==== TABLE OF CONTENTS ====
Finite-Smith-Toda-Complexes-at-Varying-Primes//00-introduction.tex
Finite-Smith-Toda-Complexes-at-Varying-Primes//01-foundations.tex
Finite-Smith-Toda-Complexes-at-Varying-Primes//02-uniform-power.tex
Finite-Smith-Toda-Complexes-at-Varying-Primes//03-layer.tex
Finite-Smith-Toda-Complexes-at-Varying-Primes//04-universal.tex
Finite-Smith-Toda-Complexes-at-Varying-Primes//05-connectivity.tex
Finite-Smith-Toda-Complexes-at-Varying-Primes//06-normalization.tex
Finite-Smith-Toda-Complexes-at-Varying-Primes//07-cobordism.tex
Finite-Smith-Toda-Complexes-at-Varying-Primes//08-realization.tex
Finite-Smith-Toda-Complexes-at-Varying-Primes//main.tex
Finite-Smith-Toda-Complexes-at-Varying-Primes//preamble.tex
Finite-Smith-Toda-Complexes-at-Varying-Primes//towers.tex
Finite-Smith-Toda-Complexes-at-Varying-Primes//references.bib
===========================

==== BEGIN Finite-Smith-Toda-Complexes-at-Varying-Primes//00-introduction.tex ====
\section{Introduction}
\label{sec:introduction}

The Smith--Toda realization problem asks whether the first chromatic
parameters can be killed in a finite spectrum. Algebraically these
parameters form a regular sequence in Brown--Peterson homology.
Topologically, killing the next parameter requires an actual map of
finite spectra that induces its multiplication on homology. Regularity
computes the homology of the cofiber once such a map exists; it does
not supply the map. At a prime $p$, write
\[
 \BP_* = \Z_{(p)}[v_1,v_2,\ldots],
 \qquad |v_i|=2(p^i-1),
\]
using Hazewinkel generators. A Smith--Toda complex $V(n)$ is a finite
$p$-local spectrum $X$ with
\[
 \BP_*X\cong\BP_*/(p,v_1,\ldots,v_n)
\]
as graded $\BP_*\BP$-comodules, with the quotient coaction and its
generator in degree zero. Here finite $p$-local means a retract of the
$p$-localization of a finite CW spectrum. The question concerns first
powers. Suitable higher powers can be realized at every finite height
by the periodicity methods of Hopkins--Smith
\cite[Theorem~9 and Proposition~5.14]{HopkinsSmith1998}. Those
exponents are chosen as part of the construction; the question here
prescribes every exponent to be one.

\begin{theorem}\label{thm:main}
For every integer $n\geq0$, there exist a prime $p$ and a finite
$p$-local spectrum $X$ such that
\[
 \BP_*X\cong\BP_*/(p,v_1,\ldots,v_n)
\]
as graded $\BP_*\BP$-comodules, with the canonical quotient coaction
and generator in degree zero.
\end{theorem}

Thus the all-height Smith--Toda existence problem has a positive answer
when the prime may depend on the height. The construction takes place
in ordinary $p$-local spectra, before any chromatic localization. It
does not assert that one prime works at every height, provide an
explicit bound on the prime, or require a multiplication on the final
spectrum.

\subsection{Context and earlier methods}
The realization problem grew out of the study of complex cobordism
modules and periodic phenomena in stable homotopy. Milnor's work on
complex cobordism and Novikov's coefficient theorem established its
polynomial coefficient structure \cite{Milnor1960,Novikov1962}.
Brown and Peterson introduced the spectrum $\BP$ \cite{BrownPeterson1966}.
Quillen
identified the formal group law underlying that theory and its
$p$-typical projection \cite{Quillen1969}; Hazewinkel supplied
the integral generators and coefficient formulas used here
\cite{HazewinkelIII}. These results describe the algebraic quotients
whose finite topological realization is at issue.

