_{j+1}\longrightarrow F_j[1],
 \qquad 0\le j<s,\qquad P_s=0,
\]
where each $F_j$ is a finite sum of arbitrary integer shifts of $T_0$.
\end{lemma}

\begin{proof}
Call an object a finite extension of free layers if it admits such a tower.
Zero, finite sums of shifts of $T_0$, and their shifts have this property. The
octahedral axiom concatenates two towers: suppose $A\to E\to B$ is a triangle,
and the first layer of a tower for $A$ is $F_0\to A\to A_1$. The cofiber $E_1$
of the composite $F_0\to A\to E$ fits into triangles
\[
 F_0\longrightarrow E\longrightarrow E_1,
 \qquad A_1\longrightarrow E_1\longrightarrow B.
\]
Induction on the number of layers in $A$, ending with the tower for $B$, gives a
tower for $E$. In particular, if $A$ and $B$ have towers and $f:A\to B$, the
triangle $B\to\cofib(f)\to A[1]$ gives a tower for $\cofib(f)$. Finite sums of
towers give the same closure for finite sums.

Closure under shifts is immediate. Closure under retracts: a retract of a tower
is a retract of its total; a retract of a finite extension of free layers is
still a finite extension of free layers, because the idempotent splits
levelwise in the idempotent-complete $\mathcal C$. Hence every $P\in\Thick$
qualifies.
\end{proof}

We use these two lemmas in the proof of the criterion. Let $U_0\in\Thick(T_0)$
be a witness. By Lemma~\ref{rew:obs:tower-normal-form}, take $P_0$ a retract
extension with a finite tower and a retract insertion $U_0\to P_0$. Let
$Y_0=\Psi(U_0)$ and consider the tower $Y_j$ induced by $\Psi$ on the free
layers.

Write one extended step as
\[
 F\longrightarrow P_1'\longrightarrow P_2'\longrightarrow F[1],
 \qquad g_1':Y_0\to P_1',\quad g_2':Y_0\to P_2',
\]
where $F$ is a finite sum of free shifts of $\Psi(T_0)$, and suppose the two
vanishing assertions hold for $g_2'$. By Lemma~\ref{obs:periodic-splitting},
choose $P_2'\simeq V'\oplus L'$ with $V'$ a finite sum of free shifts and
$H(L')$ torsion. Each component of the projection $P_2'\to V'$ is a class in
the homology of $(P_2')^*$. Dual-homology vanishing for $g_2'$ says that its
composite with each of these classes is zero. Hence
\[
 Y_0\xrightarrow{g_2'}P_2'\longrightarrow V'
\]
is zero as a derived map. The exact sequence of maps into the fiber triangle
therefore lifts $g_1'$ to
\[
 Z'=\fib(P_1'\longrightarrow V').
\]
The fiber construction, equivalently the octahedral axiom, gives a triangle
\[
 F\longrightarrow Z'\longrightarrow L'\longrightarrow F[1].
\]

In its homology sequence, the map preceding $H(F)\to H(Z')$ has torsion source
and free target in each degree. It is zero, so $H(F)\to H(Z')$ is injective. The
lifted map $Y_0\to Z'$ is zero on homology after composition with $Z'\to L'$,
because that composite is the corresponding summand of $g_2'$. Its homology
therefore factors through the submodule $H(F)\subset H(Z')$. The source
$H(Y_0)$ is torsion, so its map into the free module $H(F)$ is zero. It follows
that $H(g_1')=0$.

The map $g_1'$ is the image of a map $U_0\to P_j$ in $\mathcal C$, and this
tower tail belongs to $\Thick_{\mathcal C}(T_0)$. Hypothesis
\eqref{rew:obs:two-tests} gives $H((g_1')^*)=0$ as well. This completes the
backward induction. In particular $\Psi(g_0)$ is zero on homology. It has a
left inverse because $g_0$ is a retract insertion and every functor preserves its
retraction equation. Thus $H(\Psi(g_0))$ is also a split injection, contradicting
the nonzero homology of $Y_0$.

\subsection{A flat coefficient extension}

We now construct the coefficient ring required by the criterion. The extension
must be flat on both sides because homology of right modules and homology of
their left-module duals will both be tested after extending coefficients. It
must also keep the cone of $a$ nonzero while making its homology torsion.

The base is the two-sided Ore localization $A_{\mathrm{loc}}=\Delta[x^{-1}]$
from Proposition~\ref{complete:operations}. It is noetherian, and its associated
graded (from Section~\ref{geom:section}) is a regular ring. The element
$a\in\mathscr E_0$ is a homogeneous element of positive symbol in
$A_{\mathrm{loc}}$. The filtered structure we use comes from the $x$-adic and
digit filtrations, combined as follows.

Introduce the compact Laurent coefficient field
\[
 K_0=k((x))^\Lambda\simeq k((x^q)),
\]
on which $A_{\mathrm{loc}}$ acts by the left-coefficient skew convolution of
Proposition~\ref{complete:operations}. Writing
$F_K:\Phi\to K_0$ for the obvious inclusion, $K_0$ is the exact two-sided flat
localization of $A_{\mathrm{loc}}$ at $x$.

Give $A_{\mathrm{loc}}$ its good filtered structure from
\eqref{geom:graded-operations}: build $\mathrm{Rees}(A_{\mathrm{loc}})$ from the
associated graded $k[x,A,B,C]$ of Section~\ref{geom:section}, with the weight
grading of $x,A,B,C$ and the Ore localization of the graded pieces. This is a
noetherian graded ring by the same argument as the associated graded of the
complete ring.

For a finitely generated $A_{\mathrm{loc}}$-module $M$, choose a finite free
presentation $F\to M$ over $A_{\mathrm{loc}}$ and lift its filtration to
$F\to M$. The associated graded $\gr M$ is finitely generated over $\gr
A_{\mathrm{loc}}$. Define
\[
 \mu(M)=\varprojlim_j M/s^jM.
\]
via the Rees module $R(M)=\bigoplus_j s^jM_{\le j}$ and $M_j=M/s^jM$ for
$j\geq0$. The Rees module is a module over $R(A_{\mathrm{loc}})$.

\begin{lemma}[Good filtrations and flatness]\label{obs:filtration-properties}
If $N$ is a submodule of a finite shifted filtered free module over
$A_{\mathrm{loc}}$, then $N$ has a good filtration, and the induced surjection
from a finite shifted free module is strict for that filtration. Each quotient
$A_{\mathrm{loc}}/F^nA_{\mathrm{loc}}$ is finite, so $A_{\mathrm{loc}}$ and its
finite products are compact. Any two good filtrations on a finite module differ
by bounded shifts. A map of finite good filtered modules is bounded up to a
filtration shift.
\end{lemma}

\begin{proof}
Choose homogeneous generators of $\gr N$ over the noetherian ring
$\gr A_{\mathrm{loc}}$ and lift them to $N$. For an element of $N$, subtract a
linear combination of the lifts matching its initial symbol, and repeat. The
remainder increases in filtration at each step, and the coefficients form a
convergent series in $A_{\mathrm{loc}}$, so the element is a finite linear
combination of the lifts. This also proves the surjection from a finite shifted
free module is strict for the induced filtration.

For finiteness of the quotients: the graded pieces below $n$ are
finite-dimensional over the finite field $k$, by the associated graded of
$\Delta$ from Lemma~\ref{geom:graded-operations}. Consequently $A_{\mathrm{loc}}$
and its finite products are compact. A submodule of a finite shifted filtered
free module is the image of a compact finite free module, hence compact and
closed. The boundedness of map shifts is seen on strict generators. Bounded
difference between two filtrations is the standard fact that a good filtration
generates a good filtration of its subquotient.
\end{proof}

Products at the truncated levels give
$\mathcal S=\mu(A_{\mathrm{loc}})$ its ring structure and give $\mu(M)$ its
module structure. The natural maps from $A_{\mathrm{loc}}$ and $M$ are obtained
by taking classes at each level. A bounded-shift map of good filtered modules
gives a map on these unions; changing the bound does not change the map, by the
transition inclusions. The underlying module $\mu(M)$ is therefore functorial
and independent of the chosen good filtration.

\begin{lemma}[Flatness of the extension]\label{obs:flatness}
$\mathcal S$ is left and right flat over $A_{\mathrm{loc}}$, and the natural
map $M\otimes_{A_{\mathrm{loc}}}\mathcal S\to\mu(M)$ is an isomorphism for
every finite $A_{\mathrm{loc}}$-module $M$.
\end{lemma}

\begin{proof}
This functor is exact on finite modules. Use induced and quotient filtrations in
a short exact sequence to make it strict. Its Rees sequence remains exact modulo
$s^j$, because the quotient Rees module is $s$-torsion-free. Ore localization
preserves exactness. Inverse limits preserve it here because the transition maps
in each module system are surjective, and taking the filtered union preserves it
as well. The natural map $M\otimes_{A_{\mathrm{loc}}}\mathcal S\to\mu(M)$ is an
isomorphism for a free module, hence for every finite module by a finite
presentation. This proves left flatness; over a noetherian ring, exactness on
finite modules suffices. The identical construction for left modules proves
right flatness.
\end{proof}

The actual coefficient algebra used in the criterion is
$\mathscr B=\mathcal S[t^{\pm1}]$. Because $\mathcal S$ is a domain and $t$ is a
central element of positive even degree $D_0$, $\mathcal S[t^{\pm1}]$ is again a
domain with principal left and right ideals, and its two-sided Ore localization
is the field $\mathcal S_{(0)}[t^{\pm1}]$ obtained from $\mathcal S$'s own Ore
field. The periodic version of the criterion uses $S_0=\mathcal S$ and
$D_0=D$ in Lemma~\ref{obs:periodic-splitting}.

\begin{lemma}\label{obs:principal}
The ring $\mathcal S$ is a domain, every left or right ideal of $\mathcal S$ is
principal, and $\mathcal S$ has a two-sided Ore division ring of fractions. The
operator $a$ constructed in Proposition~\ref{op:operator} has no one-sided
inverse in $\mathcal S$. In particular $\mathcal S/a\mathcal S\ne0$, and this
right $\mathcal S$-module is torsion.
\end{lemma}

\begin{proof}
The graded ring of $\mathcal S$ is a domain, so multiplicativity of initial
symbols proves $\mathcal S$ is a domain. Every nonzero homogeneous element of
$\gr\mathcal S$ is a power of $C$ times a homogeneous unit. In a nonzero right
ideal choose $d_0$ minimizing the nonnegative $C$-order of its initial symbol.
For any member $d$ of the ideal, its initial symbol is a multiple of
$\sigma(d_0)$. Lift the multiplier, subtract the corresponding right multiple of
$d_0$; the remainder is still in the ideal and has strictly larger filtration, so
its initial $C$-order is still at least the chosen minimum. Repeating, and summing
the convergent multipliers, expresses $d$ as a right multiple of $d_0$. The
successive multipliers tend to zero because their filtration degrees increase by
the same amount as those of the remainders. This proves the right
principal-ideal property; the left proof is identical.

The intersection argument in the proof of Lemma~\ref{obs:periodic-splitting} now
gives the two-sided Ore division ring of fractions.

Finally, the operator $a$ has no one-sided inverse: if $ab=1$, then
$\sigma(\bar a)\sigma(\bar b)=1$ in the polynomial ring $k[x,A,B,C]$; but
$\sigma(\bar a)=C$ is not a unit there, so no such $b$ exists. The same argument
in the other factor shows there is no left inverse either. Hence $a\mathcal
S\ne\mathcal S$ and $\mathcal S/a\mathcal S$ is nonzero. It is torsion because
$\mathcal S$ is a domain and the two-sided Ore localization kills $a$:
$\mathcal S/a\mathcal S\otimes_{\mathcal S}Q_0=0$ since $Q_0$ is a division ring
and $a$ is a nonzero nonunit there.
\end{proof}

\subsection{Orbit control and the growth bound}

The rest of the proof needs one numerical input: a rank bound on the joint
kernel of two integral operators in the Laurent field. We derive it here from
the balanced orbit enumeration of the displacement lemma.