The low-height constructions are due to Adams, Smith, and Toda:
$V(k)$ exists for $k=1,2,3$ when $p>2k$, respectively, as summarized
in \cite[p.~493]{Nave2010}. Adams's periodicity construction is
part of his work on the groups $J(X)$ \cite{Adams1966}.
Smith's work treats the realization
of complex bordism modules \cite{Smith1970}, while Toda constructs
spectra realizing exterior parts of the Steenrod algebra
\cite{Toda1971}. At height zero the Moore spectrum supplies the
quotient by $p$. Each subsequent first-power attachment asks for
additional topological information. In the September 2023 version of
\cite[Remark~3.29]{CulverZhang2024}, the existence of $V(4)$ was
recorded as open at every prime. We address arbitrary fixed height
by allowing the prime to grow.

Known nonexistence results are consistent with this quantifier order.
For $p\geq7$, Nave proves that $V((p+1)/2)$ does not exist and that
$V((p-1)/2)$ cannot be a ring spectrum if it exists
\cite[Theorem~1.3]{Nave2010}. The obstructed height depends on $p$,
and the second assertion concerns additional structure. Neither is an
obstruction at one fixed height at every prime.

Ultraproducts exploit the distinction between a fixed height and a
varying prime. Barthel--Schlank--Stapleton establish asymptotic
algebraicity in chromatically localized categories and apply it to
generalized Moore objects \cite{BSS2020}. Their compactness is in
the localized category, whereas Theorem~\ref{thm:main} requires
an ordinary finite spectrum. Here the ultraproduct is formed from
ordinary $p$-local spectra. We retain varying sphere degrees and
construct a finite sequence of cofibers there. Only finitely many
maps and homotopies must subsequently descend to an individual
prime.

The other ingredients have complementary predecessors. Exactness
of the fixed-prime Tate tensor power provides a model for the
permutation construction \cite[Proposition~III.1.1]{NS2018}; our
comparison requires a single Euler exponent for the entire family
of tensor cubes as $p$ varies. The normalization uses completion
by a perfect algebra, in the setting of the Amitsur methods of
Mathew--Naumann--Noel \cite{MNN2017}. Power operations in cobordism
go back to tom Dieck and Quillen \cite{tomDieck1968,Quillen1971}.
For detection, we use the coherent Thom construction and its
Euler-product formula \cite{May1977,JohnsonNoel2010}. We retain these powers in complex
cobordism throughout the calculation, then apply the ordinary
multiplicative projection to Brown--Peterson homology.

A related use of filtrations appears in Randal-Williams's construction
of Smith--Toda analogues in graded algebra modules
\cite[Section~5, Theorem~5.6 and Lemma~5.7]{RandalWilliams2026}.
There successive cofibers are controlled through associated-graded
generators and vanishing estimates over a field of fixed positive
characteristic. The comparison here is methodological: our
connectivity bounds vary with the prime and feed the normalization
of an Euler layer in ordinary spectra. Recent local realization
results of Kato--Shimomura--Shimomura concern quotients with a
chromatic generator inverted \cite[Theorem~1.7]{KatoShimomura2025}.
They address a different realization target from the ordinary
finite, first-power quotient considered here.

\subsection{Construction and Detection}
Fix $n\geq1$ and a nonprincipal ultrafilter $\U$ on the odd primes.
The central ambient category is
\[
 \cC=\prod_{\U}\Sp_{(p)}.
\]
Its mapping spectra are rational, although sequences of spheres in
degrees growing with $p$ still carry essential information. A formal
weight $w_j$ records the varying even degree $2(p^j-1)$. Mapping out
of these spheres gives a rational graded algebra in which finite
Koszul extensions can be formed without discarding that degree data.

In this overview, $S^{2(p^j-1)}$ denotes the sequence of the indicated
primewise spheres, and $\one$ denotes the tensor unit. We construct
objects $Y_0,\ldots,Y_{n+1}$ in $\cC$, starting from $Y_0=\one$,
and, for $0\leq j\leq n$, maps
\[
 b_j:S^{2(p^j-1)}\longrightarrow Y_j,
 \qquad
 Y_{j+1}=\cofib\bigl(
 S^{2(p^j-1)}\otimes Y_j\xrightarrow{\ b_j\cdot-\ }Y_j
 \bigr).
\]
The first class $b_0$ is represented by primewise multiplication by
$p$. The multiplication used in this display
exists in $\cC$; the proof does not require its full commutative
structure to exist at any individual prime.