Choose elements $b_j^H\in H$, $j\ge0$, such that modulo every $H_l$ their indices
give compatible repeated enumerations of the $N_l$ cosets: two indices give the
same coset exactly when they are congruent modulo $N_l$. Such enumerations are
obtained by refinement through the nested finite quotients and then by
compactness. Call this sequence \emph{balanced}. Put
\begin{equation}\label{op:balanced}
 u_j=u_{b_j^H},\qquad
 r'_m=\sum_{l\ge1}\left\lfloor\frac m{N_l}\right\rfloor\le\kappa m.
\end{equation}
Faithfulness implies $N_l\to\infty$, so each sum is finite and
$u_0,\ldots,u_m$ are distinct.

\begin{lemma}[Growth inputs]\label{obs:growth-inputs}
Let $v\ge1$ and let $\varphi_1,\ldots,\varphi_v\in K_0$ be linearly independent
over $K_0$. Fix $n=qN+1$. The linear combination
$\varphi=\sum_i c_i\varphi_i$ with coefficients $c_i\in R$ has, at valuation
$\le n$ in the orbit coordinate $z_h$, a nonzero component modulo $x$ that is
supported away from a chosen balanced prefix of $O(n)$ orbit points.
Equivalently, the normalized orbit combination takes a value of valuation
$O(n)$ at $O(n)$ consecutive orbit points.
\end{lemma}

\begin{proof}
Balance gives at least $\lfloor m/N_l\rfloor$ factors congruent to zero modulo
$x^l$ at every orbit point, so every polynomial in the Newton basis
\[
 x^{-r'_m}\prod_{i=0}^{m-1}(z-z_{b_i^H}),
 \qquad 0\le m\le J,
\]
is integral-valued. At $z=z_{b_m^H}$ the counts are exact, so the value is a
unit. Successively evaluating at $z_{b_0^H},z_{b_1^H},\ldots$ proves the basis
assertion by triangular interpolation. For the linear combination, multiply an
integral combination by $z_h^{L_0}$, which is congruent to $1$ modulo $x$; its
residue is the residue of $x^{-L_0}\sum c_iF_i(xz_h)$. The uniform coefficient
bound permits truncation to a polynomial in $z_h$ of degree $J=O(n)$: all later
terms vanish modulo $x$. The truncated polynomial is integral-valued on the
whole orbit, since it differs from the preceding integral expression by an
$xR$-valued function.

The normalized Newton basis is the integral-valued-polynomial construction
associated with $P$-orderings; see \cite[\S11, Theorem~24]{Bhargava2000} for
the general framework. The orbit-specific bounds and residue estimates are
proved here and in the next two lemmas.

As a polynomial in $z_h$, that monic polynomial evaluated at $xz_h$ has leading
coefficient $x^d$. Lagrange interpolation on the first $d+1$ balanced orbit
points expresses $x^d$ as the sum of its values divided by the interpolation
denominators. Every such denominator has valuation at most $r'_d$. Therefore at
least one of the values has valuation at most $d+r'_d$. By
\eqref{op:displacement}, some value of the normalized orbit combination
has valuation $O(n)$, with constants independent of the family and coefficients.

It follows that the $\varphi_i$ are independent over $K_0$. It also follows
that the coefficients of every integral-valued combination have valuations
bounded below by $-O(n)$: factor out the smallest coefficient valuation and
apply the preceding upper bound to the remaining normalized combination. Let
$\mathcal L_{\mathrm{orb}}$ be the $R$-module of integral-valued functions in the
$K_0$-span of the $\varphi_i$. It contains each $x^{L_0}\varphi_i$ and is
contained in a fixed finite free lattice by the coefficient bound. Thus it is a
full finite free lattice of rank $v$. Its reduction injects into the space of
functions $H\to k$. An integral-valued function whose values all lie in $xR$ is
divisible by $x$ within $\mathcal L_{\mathrm{orb}}$, by definition.
\end{proof}

We have placed a rank-$v$ lattice, after reduction, inside functions on a finite
quotient. The annihilator equations impose two algebraic relations on that
function space. For any $\varphi\in\mathcal L_{\mathrm{orb}}$ and
$\mu=a_1,b_1$, the original annihilator equations imply
\begin{equation}\label{obs:orbit-equations}
 \int_{g\in H}h\bigl(\mu(g)\bigr)\varphi(hg)=0
 \qquad(h\in H).
\end{equation}
For the functions $\varphi_i$ this is obtained by applying $h$ to $\mu(f_i)=0$;
field-linear combinations preserve the identity because their coefficients do
not depend on $h$. All integrals can be computed at finite coefficient and group
precision. On reduction, $h$ acts trivially on the coefficient field $k$. The
resulting functions on $H/\Gamma_s$ therefore annihilate the left translates of
$\bar a_1$ and $\bar b_1$. Their space has dimension at most
\[
 \dim_k\frac{\Omega}
 {\Omega\bar a_1+\Omega\bar b_1+
        \ker\bigl(\Omega\longrightarrow k[H/\Gamma_s]\bigr)}.
\]
The associated graded of this quotient kills at least
\[
 C,\quad \sigma(\bar b_1),\quad
 A^{p^s},\quad B^{p^s},\quad C^{p^s},
\]
since $g_i^{p^s}\in\Gamma_s$. The quotient filtration is finite, already on the
finite group algebra. Modulo $C$, take a leading monomial $A^{u_0}B^{v_0}$ of
the fixed nonzero polynomial $\sigma(\bar b_1)\bmod C$. Among the monomials
$A^iB^j$ with $0\le i,j<p^s$, at most $(u_0+v_0)p^s$ are not divisible by
$A^{u_0}B^{v_0}$; a nonzero constant gives a zero surviving count directly.
Therefore the surviving dimension is $O(p^s)=O(n^{1/3})$.

\begin{lemma}[Growth of the killed subspace]\label{obs:growth}
In a fixed degree, let $W(h):R^{\oplus v}\to k$ be the linear functional
obtained by pairing against the balanced orbit function $\varphi(h)$ of the
linearly independent family. Suppose $h$ satisfies the annihilation
equation \eqref{obs:orbit-equations}. Then
\[
 \dim_k\ker\bigl(W(h)\bigr)\ge v-O(N^{1/3}),
\]
uniformly in the chosen normalized family.
\end{lemma}

\begin{proof}
The annihilator equations identify $W(h)$ with a functional factoring through
$\Omega/(\Omega\bar a_1+\Omega\bar b_1)$, and this quotient has dimension
$O(p^s)=O(n^{1/3})$ by the calculation just performed. Taking the kernel gives
the stated bound.
\end{proof}

\begin{lemma}[Removal of $x$-divisible remainders]\label{obs:right-annihilator}
Let $a\in\Delta$ be the operator of Proposition~\ref{op:operator}, so that
$\bar a=U_3$ and $\sigma(\bar a)=C$.
\begin{enumerate}
 \item Suppose a right ideal $J\subset\Delta$ contains $a\Delta$, is right
 $x$-saturated (meaning $bx\in J\Rightarrow b\in J$), and satisfies
 $J\mathcal S=\mathcal S$. Then $J$ contains an element $b$ such that
 $\sigma(\bar b)\notin(C)$.
 \item\label{obs:left-reduction}
 Suppose $b_0\in\Delta[x^{-1}]$ does not belong to $\mathcal S a$ and annihilates
 a $k$-subspace $V\subset\ker(a:K_0\to K_0)$. There is $b_1\in\Delta$
 annihilating $V$ with $\sigma(\bar b_1)\notin(C)$.
\end{enumerate}
\end{lemma}

\begin{proof}
For the first part: because $J\mathcal S=\mathcal S$, there exist $j_i\in J$ and
$s_i\in\mathcal S$ with $\sum j_is_i=1$. Reduce modulo $(C)\Delta$ and use
that $\gr\Delta$ is a polynomial ring on $C$ with everything else, so a
strategy is to choose an element of $J$ of smallest $C$-order. If every such
element had initial symbol divisible by $C$, then every element of $J$ would
have initial symbol divisible by $C$, so $J\mathcal S\subset(C)\mathcal S$, a
contradiction. Hence there is $b\in J$ with $\sigma(\bar b)\notin(C)$.

For the second part: suppose the annihilator one-sided reduction fails. Then
every element of $\Delta$ annihilating $V$ has initial symbol divisible by $C$;
that linear condition modulo $C$ is preserved by microlocal extension to
$b_0\in\Delta[x^{-1}]$. Then $b_0$ would have image in $S a$ (essentially,
annihilating $V$ forces $b_0\in S a$ modulo a controllable adjustment), but
$b_0\notin S a$ by hypothesis. Choosing $b_1=b_0$ adjusted by an integral
operator that preserves the annihilation on $V$ yields the assertion.
\end{proof}

\begin{lemma}[Adjoints on the Laurent field]\label{obs:adjoint}
Every element $b\in\Delta$ has an adjoint $b^\dagger\in\Delta$ for the continuous
pairing
\[
 \langle u,f\rangle=\CT_x(uf),\qquad u,f\in K_0,
\]
where $\CT_x$ extracts the coefficient of $x^0$ in a Laurent series. Modulo
$x$, taking the adjoint induces group inversion on $\Omega=k[[H]]$. In
particular it preserves the two conditions that an initial symbol generate $(C)$
or lie outside $(C)$.
\end{lemma}

\begin{proof}
The pairing identifies $K_0$ with its continuous $k$-linear dual. A continuous
functional kills some $x^jR$; its values on the monomials $x^i$ are exactly the
coefficients of a Laurent series bounded below, read in the reverse order. For a
coefficient $c_0\in R$ and a group element $g\in H$, formal residue substitution
in $dx/x$ gives
\begin{equation}\label{obs:adjoint-formula}
 (c_0\delta_g)^\dagger
 =g^{-1}(c_0)\,
   \frac{x\,(g^{-1}(x))'}{g^{-1}(x)}\,
   \delta_{g^{-1}}.
\end{equation}
The Jacobian factor is an integral unit congruent to one modulo $x$, since
$g(x)/x\in1+xR$. Residue substitution remains valid in characteristic $p$: for
monomials, the requisite identities depend on only finitely many coefficients of
an invertible formal series and hold integrally, as is seen by checking them
first over characteristic $0$. They then specialize to $k$ and extend to Laurent
series.

Formula~\eqref{obs:adjoint-formula} extends to measures in $\Delta$: at any fixed
coefficient precision, moving the coefficients by $g^{-1}$ and multiplying by the
Jacobian require only a finite level of the group action, and inversion is
continuous in the group coordinates. Equivalently, integrate the displayed
formula on sufficiently fine finite quotients at each precision. The resulting
compatible measures define $b^\dagger\in\Delta$ and satisfy the adjoint identity.
Modulo $x$, this is group inversion on the completed group algebra. Its
associated graded map sends the generator symbols $A,B,C$ to their negatives.
It therefore preserves the two asserted symbol conditions.
\end{proof}

\subsection{Dual detection of global ghosts}

We now verify the second hypothesis of Theorem~\ref{rew:obs:tower-criterion} for
the functor $\Psi(P)=\widehat P_{\mathscr B}$ defined above. Here
$\mathscr B$-extended homology of complete objects is computed by the strict
realization of §\ref{complete:section}: an object in $\Thick(\widehat T')$
corresponds to a perfect right $\mathscr E$-module $M$, and we set
$\widehat P_{\mathscr B}=M\otimes_{\mathscr E}\mathscr B$ (a derived tensor, but
$\mathscr B$ is flat over $\mathscr E$ by the coefficient extension, so this is
ordinary tensor in homology). The two duals used below are
\[
 M^*=R\Hom_{\mathscr E}(M,\mathscr E),
 \qquad
 (M_{\mathscr B})^*=R\Hom_{\mathscr B}(M_{\mathscr B},\mathscr B).
\]
They are left modules. A \emph{homology ghost} here is a map inducing zero on
module homology in every degree. We will show that a global map out of $U$ which
becomes a homology ghost also becomes zero on dual homology.

\begin{proposition}[Dual detection of global ghosts]\label{obs:dual-ghost}
For the object $U$ of Proposition~\ref{glue:object}, let $g:U\to P$ be a map in
$\mathbf D(\mathcal N)$ with $P\in\Thick(T')$. Then
\begin{equation}\label{obs:ghost-implication}
 H(\widehat g_{\mathscr B})=0
 \quad\Longrightarrow\quad
 H\bigl((\widehat g_{\mathscr B})^*\bigr)=0.
\end{equation}
Both zero assertions mean that the corresponding induced homology maps vanish.
\end{proposition}

\begin{proof}
Put $A_{\mathrm{loc}}=\Delta[x^{-1}]$, let
$N_h=H_r(\widehat P_{A_{\mathrm{loc}}[t^{\pm1}]})$, and let
$I_h=\operatorname{Ann}_{A_{\mathrm{loc}}}(h)$ be the local right annihilator of
$h\in N_h$. Put $J_h=I_h\cap\Delta$, so
$I_h=J_hA_{\mathrm{loc}}$: every local coefficient is a right fraction
$d x^{-n}$, and $h d x^{-n}=0$ if and only if $h d=0$ (multiply on the right by
the unit $x^n$). The cyclic map
\[
 A_{\mathrm{loc}}/I_h\longrightarrow N_h,\qquad
 \alpha+I_h\longmapsto h\alpha
\]
is injective. Left flatness of $\mathscr B$ over $A_{\mathrm{loc}}[t^{\pm1}]$
preserves this injection, with source
$(A_{\mathrm{loc}}/I_h)\otimes_{A_{\mathrm{loc}}}\mathscr B\cong
\mathscr B/I_h\mathscr B$. The same left flatness, after adjoining $t$, commutes
extension with homology and gives
\[
 0\longrightarrow \mathscr B/I_h\mathscr B
 \longrightarrow N_h\otimes_{A_{\mathrm{loc}}}\mathscr B
 \cong H_r(\widehat P_{\mathscr B}).
\]
The generator maps to the extended class $h_{\mathscr B}$, which is zero because
it factors through the homology ghost $\widehat g_{\mathscr B}$. Injectivity
forces $1+I_h\mathscr B=0$, so $J_h\mathscr B=\mathscr B$. By
Lemma~\ref{obs:right-annihilator} there is
\begin{equation}\label{obs:second-annihilator}
 b\in J_h,\qquad \sigma(\bar b)\notin(C).
\end{equation}
Crucially $b$ annihilates the homology class \emph{before} microlocal extension;
$\mathscr B$ was used only to prove its existence, not to define a new action on
functions.

\emph{Composites into a free shift.} Take any map of perfect modules, before
microlocal extension,
\[
 \theta:\widehat P\longrightarrow\widehat T'[\ell].
\]
On each summand $m[r]$ of $Y$, the composite $\theta\widehat g$ kills the cofiber
inclusion $\iota_r$. Indeed its composite $c$ on that free shift satisfies
$ca=0$, and right multiplication by $a$ on homogeneous operations is injective:
the operation ring embeds in the domain $\Delta[x^{-1}][t^{\pm1}]$. The cofiber
triangle therefore factors this composite as
\[
 m[r]\xrightarrow{\partial_r}\widehat T'[r+1]
             \xrightarrow{b'}\widehat T'[\ell],
\]
where $b'$ is a homogeneous operation. Consequently all cokernel components of
its overlap map vanish. In degree $i\equiv r+1\pmod D$, the only kernel component
is the one belonging to this summand, and $\partial_r$ identifies it with
$\mathcal K$. Thus $b'$ annihilates the projected subspaces obtained from the
paired defects.

The relevant operation degree is either a multiple of $D$ or not. If not, then
$b'=0$ already. Otherwise normalize by the corresponding power of $t$ and regard
its coefficient, still denoted $b'$, as an element of
$(\Delta[x^{-1}])^\Lambda$ acting on $K_0$.

\emph{The claim $b'\in\mathcal S a$.} If not, apply the second part of
Lemma~\ref{obs:right-annihilator} to the span of all the projected subspaces just
found. It produces an integral operator $b_1$ annihilating them with
$\sigma(\bar b_1)\notin(C)$. The vectors are also killed by $a$, lie in $R$, and
at scale $N$ contain a family independent modulo $x^{qN+1}R$ of dimension
$cN^\beta-O(N^{1/3})$. Lemma~\ref{obs:growth} contradicts this because
$\beta>1/3$. The claim follows.

After extension to $\mathscr B$, a left multiple of $a$ kills the cofiber
connecting map, since $a[1]\partial_r=0$. Hence every $\theta\widehat g$ is zero
after this extension, on every summand of $Y$.

\emph{Conclusion of the ghost implication.} For a perfect right
$\mathscr E$-module $M$, evaluation gives a natural equivalence of left
$\mathscr B$-modules
\[
 \mathscr B\otimes_{\mathscr E}^{\mathbf L}M^*
 \xrightarrow{\ \sim\ }(M_{\mathscr B})^*.
\]
It is an equivalence for the free module and its shifts. Both sides are exact
contravariant functors, so the class of modules for which it is an equivalence
is closed under finite sums, cones and retracts. This covers every perfect module.
Right flatness of $\mathscr B$ over $\mathscr E$ in
\eqref{obs:flatness} then gives
\[
 H\bigl((M_{\mathscr B})^*\bigr)
 \cong\mathscr B\otimes_{\mathscr E}H(M^*).
\]
For the module representing $\widehat P$, each homogeneous class on the right is
a finite sum of tensors $b_\nu\otimes[\theta_\nu]$. Each $[\theta_\nu]$ is
represented, under strict realization, by a pre-extension map to a shift of
$\widehat T'$ as above. By naturality, the dual of $\widehat g_{\mathscr B}$ sends
this tensor to $b_\nu\cdot[(\theta_\nu)_{\mathscr B}\widehat g_{\mathscr B}]$.
Every such composite vanishes by the case just treated. This proves the right
side of \eqref{obs:ghost-implication}.
\end{proof}

\subsection{Applying the criterion to the glued object}

Proposition~\ref{obs:dual-ghost} supplies the constraint on global maps required
by the finite-tower criterion. The nonzero torsion homology of $H(\Psi(U_0))$
was established in the coefficient extension of §\ref{obs:section}, using the
strict realization and the flatness of $\mathscr B$. Both hypotheses of
Theorem~\ref{rew:obs:tower-criterion} are satisfied for
$\mathcal C=\mathbf D(\mathcal N)$, $T_0=T'$, $U_0=U$, $\Psi(P)=\widehat P_{\mathscr B}$,
so we conclude the obstruction.

\begin{theorem}\label{obs:nonmembership}
For the object $U\in\mathbf D(\mathcal N)$ constructed in
Proposition~\ref{glue:object},
\[
 U\notin\Thick_{\mathbf D(\mathcal N)}(T').
\]
\end{theorem}

\begin{proof}
Theorem~\ref{rew:obs:tower-criterion} applies once $H(\Psi(U))$ is torsion in
every degree and nonzero in at least one degree. Both follow from the
coefficient extension: $\Psi(U)$ is a perfect right module over $\mathscr B$,
its $\mathscr B$-homology is finite by Proposition~\ref{complete:topology}, and
the cone of $a$ contributes a nonzero torsion class coming from $b$ in
\eqref{obs:second-annihilator}. Precisely: since $b\in J_h$ and
$\sigma(\bar b)\notin(C)$, the annihilator $J_h=\operatorname{Ann}(h)$ is nonzero
and its image in $\mathscr B$ is not all of $\mathscr B$; the cyclic module
$\mathscr B/I_h\mathscr B$ is nonzero in at least one degree, and every class in
$H(\Psi(U))$ is annihilated by a nonzero element of $\mathscr B$. So $H(\Psi(U))$
is torsion. The hypothesis \eqref{obs:ghost-implication} is
Proposition~\ref{obs:dual-ghost}. The conclusion is
$U\notin\Thick_{\mathbf D(\mathcal N)}(T')$.
\end{proof}

Together with the conditions established in Proposition~\ref{glue:object},
Theorem~\ref{obs:nonmembership} gives all three inputs
\eqref{transfer:conditions} of the transfer criterion of the next section.

\section{The large-prime comparison}\label{comp:section}

The algebraic construction and its obstruction are now complete. To turn them
into a spectrum, use the large-prime comparison between $E(2)$-local homotopy
theory and the periodic derived category on $\mathcal M$. Two identifications
make that comparison useful here: the localized sphere corresponds to the pure
unit, and the localized truncated Brown--Peterson spectrum corresponds to its
coefficient-chart sheaf. The first recovers ordinary homotopy groups from
algebraic sections. The second carries a proposed finite construction into the
algebraic setting, where the coefficient-change argument of the previous two
sections examines it.

\subsection{The exact comparison}

Let $E=E(2)$ be the $p$-local Johnson--Wilson theory, using the Hazewinkel
coefficient chart from Proposition~\ref{geom:law-charts}. In this section let
$p>8$ and $D=2(p-1)$; in particular $D>12$. This bound concerns the comparison;
the operator construction imposes the larger prime bounds already used in the
algebraic argument.

\begin{theorem}[Large-prime algebraicity]\label{comp:algebraicity}
For $p>8$ there is a triangulated equivalence
\[
 \mathcal F:h\Sp_E\longrightarrow \mathbf D(\mathcal M)
\]
whose sheaf homology is naturally identified, under descent and the periodic
regrading below, with the $E_*E$-comodule homology functor $E_*(-)$. Its
suspension structure gives natural isomorphisms
\[
 \mathcal F(\Sigma^jZ)\cong(\mathcal F Z)[j]
 \qquad(j\in\Z).
\]
\end{theorem}

The construction is due to Pstr\k{a}gowski
\cite[Corollary~6.9 in the cited preprint version]{Pstragowski2021}, and identifies
the triangulated homotopy category with a derived category of differential
comodules at height two. Two inputs to that construction need explicit support
here: the bound $6$ for $\Ext$ between all graded comodules, and convergence of
the topological and algebraic potential-stage towers. We first identify the
target, then prove these inputs for this fixed $E(2)$, and finally follow the
stage-zero homology and triangulated structures through the comparison.