There are two separate tasks. The first is to construct $b_{j+1}$
from $b_j$, including the choices of boundaries that permit iteration.
The second is to show that these classes kill the intended first
powers in homology. Keeping the tasks separate avoids assuming the
homology calculation in order to construct the next class.

For the construction, take the $p$-fold permutation power with the
affine group $C_p\rtimes\F_p^{\times}$ acting on the factors. The
main difficulty is uniformity: the diagrams comparing powers with
cofibers have dimensions that grow with $p$. We prove that one Euler
map, associated to twice the reduced permutation representation,
annihilates the entire comparison object. In a suitable auxiliary
category, inverting that map consequently makes the power exact.
Before inversion its Euler parameter $t$ has a quotient that detects
the next actual sphere degree. The argument controls the whole
comparison object, rather than merely its individual filtration pieces.

The power must be normalized before it can be iterated on a Koszul
cycle. We do this on a universal algebra $R$ of cycle coefficients,
with a finite-cell Koszul extension $K$. Unlike the constructed $Y_j$,
these universal coefficient algebras have actual primewise commutative
models to which the power applies. At weights proportional to
$p$, a symmetric-group connectivity estimate gives the necessary
vanishing only after tensoring over $R$ with $K$. Such a tensor test
need not detect equivalences on arbitrary modules. We show that it
does detect the comparisons needed by our actual completed targets,
using the Amitsur totalization associated to $R\to K$. This permits
normalization by powers of $t$ while preserving algebra maps and
chosen nullhomotopies, not just operations on homotopy classes.
For the attaching cycle, it removes a prescribed Euler factor while
retaining a class before inversion, where reduction modulo $t$ is
still available.

For detection, let $x_j$ be the coefficient of $Z^{p^j}$ in the
$p$-series of complex cobordism, with $x_0=p$. A primewise
Thom-corrected power calculation identifies the leading $t$-term
of the powered $x_j$ with a unit times $x_{j+1}$. Comparison with
the normalized construction proves that the Hurewicz image of
$b_j$ is a unit times $x_j$ modulo its predecessors. Under the
ordinary multiplicative projection to $\BP$, the ideals generated
by the $x_j$ become the Hazewinkel ideals. No compatibility of that
projection with power operations is needed.

Finally, the attaching maps, unit maps, cofiber sequences, and their
specified homotopies descend to a common set of primes, together with
the finitely many Hurewicz equalities.
At one such prime, regularity computes homology in all degrees at
once. The sphere map onto the resulting cyclic $\BP$ module fixes
its comodule structure. This last step is what produces an ordinary
finite Smith--Toda complex, rather than only an ultraproduct or
chromatically localized realization.

Three different forms of finite control enter this proof. Rationality
is tested one fixed homotopy degree at a time. The estimate at weights
growing with $p$ uses a finite-cell comparison of whole primewise
modules. The final realization retains only a finite diagram of
attaching maps and homotopies. Each has its own justification; no
step requires an intersection of infinitely many large prime sets.

\subsection{Organization}
Section~\ref{sec:foundations} constructs the rational graded models
and their finite realizations. Sections~\ref{sec:power}
and~\ref{sec:layer} establish the uniform permutation power and its
Euler quotient. Sections~\ref{sec:universal}
and~\ref{sec:connectivity} give universal coefficient algebras and
the large-weight estimate. Section~\ref{sec:normalization}
normalizes the power and constructs the successive attaching classes.
Section~\ref{sec:cobordism} proves the coefficient identity needed
for detection, and Section~\ref{sec:realization} completes the
Hurewicz induction and finite descent.
==== END Finite-Smith-Toda-Complexes-at-Varying-Primes//00-introduction.tex ====

==== BEGIN Finite-Smith-Toda-Complexes-at-Varying-Primes//01-foundations.tex ====
\section{Ultraproducts, gradings, and finite cells}
\label{sec:foundations}

Fix the integer $n\geq 1$ for which we seek a finite realization.
This section provides a rational algebraic description of finite
constructions in an ultraproduct of ordinary $p$-local spectra.  The
description keeps track of sphere degrees that vary with $p$.  In
particular, it will allow us to interpret a finite sequence of exterior
algebra extensions as cofiber sequences of spectra.

All categories, tensor products, limits, and colimits are understood
homotopically.  The notation $\map$ denotes a mapping spectrum and
$\Map=\Omega^\infty\map$ its mapping space.  A commutative algebra is
an $E_\infty$-algebra unless a strict differential graded model is
specified.  Chain degrees are homological: the differential, denoted
$\partial$, lowers degree by one.  For a discrete abelian group
$\Gamma$, a $\Gamma$-graded object is a family indexed by $\Gamma$;
its tensor product is convolution,
\[
 (U\otimes V)_\eta
   =\bigoplus_{\alpha+\beta=\eta}U_\alpha\otimes V_\beta.
\]
Formal weights introduce no symmetry signs.  In a chain model the
symmetry sign is determined only by chain degree.

\subsection{The ordinary categorical ultraproduct}

Choose a nonprincipal ultrafilter $\U$ on the set of odd primes.
A set belonging to $\U$ will be called \emph{large}.  Define
\begin{equation}\label{eq:found-ultraproduct}
 \cC=\prod_{\U}\Sp_{(p)},\qquad
 D=\prod_{\U}\Z_{(p)},\qquad
 \varpi=[(p)_p]\in D.
\end{equation}
Here the category in Equation~\eqref{eq:found-ultraproduct} is
\[
 \cC=\colim_{U\in\U}\prod_{p\in U}\Sp_{(p)},
\]
with transition functors given by restriction to smaller large sets.
Thus a sequence represents an object up to restriction to a large set.
We use successively larger universes when taking this colimit and its
Ind-completion.  This is a colimit of categories; no presentability or
preservation of infinite primewise colimits is part of the definition.

\begin{proposition}\label{found:ultraproduct}
The category $\cC$ is stable and symmetric monoidal, with finite
stable constructions and tensor products computed primewise.  If
$X=[(X_p)]$ and $Y=[(Y_p)]$, then
\begin{equation}\label{eq:found-mapping-spectrum}
 \map_{\cC}(X,Y)
  \simeq\colim_{U\in\U}\prod_{p\in U}
                   \map_{\Sp_{(p)}}(X_p,Y_p).
\end{equation}
Every mapping spectrum is rational, and the endomorphism algebra of
the tensor unit is $HD$.  The ring $D$ contains $\Q$, and $\varpi$ is
a nonzerodivisor.

Objects, arrows, and homotopies forming a diagram indexed by a finite
simplicial set can be represented simultaneously on a large set of
primes.  The same holds for specified finite cofiber diagrams.  Sequence
objects become compact in $\Ind(\cC)$.
\end{proposition}

\begin{proof}
Products and filtered colimits of stable categories preserve the
stable structure.  To see the mapping formula and the assertion about
finite diagrams directly, use simplicial presentations of the
categories.  Inner horn fillers and any prescribed finite diagram
involve only finitely many simplices.  Such data therefore pass through
the filtered colimit.  Mapping spaces are obtained from finite limit
tests on these presentations, and filtered colimits of spaces commute
with finite limits.  This gives the mapping-space version of
Equation~\eqref{eq:found-mapping-spectrum}; applying it with every
fixed integer suspension gives the assertion for mapping spectra.
The same mapping-space tests show that primewise finite limits and
colimits have the required universal properties.  The tensor product
and its coherent finite-arity structure pass through the construction
as well.  These facts also follow from the general categorical
ultraproduct results in the author version of
\cite[Proposition~3.12, Corollary~3.16, and Lemma~3.17]{BSS2020}.

For every fixed nonzero integer $m$, multiplication by $m$ is an
equivalence at every prime not dividing $m$.  It is consequently an
equivalence on each mapping spectrum in
Equation~\eqref{eq:found-mapping-spectrum}.  These spectra are therefore
rational.  Homotopy groups commute with products of spectra and with
filtered colimits, so their groups are the corresponding group
ultraproducts.  Each fixed positive stable stem of the sphere is finite;
this is the stable consequence of Serre's finiteness theorem
\cite[Chapter~V, \S3, Proposition~3]{Serre1951}.  Its $p$-localization
therefore vanishes for all but finitely many primes.  Negative stable
stems vanish, and the zeroth stem gives $D$.  Hence the endomorphism
algebra of the unit is the discrete commutative ring spectrum $HD$.