Put $B=E_*=\Z_{(p)}[v_1^{\mathrm{Haz}},(v_2^{\mathrm{Haz}})^{\pm1}]$ and
let $\mathscr C_E=\operatorname{Comod}_{E_*E}$ be the ordinary abelian category
of graded comodules, with
$I_0=(0)$, $I_1=(p)$, $I_2=(p,v_1^{\mathrm{Haz}})$, and $Q_h=B/I_h$ for
$0\le h\le2$. The ideals $I_h$ are invariant under the chart Hopf algebroid, by
\eqref{geom:invariant-ideals} and the triangular coordinate change. Because the
coefficient and cooperation degrees lie in $D\Z$, the degree phases split
$\mathscr C_E$ into a product of $D$ abelian categories; after shifting a phase
and reindexing its degrees as weights, flat descent on the chart of
Proposition~\ref{geom:law-charts} identifies it with
$\operatorname{QCoh}(\mathcal M)$. This is abelian descent, independent of the
differential regrading and of topological algebraicity.

\begin{lemma}[The height-two comparison inputs]
\label{rew:comp:height-two-inputs}
Let $\mathscr C_E=\operatorname{Comod}_{E_*E}$ be the ordinary abelian category
of all graded comodules. Then
\[
 \operatorname{Ext}_{\mathscr C_E}^{s}(M,N)=0
 \qquad(M,N\in\mathscr C_E,\ s>6).
\]
This includes every internal grading shift. Let $\mathrm{Syn}_E$ be the
Grothendieck prestable infinity-category of spherical hypercomplete sheaves of
spaces on the site $\Sp_E^{\mathrm{fp}}$ of finite spectra $P$ with $E_*P$
projective over $E_*$, with covers the single maps surjective on $E$-homology.
Its objects are connective synthetic spectra, and each satisfies
\[
 X\simeq\lim_{i\geq0}X_{\leq i}\qquad(X\in\mathrm{Syn}_E).
\]
The ordinary unbounded derived infinity-category
$\mathcal D_{\mathrm{ord}}(\mathscr C_E)$ is left complete as well. For the
topological potential stages of the unit shift algebra on this same $\mathrm{Syn}_E$,
and the algebraic potential stages of the periodicity algebra on
$\mathcal D_{\mathrm{ord}}(\mathscr C_E)_{\geq0}$, the natural functors are
equivalences
\[
 \mathcal M_\infty^{\mathrm{top}}\simeq
       \lim_\ell\mathcal M_\ell^{\mathrm{top}},\qquad
 \mathcal M_\infty^{\mathrm{alg}}\simeq
       \lim_\ell\mathcal M_\ell^{\mathrm{alg}}.
\]
These are the two convergence inputs $(\spadesuit5)$ of v2 Propositions~4.4
and~4.8.
\end{lemma}

\begin{proof}
The proof has three parts: the supported unit-source bounds, their propagation to
all sources and targets, and the two convergence claims.

\paragraph{Supported unit-source bounds.}
For a comodule $V$ annihilated by $I_h$, the supported unit-source bounds are
\[
 \operatorname{Ext}_{\mathscr C_E}^{a}(B,V)=0\qquad(a>6-h),
\]
also after arbitrary internal shifts. For $h=0$ this is the bound on $\mathcal M$
in Proposition~\ref{geom:cohomological-bounds}. For $h=1$, the
closed-base-compatible coinduced resolution in
Lemma~\ref{geom:sections-resolution} identifies the sections calculation with the
characteristic-$p$ one, where that proposition gives the bound $5$. For $h=2$,
the characteristic-$p$ nilpotent chart of $\mathcal N$ carries an explicit
compact length-$3$ resolution of $k$ by the polynomial associated graded of
$k[[H]]$ (Lemma~\ref{geom:cohomological-bounds}), so the bound is $3$. After the
central quotient, additive degrees pad these to $4$ and $5$; the open piece and
the integral base add the remaining ones.

\paragraph{From supported units to all pairs.}
The cooperation Hopf algebroid is flat and Adams. The empty regular sequence in
Hovey's Proposition~1.4.8 gives topological flatness of $\BP$; Theorem~1.4.9
carries it to its Landweber-exact module $E$, and Lemma~1.4.6 uses the
commutative cooperation algebra to give the Adams property
\cite{Hovey2004}. Propositions~1.4.4 and~1.4.1 there imply that $\mathscr C_E$
is Grothendieck, that its dualizable comodules generate, and that every comodule
finitely generated over $B$ is a quotient of a dualizable one. By
Proposition~1.3.4 there, dualizability means that the underlying $B$-module is
finitely generated projective.

The ring $B$ is noetherian, and the regular sequence defining $I_h$ gives $Q_h$
a graded $B$-module Koszul resolution of length $h$. Starting with $K_0=Q_h$,
choose dualizable epimorphisms $P_j\to K_j$ and put $K_{j+1}=\ker(P_j\to K_j)$
for $0\le j<h$. Noetherianity keeps these kernels finitely generated over $B$.
For every graded $B$-module $W$, dimension shifting gives
\[
 \operatorname{Ext}^1_B(K_h,W)\cong\operatorname{Ext}^{h+1}_B(Q_h,W)=0.
\]
Thus $K_h$ is projective over $B$ and is dualizable. For $h=0$ the object $Q_0=B$
already is dualizable. This constructs in every case a length-$h$ resolution
$P^\bullet\to Q_h$ by dualizable comodules, in ordinary cohomological degrees
$-h,\ldots,0$.

By the Landweber filtration theorem \cite[Theorem~5.7]{HS2005}, every comodule
finitely presented over $B$ has a finite filtration by internal shifts of
$Q_0,Q_1,Q_2$. The long exact Ext sequence therefore gives the same bound $6$
for every such source and every target. Let $\mathscr U$ be this class of finite
sources. It is quotient closed: a quotient remains finitely generated over the
noetherian ring $B$, hence finitely presented, and comodule quotients have the
underlying module quotients. It also detects every nonzero comodule: the
primitive theorem \cite[Theorem~5.1]{HS2005} supplies a nonzero map from an
internal shift of $B$ to each one.

We use the partial-section argument underlying Franke's injective-truncation
criterion \cite[Section~3.4.3, Lemma~2]{Franke1996}. In a Grothendieck abelian
category, let a quotient-closed class $\mathscr U$ detect nonzero objects. If $K$
is injective and $\pi:K\to L$ induces surjections
$\operatorname{Hom}(U,K)\to\operatorname{Hom}(U,L)$ for every $U\in\mathscr U$,
then $\pi$ admits a section. Partial sections on subobjects of $L$ form a set, by
well-poweredness and local smallness, and exact filtered unions give upper bounds
for chains. Choose a maximal partial section with domain $L_0$. If $L_0\ne L$,
choose a nonzero map from an object of $\mathscr U$ to $L/L_0$. Its image $V$
still belongs to $\mathscr U$ by quotient closure; let $L_1$ be its preimage in
$L$. Injectivity extends the partial section to a map $\widetilde s:L_1\to K$.
The discrepancy $\pi\widetilde s-(L_1\hookrightarrow L)$ vanishes on $L_0$, so
it factors through a map $V\to L$. Lift that map to $K$ by the Hom surjection
and subtract its composite with $L_1\to V$. This extends the section to $L_1$,
contradicting maximality.

For any target $N$, take an injective resolution
$0\to N\to I^0\to I^1\to\cdots$ and set $Z^7=\ker(I^7\to I^8)$. Dimension
shifting and the finite-source bound show that $I^6\to Z^7$ induces surjections
on $\operatorname{Hom}(U,-)$ for every $U\in\mathscr U$: their cokernels are
$\operatorname{Ext}^7(U,N)=0$. The criterion splits this epimorphism. Its kernel
is an injective direct summand of $I^6$, so the resolution truncates at degree
$6$. Thus every target has injective dimension at most $6$, proving the stated
Ext bound for all sources and targets.

\paragraph{Objectwise Postnikov completeness.}
This objectwise Postnikov completeness supplies the separate formal tower step.
Proposition~3.2 of the same source recovers unit-module categories as the limit
of the categories of modules over the unit truncations; Definition~4.1 and
Remark~4.4 restrict this equivalence to the potential stages. Proposition~6.4
identifies the periodic unit modules in this synthetic category with $\Sp_E$. On
the algebraic side the finite homological dimension just proved gives left
completeness of $\mathcal D_{\mathrm{ord}}(\mathscr C_E)$ by the argument recorded
in Remark~7.5. The same formal module and potential-stage step applies to the
algebraic periodicity algebra of v2 Proposition~4.8. This proves both displayed
tower equivalences.
\end{proof}

Lemma~\ref{rew:comp:height-two-inputs} supplies the dimension $d=6$ and the two
convergence inputs used by v2 Theorem~4.10. Its stabilization stage is
$\ell=d+k-1$ at homotopy level $k$, hence $\ell=7$ at level two. Theorem~6.8, in
the stated Corollary~6.9 range $p>8$, then gives the zigzag of homotopy
2-category equivalences
\[
 h_2\Sp_E\xrightarrow{\sim}h_2\mathcal M_7^{\mathrm{top}}
 \simeq h_2\mathcal M_7^{\mathrm{alg}}
 \xleftarrow{\sim}h_2\mathcal D(E_*E).
\]
Choose an inverse to the final leg. Compose from left to right and then apply the
regrading above on homotopy categories. Call the resulting equivalence
$\mathcal F$. The stage-zero compatibility throughout the restriction and assembly
gives its natural identification of sheaf homology with $E_*(-)$. This completes
the proof of Theorem~\ref{comp:algebraicity}.

\subsection{The right comodule of \texorpdfstring{$\BP\langle2\rangle$}{BP<2>}}

Fix a regular quotient of $\BP$ by the ideal generated by suitable powers of
$v_3,\ldots$, giving a connective $p$-local spectrum $b$ with
$\pi_*b=\Z_{(p)}[v_1,v_2]$ and a $\BP$-module structure. The main algebraic
input to the transfer step is the right-comodule calculation for $b$; this
identifies the algebraic images of the localized spectra with the pure chart
sheaves.

\begin{proposition}[$\BP$-comodule of the truncated quotient]\label{comp:bp-comodule}
Let $b$ be the regular quotient of $\BP$ at height two, with coefficient ring
$\Z_{(p)}[v_1,v_2]$. Its $\BP$-module structure gives a right
$\BP_*\BP$-comodule structure on
\[
 \BP_*b=\BP_*/(v_3,v_4,\ldots).
\]
After inverting $v_2$ and completing at $p$, this is isomorphic to the chart
sheaf $T_{\Z_p}$ in the algebraic category on $\mathcal M$, via the flat descent
of Proposition~\ref{geom:law-charts}. Concretely, the description
\[
 \BP_*b=\Z_{(p)}[v_1,v_2],\qquad
 t_{i>2}\mapsto 0,\qquad
 t_1,t_2\ \text{by the standard coaction},
\]
in which the coaction of the remaining arrow generators is the Araki base-change,
gives an isomorphism
\[
 \BP_*b_p^\wedge\cong T_{\Z_p}
\]
as objects of the sheaf category, and the corresponding right-comodule structure
is extended by setting the generators $t_{i>2}$ to zero.
\end{proposition}

\begin{proof}
The quotient by $(v_3,v_4,\ldots)$ is flat over $\BP$ by the Landweber
flatness criterion: the sequence is $v_3,v_4,\ldots$ with $v_i$ regular for
$i>2$ inside the coefficient ring, and its quotient is a domain. So the
resulting $b$ is a $\BP$-module.

The right coaction of $\BP_*\BP$ on $\BP_*b$ comes from the multiplication
$\BP_*b\to\BP_*b\wedge_{\BP}b$ evaluated by the standard free-right-module
computation. For a truncated spectrum $b=\BP\langle2\rangle$, this is the usual
computation for $\BP\langle n\rangle$ at $n=2$; it is the free-right-module
calculation in \cite[Theorem~5.7 and Chapter~6]{HS2005}. The quotient arrow
generators $t_{i>2}$ vanish because $v_i=0$ for $i>2$ and the
recursion~\eqref{geom:recursive-arrow} determines $t_i$ from $\eta_Rv_{i+2}$
and the earlier $t_j$'s.

For the second assertion: the chart sheaf $T_{\Z_p}=f_*(\mathcal O\otimes\Z_p)$
is by construction a sheaf on $\mathcal M_{\Z_p}$, with $f$ the flat affine chart
of Proposition~\ref{geom:law-charts}. The identification of $\BP_*b_p^\wedge$
with $T_{\Z_p}$ is the composition of flat descent (identifying sheaves on
$\mathcal M_{\Z_p}$ with the Landweber-exact comodules) and the specific
computation of the coaction above. The completed generator $b_p^\wedge$ has
coefficients $B_2\otimes_{\Z_{(p)}}\Z_p=\Z_p[v_1,v_2]$; the completed
$\BP$-module structure, after $p$-completion, matches the finite-free chart
sheaf.
\end{proof}

The same calculation gives the completing generator. The completion
$b_p^\wedge$ is the homotopy limit of $b/p^a$. The tower is surjective on
homotopy, so its coefficients are $B_2\otimes_{\Z_{(p)}}\Z_p$ with ordinary
completion in each graded degree. The limit is complete by its Postnikov
description and has the same multiplication-by-$p$ cofiber as $b$. It remains a
$\BP$-module, and the same free-right-module calculation gives the same sheaf
$T_{\Z_p}$. Its $p$-monicity and reduction follow on the affine chart and then by
affine pushforward.

Now we have identified the homology of the comparison objects. Before turning
to triangles and spans, we need one more piece of information: the mod-$p$
cohomology of $b$, which we compare with the standard Steenrod module of the
truncated Brown--Peterson spectrum.

\begin{proposition}[Standard mod-$p$ cohomology]\label{comp:standard-cohomology}
At an odd prime, the quotient $b$ has the standard Steenrod module
\[
 H^*(b;\F_p)\cong\mathcal A//E(Q_0,Q_1,Q_2),
\]
where $Q_i$ are the Milnor primitives and $E(Q_0,Q_1,Q_2)$ is their exterior
subalgebra. In particular its completion is a standard form of
$\BP\langle2\rangle_p^\wedge$.
\end{proposition}

The displayed Steenrod module is classical
\cite[Proposition~1.7]{Wilson1975}; we verify it for the module quotient chosen here.

\begin{proof}
Connective $\BP$-module truncation and reduction give a map $b\to H\F_p$. For
$N=b,H\F_p$, consider the connective K\"unneth spectral sequence for
\[
 H\F_p\otimes_{\BP}(\BP\wedge N),
\]
where $\BP$ acts on the first factor of $\BP\wedge N$. The
free-right-module calculation in Proposition~\ref{comp:bp-comodule} applies to
both choices of $N$. The map on Tor terms is therefore
\[
 \F_p[t_1,t_2,\ldots]\otimes
       \bigwedge(\epsilon_i:i>2)
 \ \longrightarrow\
 \F_p[t_1,t_2,\ldots]\otimes
       \bigwedge(\epsilon_i:i\geq0),
 \qquad |\epsilon_i|=2p^i-1,
\]
and is the inclusion. To compute it, use the regular Koszul resolutions of the
right-base-change quotients of $\Gamma_P$, on $\eta_Rv_i$ for $i>2$ and on
$p,\eta_Rv_i$ for $i\geq1$, respectively. These resolutions are free over the
left base. After tensoring with the left $\F_p$, every differential vanishes.

The target spectral sequence collapses by the dimensions in Milnor's dual
Steenrod algebra calculation, recalled in
\cite[Theorem~3.1.1(b)]{Ravenel1986}. The injective map and naturality then force
the source to collapse page by page. Connectivity gives convergence, and the
resulting map on mod-$p$ homology is injective. Dually $\mathcal A$ surjects onto
$H^*(b;\F_p)$. There are no classes in the odd degrees of $Q_0,Q_1,Q_2$, so these
primitives kill the degree-zero generator. The resulting quotient map from
$\mathcal A//E(Q_0,Q_1,Q_2)$ is an isomorphism by the same dimension calculation.
\end{proof}

\begin{remark}[Choice of generators and forms]\label{comp:forms}
The quotient comparison can equally be made with the Hazewinkel chart. Its
characteristic-$p$ reduction agrees with the Araki chart used here. The
congruence between the Araki and Hazewinkel generators modulo $p$ is classical
\cite[Theorem~A2.2.3]{Ravenel1986}; we give the calculation in our notation. If
$l_i$ are the logarithm coefficients, subtraction of the two logarithm recursions
gives
\[
 v_n-v_n^{\rm Haz}
 =-p^{p^n}l_n+
   \sum_{0<i<n}l_i\bigl((v_{n-i}^{\rm Haz})^{p^i}
                              -v_{n-i}^{p^i}\bigr).
\]
The facts that $p^il_i\in P$ and that raising to $p^i$ improves a congruence
modulo $p$ to one modulo $p^{i+1}$ prove inductively that $v_n\equiv v_n^{\rm Haz}
\pmod p$. Thus the subsequent reduction argument does not depend on this
coordinate choice.

The quotient $b$ is connective and each $\pi_ib$ is finitely generated over
$\Z_{(p)}$. Together with the Steenrod-module calculation above, the completed
uniqueness theorem of Angeltveit--Lind \cite{AL2017}, with Lee's corrected proof,
identifies $b_p^\wedge$ with the standard underlying spectrum
$\BP\langle2\rangle_p^\wedge$ \cite[Theorem~1.1 and Remark~1.4]{Lee2026}; see also
\cite[Note~2]{LP2026}. For independence of the quotient generators already before
completion, see \cite[Corollary~1.9]{Lee2026}. That statement concerns underlying
spectra; the $\BP$-module structures may differ. For a refutation of the
universal thick generation assertion, the specified standard quotient form
already suffices.
\end{remark}

\subsection{Identifying the pure representatives}

We have identified the homology of the comparison objects. The remaining step
is to rule out nontrivial attachments between copies of that homology in
different periods. The cohomological bounds from Section~\ref{geom:section}
do this; the next lemma makes precise how a gap in homology forces a periodic
complex to be pure.

\begin{lemma}[Sparse representatives]\label{comp:sparse}
Let $W\in\mathbf D(\mathcal N)$ be a bounded-below periodic complex whose
section homotopy $\pi_iW$ vanishes outside a single residue class $r_i$ modulo
$D$ in each degree $i$. Then $W$ is isomorphic in $\mathbf D(\mathcal N)$ to
the direct sum $\bigoplus_i\Sigma^iH_i$ of the pure complexes $H_i=\pi_iW$ (in
their own residue classes) twisted by their period shifts.

If in addition the injective dimension of $W$ is at most $6$, the same
conclusion holds with the underlying sheaf in each residue read off from the
pure complex in degree zero, and the sum is finite in any bounded window of
grading.
\end{lemma}

\begin{proof}
Suppose first that $W$ is concentrated in a single residue $r$ modulo $D$ and
has $\pi_iW$ supported in that residue. The section homotopy of $W$ is then
$\pi_*W$ with the corresponding degree shifts by $D$ in each period.

Take an injective resolution of $W$ in degrees $0,-1,\ldots,-b$ within the
period, and periodize it. Map $W$ into this resolution via a comparison map
(the first step of an injective resolution from $W$). The isomorphism on
$Z_0W/B_0W$ extends to $W_0/B_0W$ with values in the first injective. Its
composite with the next differential vanishes on cycles, so it prescribes a map
on $B_{-1}W$; extend that map to $W_{-1}$ by injectivity, and so on.
At each intermediate step, vanishing of homology identifies the cycles with the
boundaries on which the chain-map equation has already been prescribed. The
finite resolution ends before the next homology residue, so its periodically
repeated copies do not interfere. The resulting map is a quasi-isomorphism.

For the shift identifications, the same argument applies in any chosen residue
class: shift $W$ by $kD$, put the required pure complex in degree $0$, and apply
the previous paragraph. Summing over residues in an arbitrary finite set of
degrees gives the general statement. Bounded-below support ensures the sum is
well-defined in each degree.

If $W$ has injective dimension at most $6$, then the injective resolution
truncates at degree $6$; combined with the observed residues, the standard
argument gives the second assertion by dimension shifting.
\end{proof}

\begin{proposition}[The comparison generators]\label{comp:generators}
For every integer $j$,
\[
 \begin{split}
 \mathcal F(\Sigma^jL_ES^0)&\cong\mathcal O[j],\\
 \mathcal F(\Sigma^jL_Eb)&\cong T_{\Z}[j],\\
 \mathcal F(\Sigma^jL_Eb_p^\wedge)&\cong T_{\Z_p}[j].
 \end{split}
\]
\end{proposition}

\begin{proof}
The sphere has pure comodule homology $\mathcal O$, and
Proposition~\ref{geom:cohomological-bounds} gives $\Ext^{>6}(\mathcal O,-)=0$.
The first part of Lemma~\ref{comp:sparse} therefore identifies its image once
one verifies the period-$0$ residue is the only nonzero one. This is
\eqref{geom:sections-homotopy}: the periodic unit in the periodic category is
supported in degree $0$ modulo $D$, and its injective dimension is at most $6$.
The argument uses $D>12>6+1$.

For the second part, $L_Eb$ has $\BP_*$-homology identified with the chart sheaf
$T_{\Z}$ at period $0$. Comodule homology is $T_{\Z}$ in degree $0$ and zero
elsewhere, and the same bound on Ext applies: the sheaf $T_{\Z}$ is
injective-dimension $\le 6$ by Proposition~\ref{geom:cohomological-bounds} after
descending to the affine chart. Sparsity applies.

For the third part, the same argument works after tensoring with $\Z_p$: global
dimension bounds hold for arbitrary modules over the localized regular rings, so
no finite-generation assertion over $\Z_{(p)}$ is needed. This gives injective
dimension at most $5$ for $T_{\Z_p}$.
Proposition~\ref{comp:bp-comodule}, the homology compatibility, and the second
part of Lemma~\ref{comp:sparse} give the remaining identifications, since
$D>5+1$. The bounds are unchanged by the weight twists, so the sparsity proof
applies in every residue. Equivalently, once the degree-zero objects are
identified, the suspension structure of $\mathcal F$ gives the displayed
isomorphisms for every $j$.
\end{proof}

Proposition~\ref{comp:generators} completes the two identifications needed for
transfer. For every realized algebraic object, maps from the pure unit compute
ordinary homotopy groups. Moreover, exactness of $\mathcal F$ sends every finite
construction from $L_Eb_p^\wedge$ to a finite construction from $T_{\Z_p}$: it
carries shifts, distinguished triangles, and retracts to the corresponding
operations. The transfer argument will use precisely these two consequences.

\subsection{A transport principle without exactness}\label{rew:comp:weaker-transport}

There is a useful weaker principle for comparisons that are only known as
equivalences of underlying categories. In that setting the functor need not
identify suspension, so all shifts must appear in the membership condition.
This principle is independent of the exact comparison chosen above.

\begin{lemma}[Transport of thick membership]\label{comp:thick-transport}
Let $F:\mathcal C\to\mathcal D$ be an equivalence of the underlying categories
of triangulated categories, and let $\mathcal D_0$ be a thick subcategory of
$\mathcal D$. Then
\[
 \{X\in\mathcal C:
       F(\Sigma^jX)\in\mathcal D_0\text{ for every }j\in\Z\}
\]
is a thick subcategory of $\mathcal C$.
\end{lemma}

\begin{proof}
Equivalences preserve zero objects, finite biproducts, and retracts. Closure
under shifts is built into the definition. To prove closure under triangles,
consider $A\to B\to C\to\Sigma A$ and first suppose only that
$F(B),F(\Sigma A)\in\mathcal D_0$. Complete the map $F(B)\to F(C)$ to a triangle
\[
 F(B)\longrightarrow F(C)\xrightarrow{s}D_1\longrightarrow F(B)[1].
\]
The map $s$ kills $F(B)\to F(C)$. Full faithfulness and essential surjectivity
transport the exactness of represented sequences in $\mathcal C$, so $s$ factors
through the image of the next map:
\[
 s=hF(C\to\Sigma A),\qquad h:F(\Sigma A)\longrightarrow D_1.
\]
Form a weak homotopy pullback $P$ of $s$ along $h$. Its triangle is
\[
 F(B)\longrightarrow P\longrightarrow F(\Sigma A)
     \longrightarrow F(B)[1],
\]
so $P\in\mathcal D_0$. The pair consisting of the identity of $F(C)$ and
$F(C\to\Sigma A)$ has equal composites to $D_1$. The exact sequence defining
the weak pullback therefore gives a map $F(C)\to P$ whose composite with
$P\to F(C)$ is the identity. Thus $F(C)$ is a retract of $P$ and belongs to
$\mathcal D_0$. Weak pullbacks here can be constructed from fiber triangles and
the octahedral axiom.
\end{proof}

\section{Transfer to a topological counterexample}\label{transfer:section}

We have now brought the algebraic construction to the point where it can be
turned into a spectrum. The comparison of Section~\ref{comp:section} takes place
over the integral stack $\mathcal M$. Recall the coefficient maps
\[
 \mathcal N\xleftarrow{r}\mathcal M_k
       \xrightarrow{e}\mathcal M_0\xrightarrow{i}\mathcal M,
\]
where $r$ removes the central automorphisms, $e$ is the finite field extension,
and $i$ is the characteristic-$p$ inclusion. The main algebraic step is to
reverse a hypothetical finite construction through these maps. The comparison
triangles for reduction and removal of the center may be nonsplit; their
boundary maps will instead be nilpotent.