The inverse of any fixed nonzero integer exists in $D$, giving the
inclusion $\Q\subset D$.  If $\varpi a=0$, choose a sequence
representing $a$.  On a large set one has $p a_p=0$ in the domain
$\Z_{(p)}$, so $a_p=0$ there.  Thus $a=0$.  Finally, compactness of
the sequence objects in the Ind-completion is part of the defining
Yoneda embedding into that completion.
\end{proof}

The same rationality argument applies to ultraproducts of categories
of spectra with naive actions of finite groups that may vary with
$p$: multiplication by a fixed integer prime to $p$ is still an
equivalence.  We often omit $H$ when writing a discrete scalar ring
as a ring spectrum.  Notice that rationality in Proposition~\ref{found:ultraproduct}
concerns fixed integer degrees.  It does not discard homotopy in the
prime-dependent degrees used below.

\subsection{Rational algebra models and specified nullhomotopies}

\begin{lemma}\label{found:rectification}
Let $R$ be a discrete commutative ring containing $\Q$, and let
$\Gamma$ be a discrete abelian group.  Commutative algebras in
$\Gamma$-graded $HR$-module spectra admit strict commutative differential
graded $R$-algebra models, with no boundedness requirement on chain
degree.  This description is compatible with diagrams and with
derived base change under specified scalar algebras.
\end{lemma}

\begin{proof}
The symmetric monoidal equivalence between $HR$-module spectra and
the derived category of $R$-modules is
\cite[Theorem~7.1.2.13]{LurieHA}.  The commutative differential graded
algebra model and its comparison with $E_\infty$-algebras require
$\Q\subset R$, precisely our hypothesis
\cite[Propositions~7.1.4.10--7.1.4.11]{LurieHA}.
For clarity, the comparison uses a resolution of the commutative
operad by an $R$-linear $E_\infty$-operad.  Since the order of each
finite symmetric group is invertible in $R$, its coinvariant functor
is exact.  Comparison of free algebras is therefore a quasi-isomorphism;
the filtration by free cells extends this comparison to cellular
resolutions.  With the additional $\Gamma$-grading the same argument
is applied weight by weight, summing the weights of the inputs.  Under
a prescribed scalar algebra one first chooses a cofibrant model of
that algebra and then uses derived base change.  All these comparisons
are equivalences of categories with their mapping spaces, so they also
apply to diagrams and their specified homotopies.
\end{proof}

We will need more than the equivalence class of an algebra that kills
a cycle: its universal nullhomotopy must induce the intended map of
underlying cofibers.  The following formulation records that choice.

\begin{lemma}\label{found:free-null}
Let $B$ be a $\Gamma$-graded commutative differential graded algebra
over a discrete ring $R\supset\Q$, and let $x\in B_\eta$ be a cycle
of chain degree $r$.  The free extension
\[
 B//x:=B[z],\qquad
 |z|_{\mathrm{chain}}=r+1,\quad \wt(z)=\eta,\quad
 \partial z=x
\]
models the derived algebra pushout universally adjoining a null of
$x$.  In contrast, write $B/x$ for the underlying module cofiber of
multiplication by $x$, with its degree and weight shifts.

When $r=0$, there is an identification of underlying $B$-modules
\begin{equation}\label{eq:found-null-cofiber}
 B//x\simeq B/x.
\end{equation}
It uses the specified universal null $z$.  Consequently, an algebra
map from $B//x$ defined by a map from $B$ and a specified null of the
image of $x$ induces the corresponding map of module cofibers using
that null multiplied by the base input.
\end{lemma}

\begin{proof}
Let $F$ be the free commutative $R$-algebra on a cycle $u$ of degree
$r$ and weight $\eta$, and map $u$ to $x$.  The required pushout is
$B\otimes_F R$, where $F\to R$ sends $u$ to zero.  Resolve this
augmentation by the free disk algebra
\[
 F[z],\qquad \partial z=u.
\]
The complex spanned by $u,z$ is contractible.  Every positive symmetric
power of this disk is acyclic, since tensor powers are contractible
and symmetric-group coinvariants are exact over $R$.  Thus the disk
algebra maps by a quasi-isomorphism to $R$.
As an $F$-module it has an increasing filtration by the number of
copies of $z$, whose successive quotients are free shifted
$F$-modules.  This filtration computes the derived tensor with $B$.
After base change the result is exactly $B[z]$ with $\partial z=x$.
The argument respects the specified weight throughout.