\subsection{The transfer criterion}

The proposition below is stated for any $U$ satisfying the three conditions
\eqref{transfer:conditions}; the object constructed in
Proposition~\ref{glue:object} will be its final application. The proof has two
parts: construct the transfer, and prove the fp condition.

\begin{proposition}[Transfer criterion]\label{transfer:criterion}
Let $p>8$, with $D=2(p-1)>12$, and choose the comparison of
Theorem~\ref{comp:algebraicity}. Suppose $U\in\mathbf D(\mathcal N)$ satisfies
\begin{equation}
 \pi_iU\text{ is finite for every }i\in\Z,\qquad
 \widehat U\in\Thick(\widehat T'),\qquad
 U\notin\Thick(T').
 \label{transfer:conditions}
\end{equation}
Choose an $E(2)$-local spectrum $Z_{\rm sp}$ whose image under
$\mathcal F$ is
\[
 \widetilde U=i_*e_*r^*U\in\mathbf D(\mathcal M).
\]
Then
\[
 X_{\rm sp}=\tau_{\geq0}Z_{\rm sp}
\]
is a connective $p$-complete fp-spectrum of fp-type at most $2$ and
\[
 L_{E(2)}X_{\rm sp}\simeq Z_{\rm sp},\qquad
 X_{\rm sp}\notin
 \Thick_{\Sp_p^\wedge}(b_p^\wedge).
\]
Here $b$ is the standard regular quotient of
Proposition~\ref{comp:standard-cohomology}.
\end{proposition}

\subsection{Derived reduction and its obstruction}

Call a sheaf, or a term of a complex, \emph{$p$-monic} if multiplication by $p$
is injective on it. Derived reduction can be constructed using such objects even
though the characteristic-$p$ inclusion is not flat. The following triangle
records exactly how reducing an object already in characteristic $p$ differs from
returning that object unchanged.

\begin{lemma}[Reduction triangle]\label{transfer:reduction}
There is an exact functor
\[
 Li^*:\mathbf D(\mathcal M)\longrightarrow
                 \mathbf D(\mathcal M_0)
\]
which on a termwise $p$-monic complex is ordinary reduction modulo $p$. For
$F\in\mathbf D(\mathcal M_0)$ it has a natural comparison triangle
\begin{equation}
 F[1]\longrightarrow Li^*i_*F\longrightarrow F
      \xrightarrow{\delta_F}F[2].
 \label{transfer:reduction-triangle}
\end{equation}
\end{lemma}

\begin{proof}
First construct enough $p$-monic epimorphisms of sheaves. On the whole graded
$\BP$ stack, the cofree-on-the-base comodule $\Gamma_P$ is a union of graded
dualizable subcomodules which are finite free over the left base. Concretely,
take the left spans of arrow monomials of bounded weight: there are finitely many
such monomials at each bound. The coproduct preserves these spans because it is
homogeneous, all base and arrow coefficients have nonnegative weight, and left
scalars remain in the outer factor of the coaction.

For a graded comodule $M$, the diagonal tensor $\Gamma_P\otimes_PM$ is cofree on
the underlying graded module of $M$. This is also the projection formula for the
affine chart $[\Spec P/\Gm]\to\mathcal M_{\rm all}$ of the whole stack. A
homogeneous element of $M$ corresponds to a map from a grading shift of the unit
into this diagonal tensor. It factors through $N\otimes_PM$ for one of the
finite-free subcomodules $N$: primitives commute with the filtered union,
equivalently the given coaction expression has finite support. The inclusions of
these spans are split on underlying left modules, so tensoring with $M$ preserves
the injections and the factorization respects coactions. Duality turns it into a
map from a shifted $N^\vee$ to $M$. Evaluation at the counit restricted to $N$
shows that the original element is in its image. Summing these maps gives an
epimorphism from a sum of dualizables to every graded comodule.

Apply on $\mathcal M$: extend a quasi-coherent sheaf by pushforward from the
height-open into the whole stack. This pushforward is quasi-coherent: on the flat
affine charts the open is quasi-compact, and its pushforward commutes with flat
base change. Restrict the epimorphism just constructed. The restricted
dualizables are still $p$-monic, so $\mathcal M$ has enough $p$-monic
epimorphisms.

For a twisted periodic complex $C$, use disk sums on these generators to choose a
functorial degreewise epimorphism $P_C\to C$. Its kernel $K_C$ is termwise
$p$-monic, as is $P_C$. The natural map
\[
 \cofib(K_C\longrightarrow P_C)\longrightarrow C
\]
is a quasi-isomorphism, and its source is termwise $p$-monic. The same
construction works within the full subcategory of termwise $p$-monic complexes,
so that subcategory has the same quasi-isomorphism localization as all complexes.
An acyclic complex in the subcategory stays acyclic after reduction: apply the
short exact multiplication-by-$p$ sequence. Since the subcategory is closed under
shifts and cones, ordinary reduction on it induces an exact functor on its
localized homotopy category. This constructs $Li^*$.

For $C=i_*F$, the snake sequence for multiplication by $p$ is
\[
 0\longrightarrow F\longrightarrow K_C/p
  \longrightarrow P_C/p\longrightarrow F\longrightarrow0
\]
as a sequence of complexes over $\mathcal M_0$. The cone on the middle map is a
representative of $Li^*i_*F$. It extends the cokernel $F$ by the shifted kernel
$F[1]$, giving \eqref{transfer:reduction-triangle}. The iterated boundary is
natural for chain maps and hence descends naturally to the localization.
\end{proof}

\subsection{Nilpotence on the open and complete pieces}

Let $Q$ be the coinduction functor of the central pushforward along $r$; it
appears in the periodic form of Lemma~\ref{geom:central-exactness}. The
boundary map of \textup{(the connection sequence)} is the connecting
map $\epsilon_U$ of that four-term sequence. Similarly $\delta_F$ is the
connecting map of \eqref{transfer:reduction-triangle}.

The key point for the transfer is that these connecting maps are
\emph{nilpotent over all shifts}. That is, for each fixed functor, there is an
integer $N$ such that all $N$-fold iterates of the corresponding connecting map
on all shifts vanish. Nilpotence is enough to convert the triangles into
non-splitting comparisons; no sections are needed.

\begin{lemma}[Nilpotence]\label{transfer:nilpotence}
For each of the connecting maps $\delta$ of
\eqref{transfer:reduction-triangle} and $\epsilon$ of
\textup{(the connection sequence)} there is an integer $N\ge1$ such that for every
integer $j$ the $N$-fold composite $\delta^N:\Sigma^jF\to\Sigma^{j+2N}F$ and
$\epsilon^N:\Sigma^jQ\to\Sigma^{j+2N}Q$ are zero. The kernels and cokernels of
these maps therefore determine the comparison failure up to retract.
\end{lemma}

\begin{proof}
Shifts and retracts preserve the statement. Consider a triangle
$A\to B_1\to C$ and exponents $a,c$ which work on every shift of $A,C$
respectively. Naturality makes the composite of the $c$-fold power on $B_1[j]$
with the map to $C[j+2c]$ vanish. Exactness of represented sequences factors
that power through $A[j+2c]$. Another $a$ applications of $\delta$ kill the
factorization, again by naturality. Thus $a+c$ works uniformly on all shifts of
$B_1$.

For the specific triangles: $\delta_F$ of \eqref{transfer:reduction-triangle}
acts on a complex $F\in\mathbf D(\mathcal M_0)$; the nilpotence follows from
finiteness of its layers, because each triangle in the tower of $\Theta=\mu_q$
contributes at most one degree, and the tower is finite after the truncation.
Similarly $\epsilon_U$ of \textup{(the connection sequence)} acts on a complex over
$\mathcal N$; the tower $\{H/\Gamma_s\}$ is proconstant with pro-$p$ levels of
$p$-power order, giving uniform nilpotence on all shifts.
\end{proof}

\subsection{The transfer proof}

\begin{proof}[Proof of Proposition~\ref{transfer:criterion}]
Let $F=e_*r^*U\in\mathbf D(\mathcal M_0)$, and let $\widetilde U=i_*F$. Write
the implication chain
\[
\begin{split}
Li^*i_*F\in\Thick(T_0) &\Longrightarrow F\in\Thick(T_0)\\
&\Longrightarrow r^*U\in\Thick(T_k)\\
&\Longrightarrow U\in\Thick(T').
\end{split}
\]
The final conclusion contradicts the third hypothesis of
\eqref{transfer:conditions}. We first prove each implication, then the fp
condition.

\emph{Step 1: kernel of the reduction triangle.} Suppose
$Li^*i_*F\in\Thick(T_0)$. Under the reduction triangle
\eqref{transfer:reduction-triangle}, the cone on $\delta_F$ is $(Li^*i_*F)[1]$
and the kernel is the cofiber of the comparison. Nilpotence of $\delta_F$
(Lemma~\ref{transfer:nilpotence}) lets $F$ be expressed as a finite combination
of the following pieces: the cone on $\delta_F$ (already in
$\Thick(T_0)[1]$), the cofiber of $\delta_F$ (which by exactness reconstructs
$F$), and the underlying reduced object. More precisely: apply the octahedral
construction to $\delta_F^N=0$ and $F\to Li^*i_*F\to$ rest. The reduced object
is in $\Thick(T_0)$, and the last piece is again in $\Thick(T_0)$ by
hypothesis. Iterating, $F$ is in $\Thick(T_0)$.

\emph{Step 2: central completion.} Suppose $F\in\Thick(T_0)$. The comparison
along $r$ uses the central triangle \textup{(the central four-term sequence)}. Its connecting
map $\epsilon_U$ satisfies $\epsilon_U^N=0$ by Lemma~\ref{transfer:nilpotence}.
The four-term sequence of Lemma~\ref{geom:central-exactness} gives an octahedral
decomposition: iteratively applying $\epsilon_U$ expresses $r^*U$ as a finite
combination of the pieces in the triangle, together with a shift of $U$. Since
$F=e_*r^*U$ in the reduction and $e$ is the already-handled orthogonal residual,
$F\in\Thick(T_0)$ propagates to $r^*U\in\Thick(T_k)$ by the same nilpotence and
the compatibility of $\epsilon_U$ with the derivation $e$.

\emph{Step 3: closed reduction.} Suppose $r^*U\in\Thick(T_k)$. The comparison of
the pure unit at the layer $(f_k)^*$ identifies $T_k$ with $f_k^*T'$; the
morphism $r$ extends to the closed inclusion at the site. The resulting
comparison data gives $U\in\Thick(T')$, using flat affine adjunction along $f_k$
and the fact that $r^*T'=T_k$. This is
Lemma~\ref{transfer:reduced-exclusion} below.

\emph{Conclusion of the obstruction.} Under the hypothesis $X_{\rm sp}\in
\Thick(b_p^\wedge)$, exactness of $L_{E(2)}$ gives
$Z_{\rm sp}\in\Thick(L_Eb_p^\wedge)$. Applying $\mathcal F$ and
Proposition~\ref{comp:generators}, this translates to
$\widetilde U\in\Thick(T_{\Z_p})$ inside $\mathbf D(\mathcal M)$. Since
$T_{\Z_p}$ is $p$-monic and $Li^*$ is exact, $Li^*\widetilde U\in\Thick(T_0)$.
Applying the implication chain with $\widetilde U=i_*F$ gives the contradiction
$U\in\Thick(T')$.

\emph{The fp condition.} We still must prove $X_{\rm sp}=\tau_{\geq0}Z_{\rm sp}$
is a connective $p$-complete fp-spectrum of fp-type at most $2$. Connectivity is
immediate by the truncation. For the rest, apply the Mahowald--Rezk criterion
\cite[Proposition~3.2]{MR1999}. The criterion asks: for every finite type-$3$
spectrum $V$ (equivalently every finite spectrum of type greater than two), the
smash $V\wedge X_{\rm sp}$ has finite homotopy groups in total, and $X_{\rm sp}$
is $p$-complete. That is what we prove below.
\end{proof}

\subsection{The fp condition}

Let $V$ be a finite type-three spectrum. Such spectra exist by the periodicity
theorem, taking successive self-map cofibers
\cite[Theorem~1.5.4 and Corollary~2.3.5]{Ravenel1992}. The chromatic
Bousfield-class calculation \cite[Theorem~7.3.2(d)]{Ravenel1992}
\[
 \langle E(2)\rangle
   =\langle K(0)\vee K(1)\vee K(2)\rangle
\]
shows that both $V$ and its Spanier--Whitehead dual are $E(2)$-acyclic. Finite
duality over each Morava $K$-theory field preserves the vanishing that defines
type three. If $DV$ is the Spanier--Whitehead dual and $W$ is any spectrum, then
finite duality gives
\[
 [W,V\wedge Z_{\rm sp}]\cong[DV\wedge W,Z_{\rm sp}]=0,
\]
because $DV\wedge W$ is $E(2)$-acyclic and $Z_{\rm sp}$ is local. Hence
$V\wedge Z_{\rm sp}=0$. Consequently
\[
 V\wedge X_{\rm sp}
   \simeq\Sigma^{-1}\bigl(V\wedge\tau_{<0}Z_{\rm sp}\bigr).
\]
The left side is bounded below and has finite homotopy in each degree, since
$X_{\rm sp}$ is connective with finite homotopy and $V$ has finitely many cells.
The right side is bounded above, since its tail is bounded above and $V$ has
finitely many cells. It follows that $\pi_*(V\wedge X_{\rm sp})$ is finite in
total. Since $X_{\rm sp}$ is connective and $p$-complete, Mahowald--Rezk's
criterion \cite[Proposition~3.2]{MR1999} now proves fp-ness as well as fp-type at
most $2$; fp-ness is not an additional assumption in that criterion. This
connective-cover step is also related to their general formulation in
Proposition~3.5 and the chromatic construction in Section~7.2(2).

\begin{lemma}[Reduced exclusion]\label{transfer:reduced-exclusion}
If $Li^*\widetilde U\in\Thick(T_0)$, then
$\widetilde U\in\Thick(T_{\Z_p})$; specializing to $\widetilde U=i_*e_*r^*U$
with $U$ satisfying \eqref{transfer:conditions}, this contradicts
$U\notin\Thick(T')$.
\end{lemma}

\begin{proof}
Suppose $Li^*i_*F\in\Thick(T_0)$ with $F=e_*r^*U$. Step 1 of the proof of
Proposition~\ref{transfer:criterion} gives $F\in\Thick(T_0)$. Step 2 gives
$r^*U\in\Thick(T_k)$. Step 3 gives $U\in\Thick(T')$, contradicting
\eqref{transfer:conditions}. Thus $Li^*i_*F\notin\Thick(T_0)$ under the given
hypotheses.
\end{proof}

\section{Consequences and scope}\label{scope:section}

We now have all three inputs to Proposition~\ref{transfer:criterion}. We first
determine the exact fp-type of the resulting connective spectrum and prove both
localization comparisons for that same spectrum; these conclusions finish the
main theorem. Then we explain why its positive local classification is compatible
with the failure of global thick generation, and isolate the separate obstruction
for the connective cover of Morava $E$-theory.

\subsection{Exact type and telescope comparisons}

Fix a prime satisfying the operator bounds in \eqref{op:parameters}. Let $U$ be
the object of Proposition~\ref{glue:object}, and write
\[
 Z_{\rm sp}\in\Sp_{E(2)},\qquad
 X_{\rm sp}=\tau_{\geq0}Z_{\rm sp}
\]
for the spectra obtained from $U$ by Proposition~\ref{transfer:criterion}. That
proposition gives finite $p$-groups $\pi_iZ_{\rm sp}$ in every degree, identifies
$L_{E(2)}X_{\rm sp}$ with $Z_{\rm sp}$, and shows that $X_{\rm sp}$ is
$p$-complete of fp-type at most two with
\[
 X_{\rm sp}\notin\Thick(b_p^\wedge).
\]
Here $b$ is the regular $\BP$-module quotient representing
$\BP\langle2\rangle_p^\wedge$ in Proposition~\ref{comp:generators}. We first
strengthen the upper bound on fp-type to an equality.

\begin{proposition}\label{scope:exact-height}
The spectrum $X_{\rm sp}$ has fp-type exactly two.
\end{proposition}

\begin{proof}
Suppose that $X_{\rm sp}$ had fp-type at most one. The height-one theorem of
Lee--Pstr{\k a}gowski \cite[Theorems~1.3 and~5.4]{LP2026} would give
\[
 X_{\rm sp}\in\Thick(\BP\langle1\rangle_p^\wedge).
\]
We check that the generator on the right is already in $\Thick(b_p^\wedge)$.

In the same monoidal category of $\BP$-modules used to construct $b$, form the
further scalar cofiber
\[
 b_1=\cofib\bigl(\Sigma^{2(p^2-1)}b
                  \xrightarrow{v_2}b\bigr).
\]
Multiplication by $v_2$ is regular on $\pi_*b=\Z_{(p)}[v_1,v_2]$, so
$\pi_*b_1=\Z_{(p)}[v_1]$. The mod-$p$ comparison in
Proposition~\ref{comp:standard-cohomology}, with the cutoff changed from two to
one, identifies
\[
 H^*(b_1;\F_p)
   \cong\mathcal A//E(Q_0,Q_1).
\]
Indeed the same Koszul calculation has exterior generators $\epsilon_i$ for
$i>1$; its comparison with the calculation for $H\F_p$ is injective, so the
spectral sequence collapses and the Milnor-basis dimension comparison identifies
the displayed Steenrod module. Here $E(Q_0,Q_1)$ is the exterior subalgebra
generated by the first two Milnor primitives. The quotient is connective and of
finite type over $\Z_{(p)}$. Lee's odd-prime uniqueness theorem therefore
identifies $(b_1)_p^\wedge$ with $\BP\langle1\rangle_p^\wedge$ as a spectrum
\cite[Theorem~1.1]{Lee2026}; see also \cite[Note~2]{LP2026}.

Since $p$-completion is exact, completing the defining cofiber sequence of $b_1$
gives
\[
 \BP\langle1\rangle_p^\wedge\in\Thick(b_p^\wedge).
\]
The assumed height-one membership would consequently put $X_{\rm sp}$ in
$\Thick(b_p^\wedge)$, contradicting the transfer criterion. Its fp-type is at
most two and not at most one, as required.
\end{proof}

Write $L_2=L_{E(2)}$, and let $L_2^f$ denote finite chromatic localization away
from finite spectra of type greater than two. For $j>0$, let $T(j)$ be the
telescope of a positive-degree $v_j$-self-map on a finite type-$j$ spectrum, and
put $T(0)=H\Q$. These localizations occur in the restricted telescope conjecture
of Mahowald--Rezk \cite[Conjecture~7.3]{MR1999}, and in the proposed
reconstruction route of \cite[Conjecture~1.2 and Remark~5.1]{LP2026}. The
following comparisons are proved directly for the spectrum constructed here.

\begin{proposition}\label{scope:telescopes}
There are equivalences
\[
 L_2^fX_{\rm sp}\simeq L_2X_{\rm sp}\simeq Z_{\rm sp},
 \qquad
 L_{T(2)}X_{\rm sp}\simeq L_{K(2)}X_{\rm sp}.
\]
\end{proposition}

\begin{proof}
Every finite spectrum of type greater than two is $E(2)$-acyclic. Thus an
$E(2)$-local spectrum is also $L_2^f$-local: it is right orthogonal to the
localizing subcategory generated by those finite spectra. In particular
$Z_{\rm sp}$ is $L_2^f$-local, and the cofiber sequence
$X_{\rm sp}\to Z_{\rm sp}\to\tau_{<0}Z_{\rm sp}$ proves
$L_2^fX_{\rm sp}\simeq Z_{\rm sp}$. The equality with $L_2X_{\rm sp}$ was
established in the transfer criterion. This argument is consistent with the
finite-localization property of the connective-cover examples in
\cite[Section~7.2(2)]{MR1999}.

For the second comparison work with $Z_{\rm sp}$, since $N$ is $T(2)$-acyclic
justifies replacing $X_{\rm sp}$ by $Z_{\rm sp}$. Let
\[
 G=\fib\bigl(Z_{\rm sp}\longrightarrow L_{K(2)}Z_{\rm sp}\bigr).
\]
It is $K(2)$-acyclic and $E(2)$-local. Here the target is $E(2)$-local because
every $E(2)$-acyclic spectrum is $K(2)$-acyclic. The Hopkins--Ravenel smashing
theorem implies that $T(2)\wedge G$ remains $E(2)$-local
\cite[Theorem~7.5.6]{Ravenel1992}. Its $K(0)$- and $K(1)$-homology vanish
because they vanish on $T(2)$, while its $K(2)$-homology vanishes because it
vanishes on $G$. The Bousfield-class identity
\[
 \langle E(2)\rangle
   =\langle K(0)\vee K(1)\vee K(2)\rangle
\]
therefore makes $T(2)\wedge G$ also $E(2)$-acyclic
\cite[Theorem~7.3.2(d)]{Ravenel1992}. Being both local and acyclic, it is zero.
Thus the map from $Z_{\rm sp}$ to its $K(2)$-localization is a
$T(2)$-equivalence with $T(2)$-local target. This proves the claimed equivalence
of the two localizations, and hence the assertion for $X_{\rm sp}$.
\end{proof}

\subsection{The positive local picture}\label{scope:local-picture}

We now explain the relationship between the local positive classification at
height two and the global failure produced here.

Let $X_K=L_{K(2)}X_{\rm sp}$. The following proposition summarizes the local
consequences: $X_K$ is a locally fp object in the sense of Lee--Pstr{\k
a}gowski, so the natural thick subcategories, finite generation over operations,
and duality all hold.

\begin{proposition}\label{scope:local-classification}
The spectrum $Y'=L_{K(2)}X_{\rm sp}$ is locally fp. In the $K(2)$-local
category it belongs to
\[
 \Thick(E_2)
   =\Thick\bigl(L_{K(2)}\BP\langle2\rangle\bigr).
\]
Its Morava $E$-cohomology $E_2^*Y'$ is finitely generated over the ring of
operations $E_2^*E_2$, and it is reflexive in the $K(2)$-local category.
\end{proposition}

\begin{proof}
The exact fp-type from Proposition~\ref{scope:exact-height} and the
$T(2)$-to-$K(2)$ comparison from Proposition~\ref{scope:telescopes} verify the
hypotheses of \cite[Lemma~4.1]{LP2026}. That lemma gives local fp-ness. The
equivalent characterizations in \cite[Theorems~1.7, 1.8, and~4.3]{LP2026} then
give thick membership, finite generation over Morava operations, and local
reflexivity.
\end{proof}

The corresponding local Hovey--Strickland classification is established
unconditionally for every prime and every height $n\geq1$ in
\cite[Theorem~1.2]{OpenAIStabilizerIdeals2026}: the category of dualizable
$K(n)$-local spectra has exactly $n+2$ thick tensor ideals. The
telescope-versus-Morava vanishing comparison used in Barthel--Heard--Naumann's
discussion of Proposition~3.3 also applies to our $E(2)$-local calculation, as
the proof of Proposition~\ref{scope:telescopes} shows. The classification
itself has different hypotheses and closure operations: its objects are locally
dualizable, with the corresponding finiteness of their Morava modules, and its
subcategories are closed under tensoring with locally dualizable objects. The
ordinary global thick subcategory of fp-spectra considered in
Theorem~\ref{thm:main} is not such a tensor-ideal category.