If $r=0$, the variable $z$ is odd, hence exterior.  The underlying
module is the two-term extension of $B$ by the weight-$\eta$, degree-one
copy of $B$, with differential given by multiplication by $x$.  This
is Equation~\eqref{eq:found-null-cofiber}.  To identify the chosen
null without an ambiguity, one can first perform this calculation
over the polynomial cycle algebra $F=R[u]$.  Its augmentation is the
cofiber of multiplication by $u$, and its discrete target has no
degree-one ambiguity in the null of the generator.  Tensoring to $B$
gives the displayed comparison through the universal null.  A map
out of the disk algebra sends $z$ to the chosen target null; on the
underlying two-term module its value on a base multiple of $z$ is
therefore that same null multiplied by the image of the base element.
\end{proof}

\begin{remark}\label{found:scalar-pushouts}
The free cycle and disk algebras over $R$ are obtained by extension
of scalars from those over $\Q$.  Thus the cell in
Lemma~\ref{found:free-null} also has its stated universal property
when considered merely as an attachment over rational spectra.
More generally, a pushout of a span of algebras under common scalars
is also the pushout of that span without separately imposing those
scalars: the map from the middle algebra already supplies the common
scalar map and its compatibility.  We will use this observation for
power operations that can change the action of $D$.  Comparisons
involving specified nulls are transported together with their
specified homotopies along equivalences.
\end{remark}

\subsection{Coherent sphere gradings}

We next assign varying sphere degrees to a fixed lattice of formal
weights.  The required multiplicative coherence is imposed in the
rational category $\cC$.

\begin{lemma}\label{found:picard-grading}
Let $\mathcal D$ be a stable symmetric monoidal category with rational
mapping spectra, and let $\Gamma$ be a finite free abelian group with
a chosen basis.  Suppose the desired invertible object for each basis
element is a tensor square of an invertible object.  These objects
extend to a symmetric monoidal grading of $\mathcal D$ by the
discrete group $\Gamma$.

For two such gradings, prescribed equivalences on the basis extend
to a monoidal equivalence of gradings.  The construction can be made
relative to any specified subset of the basis.
\end{lemma}

\begin{proof}
The Picard space of $\mathcal D$ is a grouplike $E_\infty$-space and
hence is the infinite loop space of a connective spectrum $U$;
the recognition statement is
\cite[Remark~5.2.6.26]{LurieHA}.  A monoidal grading by $\Gamma$ is
equivalently a spectrum map $H\Gamma\to U$.  For $k\geq2$,
\[
 \pi_k U\cong\pi_{k-1}\map_{\mathcal D}(\one,\one),
\]
so $\tau_{\geq2}U$ is rational.

Put $Q_s=\cofib(S\to H\Z)$.  The rationalization of $S\to H\Z$
is an equivalence.  Moreover $Q_s$ is 2-connective, and
$\pi_2 Q_s=\pi_1^s=\Z/2$
\cite[Corollaries~4J.3--4J.4]{Hatcher2002}. Since maps from a rationally zero spectrum
to a rational spectrum vanish, there is an equivalence
\[
 \map(Q_s,U)\simeq\map(Q_s,\tau_{\leq1}U).
\]
Connectivity and truncation imply that its homotopy groups in degrees
at least zero vanish.  In degree minus one its only possible
contribution is
\[
 \pi_{-1}\map(Q_s,\tau_{\leq1}U)
   \cong\Hom(\Z/2,\pi_1 U),
\]
which is killed by two.  The fiber sequence
\[
 \map(Q_s,U)\longrightarrow\map(H\Z,U)
                    \longrightarrow\map(S,U)=U
\]
therefore shows that evaluation is an equivalence of spaces onto a
union of components of $\Omega^\infty U$.  The obstruction to a
component lying in that union takes values in a group killed by two.
Every double in $\pi_0 U$ lies in the image, so a specified tensor
square extends to a grading on $\Z$.