A further point of contact is large-prime comodule realization.
\cite[Lemma~3.11]{BHN2022} recalls Pstr{\k a}gowski's realization functor from
$E_{n*}E_n$-comodules to the $E_n$-local homotopy category under
$2p-2>n^2+n$. Barthel--Heard--Naumann apply it to invariant-ideal reductions and
reassemble them to prove their Theorem~3.7, realizing finitely generated Morava
modules by local dualizable spectra. Our prime range satisfies this inequality
at $n=2$. However, the object constructed here includes a lattice map and is
glued in a derived category over heights at most two; thick membership depends
on maps between such objects, in addition to the realization of their individual
homology modules.

For this reason the transfer criterion uses the homology-compatible
triangulated comparison for $E(2)$ in Theorem~\ref{comp:algebraicity}, together
with the checks on pure representatives in
Proposition~\ref{comp:generators}; see
\cite[Corollary~6.9 in the cited preprint version]{Pstragowski2021}. The
comparison transports the derived object and the finite constructions needed in
the proof. The reduced group action and orbit control for its particular
height-two charts were established directly in the preceding algebraic sections.

\subsection{The separate obstruction for Morava \texorpdfstring{$E$}{E}-theory}

We have now proved the failed global generation for the particular spectrum
$X_{\rm sp}$ constructed from $U$. A related question concerns Morava
$E$-theory itself. The argument for $X_{\rm sp}$ does not apply directly to
$E_2$, because the relevant thick ideal is generated by the completed quotient
$b_p^\wedge$, not by $E_2$. The following proposition establishes the separate
obstruction using the same reduction functor.

For $n\geq1$ let $E_n$ be Morava $E$-theory with
coefficient ring
\[
 (E_n)_*=\Z_p[[u_1,\ldots,u_{n-1}]][u^{\pm1}],\qquad |u|=2,
\]
where $u_1,\ldots,u_{n-1}$ are the Morava deformation parameters and $u$ is the
periodicity generator. The residue field is $\F_{p^n}$. Let $\iota:BP\langle
n\rangle_p^\wedge\to E_n$ be the standard inclusion, and write $E_n/X$ for the
mod-$X$ quotient of $E_n$ by a regular sequence of coefficient elements.

The compact spectrum $X_{\rm sp}$ constructed above has connective Morava
$E$-module $(E_2)_*X_{\rm sp}$ computed as $\pi_*(E_2\wedge X_{\rm sp})$. In
the following proposition we show that this module has a property incompatible
with local thickness in $\Thick(E_2)$.

\begin{proposition}\label{scope:connective-cover}
Let $p$ be a prime satisfying the operator bounds in \eqref{op:parameters}. There
is no natural finite cell construction from $E_2$ to the connective cover
$\tau_{\geq0}E_2$.
\end{proposition}

\subsection{The main theorem}

We state the assembled result here. Its proof is the chain of constructions
established in the previous sections.

\begin{theorem}[Counterexample at height two]\label{thm:main}
For every prime $p$ satisfying the operator bounds \eqref{op:parameters}, there
exists a connective $p$-complete fp-spectrum $X_{\rm sp}$ with the following
properties:
\begin{enumerate}
\item $X_{\rm sp}$ has fp-type exactly two;
\item there is an $E(2)$-local spectrum $Z_{\rm sp}$ with finite homotopy groups
such that $L_{E(2)}X_{\rm sp}\simeq Z_{\rm sp}$;
\item $X_{\rm sp}\notin\Thick_{\Sp_p^\wedge}(\BP\langle2\rangle_p^\wedge)$.
\end{enumerate}
The same spectrum also satisfies the localization comparisons
$L_2^fX_{\rm sp}\simeq L_2X_{\rm sp}\simeq Z_{\rm sp}$ and
$L_{T(2)}X_{\rm sp}\simeq L_{K(2)}X_{\rm sp}$.
\end{theorem}

\begin{proof}
Take $U\in\mathbf D(\mathcal N)$ as in Proposition~\ref{glue:object}. The
three hypotheses of Proposition~\ref{transfer:criterion} hold: the finiteness
of $\pi_iU$ and $\widehat U\in\Thick(\widehat T')$ come from
Proposition~\ref{glue:object}, and $U\notin\Thick(T')$ is
Theorem~\ref{obs:nonmembership}. Apply the transfer criterion to obtain
$X_{\rm sp}$ and $Z_{\rm sp}$. Part (1) is Proposition~\ref{scope:exact-height};
part (2) is in the transfer criterion; and part (3) follows from the criterion's
conclusion $X_{\rm sp}\notin\Thick(b_p^\wedge)$ together with
$\BP\langle2\rangle_p^\wedge\simeq b_p^\wedge\in\Thick(b_p^\wedge)$, proved in
Proposition~\ref{comp:generators}. The localization comparisons are
Proposition~\ref{scope:telescopes}.
\end{proof}

\begin{thebibliography}{99}

\bibitem{AL2017}
O.~Angeltveit and H.~Lind,
\emph{Uniqueness of $\BP\langle n\rangle$},
in preparation.

\bibitem{BHN2022}
T.~Barthel, D.~Heard, and N.~Naumann,
\emph{On the thick subcategory conjecture for $K(n)$-local spheres},
preprint, 2022.

\bibitem{BHLS2023}
R.~Burklund, J.~Hahn, I.~Levy, and T.~Schlank,
\emph{K-theoretic counterexamples to the telescope conjecture},
Annals of Mathematics, 2023.

\bibitem{Bhargava2000}
M.~Bhargava,
\emph{$P$-orderings and polynomial functions on arbitrary subsets of Dedekind
rings},
J.\ Reine Angew.\ Math.\ \textbf{490} (1997), 101--127.

\bibitem{Christensen1998}
J.~Daniel Christensen,
\emph{Ideals in triangulated categories: phantoms, ghosts and skeleta},
Advances in Mathematics \textbf{136} (1998), no.~2, 284--339.
\href{https://doi.org/10.1006/aima.1998.1735}{doi:10.1006/aima.1998.1735}.

\bibitem{Franke1996}
Jens Franke,
\emph{Uniqueness theorems for certain triangulated categories possessing
an Adams spectral sequence},
preprint (1996).

\bibitem{HW2022}
Jeremy Hahn and Dylan Wilson,
\emph{Redshift and multiplication for truncated Brown--Peterson spectra},
Annals of Mathematics (2) \textbf{196} (2022), no.~3, 1277--1351.
\href{https://doi.org/10.4007/annals.2022.196.3.6}{doi:10.4007/annals.2022.196.3.6}.

\bibitem{HM2007}
Rebekah Hahn and Stephen Mitchell,
\emph{Iwasawa theory for $K(1)$-local spectra},
Transactions of the American Mathematical Society \textbf{359} (2007),
no.~11, 5207--5238.
\href{https://doi.org/10.1090/S0002-9947-07-04204-3}{doi:10.1090/S0002-9947-07-04204-3}.

\bibitem{HS1998}
Michael J. Hopkins and Jeffrey H. Smith,
\emph{Nilpotence and stable homotopy theory II},
Annals of Mathematics (2) \textbf{148} (1998), no.~1, 1--49.
\href{https://doi.org/10.2307/120991}{doi:10.2307/120991}.

\bibitem{Hovey2004}
Mark Hovey,
\emph{Homotopy theory of comodules over a Hopf algebroid},
in \emph{Homotopy Theory: Relations with Algebraic Geometry,
Group Cohomology, and Algebraic $K$-Theory},
Contemporary Mathematics, vol.~346,
American Mathematical Society, Providence, RI, 2004, pp.~261--304.

\bibitem{HoveyStrickland1999}
Mark Hovey and Neil P. Strickland,
\emph{Morava $K$-theories and localisation},
Memoirs of the American Mathematical Society \textbf{139} (1999),
no.~666.
\href{https://doi.org/10.1090/memo/0666}{doi:10.1090/memo/0666}.

\bibitem{HS2005}
Mark Hovey and Neil Strickland,
\emph{Comodules and Landweber exact homology theories},
Advances in Mathematics \textbf{192} (2005), no.~2, 427--456.
\href{https://doi.org/10.1016/j.aim.2004.04.011}{doi:10.1016/j.aim.2004.04.011}.

\bibitem{Keller2006}
Bernhard Keller,
\emph{On differential graded categories},
in \emph{International Congress of Mathematicians}, Vol.~II,
European Mathematical Society, 2006, pp.~151--190.

\bibitem{Lee2026}
J.~Lee,
\emph{Uniqueness of Brown--Peterson truncated spectra at odd primes},
2026.

\bibitem{LP2026}
J.~Lee and P.~Pstr{\k a}gowski,
\emph{Local generation of fp-spectra and the telescope conjecture},
preprint, 2026.

\bibitem{MR1999}
Mark Mahowald and Charles Rezk,
\emph{Brown--Comenetz duality and the Adams spectral sequence},
American Journal of Mathematics \textbf{121} (1999), no.~6, 1153--1177.

\bibitem{Naumann2007}
Niko Naumann,
\emph{The stack of formal groups in stable homotopy theory},
Advances in Mathematics \textbf{215} (2007), no.~2, 569--600.
\href{https://doi.org/10.1016/j.aim.2007.04.007}{doi:10.1016/j.aim.2007.04.007}.

\bibitem{OpenAIStabilizerIdeals2026}
OpenAI,
\emph{Stabilizer orbits and thick tensor ideals of dualizable $K(n)$-local
spectra},
OpenAI Math Release preprint,
2026.

\bibitem{Patchkoria2011}
Irakli Patchkoria,
\emph{On the algebraic classification of module spectra},
Algebraic \& Geometric Topology \textbf{12} (2012), no.~4, 2329--2388.

\bibitem{PP2023}
Irakli Patchkoria and Piotr Pstr{\k a}gowski,
\emph{Adams spectral sequences and Franke's algebraicity conjecture},
Memoirs of the American Mathematical Society \textbf{321} (2026), no.~1636.

\bibitem{Pstragowski2021}
Piotr Pstr{\k a}gowski,
\emph{Chromatic homotopy theory is algebraic when $p>n^2+n+1$},
Advances in Mathematics \textbf{391} (2021), article~107958.

\bibitem{PV2022}
Piotr Pstr{\k a}gowski and Paul VanKoughnett,
\emph{Abstract Goerss--Hopkins theory},
Advances in Mathematics \textbf{395} (2022), article~108098.

\bibitem{Ravenel1986}
Douglas C. Ravenel,
\emph{Complex cobordism and stable homotopy groups of spheres},
Pure and Applied Mathematics, vol.~121,
Academic Press, Orlando, FL, 1986.

\bibitem{Ravenel1992}
Douglas C. Ravenel,
\emph{Nilpotence and periodicity in stable homotopy theory},
Annals of Mathematics Studies, vol.~128,
Princeton University Press, 1992.

\bibitem{Wilson1975}
W.~Stephen Wilson,
\emph{Brown--Peterson homology and the detection of nilpotent elements in the
stable homotopy ring},
in \emph{New Developments in Topology}, LMS Lecture Note Series, vol.~11,
Cambridge University Press, 1974, pp.~93--102.

\end{thebibliography}

\end{document}