The equivalence onto a union of components also identifies all path
spaces between eligible objects.  Hence specified equivalences of
the generator objects lift to equivalences of gradings, with their
higher compatibilities.  Finally, $H\Gamma$ is the finite direct sum
of copies of $H\Z$ indexed by the basis.  The space of its maps to
$U$ is the corresponding product.  Applying the preceding argument
factor by factor proves both the assertion and its relative form.
\end{proof}

Define the lattices and additive functions
\begin{equation}\label{eq:found-weights}
 \begin{gathered}
  \Lambda=\langle w_0,\ldots,w_{n-1}\rangle,\qquad
  \Omega=\langle w_0,\ldots,w_n\rangle,\\
  d=d_p:\Omega\longrightarrow\Z,\qquad
  d(w_h)=d_h=2(p^h-1),\qquad q=d_1=2(p-1),\\
  \nu:\Omega\longrightarrow\Z,\qquad \nu(w_h)=1.
 \end{gathered}
\end{equation}
The symbols $w_h$ and their relations are independent of $p$, whereas
$d_h$ and $q$ depend on $p$.  Each sequence of spheres
$(S^{d_p(w_h)})_p$ is a tensor square.  By
Lemma~\ref{found:picard-grading} we may choose a monoidal sphere
grading $S(\eta)$ on $\Omega$ with
\[
 S(\eta)\simeq[(S^{d_p(\eta)})_p].
\]
Since $d_p(w_0)=0$, choose this grading first on
$\Omega/\langle w_0\rangle$ and then pull it back.  The $w_0$
direction is thus the unit direction.  Fixed homological suspensions,
which may be odd, are kept separate from these even sphere degrees.

\subsection{Normalized mapping objects}

For $\Gamma=\Omega$ or $\Lambda$ and $X\in\cC$, define
\begin{equation}\label{eq:found-normalized-maps}
 h_\Gamma(X)_\eta=\map_{\cC}(S(\eta),X),
 \qquad \eta\in\Gamma.
\end{equation}
For a primewise $\Gamma$-graded system $(X_p(\eta))$, the same
notation means
\[
 h_\Gamma(X)_\eta
   =\map_{\cC}\bigl(S(\eta),[(X_p(\eta))_p]\bigr).
\]
An ungraded algebra can also be regarded as constant in the weights,
using its multiplication for the weight-additive pairings.

\begin{lemma}\label{found:mapping-pairings}
The constructions $h_\Gamma$ are exact componentwise and possess
coherent lax symmetric monoidal pairings.  On algebraic inputs they
give graded $D$-algebras and their modules.  The pairings are balanced
over $D$ whenever its action is supplied by trivial sphere scalars.
They also respect additional diagram indices, including filtrations.
\end{lemma}

\begin{proof}
Mapping out of a fixed sphere is exact.  Given two input maps, tensor
them and use the chosen equivalence
$S(\alpha)\otimes S(\beta)\simeq S(\alpha+\beta)$; then compose
with the target pairing.  This defines the pairing at the pair of
weights $(\alpha,\beta)$.  Monoidality of the sphere grading and
separate exactness of tensor make these maps coherent at every finite
tuple of weights.  Their sums give convolution pairings in the
category of graded mapping spectra.  This construction requires no
infinite coproduct in $\cC$ itself.  The endomorphism algebra of the
unit acts centrally through tensor, so the pairings are $D$-balanced.
If the targets carry another diagram index, the same construction
commutes with its structure maps and its additive pairings.
\end{proof}

\begin{remark}\label{found:whole-systems}
For graded inputs we use an ultraproduct of categories of \emph{whole}
primewise graded systems.  Evaluation at any fixed index gives a
functor into $\cC$.  Primewise graded pairings then give the diagrams
and pairings used in Lemma~\ref{found:mapping-pairings}, compatibly
with restriction to large sets.  One may first pull these systems
back along prime-dependent additive maps from a common indexing
lattice.  This only uses a functor from the ultraproduct of whole
systems to systems in $\cC$; it does not identify these categories
or assert that every diagram in $\cC$ has such a lift.  The same
distinction applies to group actions, filtrations, and module
structures.  The simultaneous coherent sphere grading was chosen
only in $\cC$.  An individual sphere or cell shift used primewise is
simply the sphere of the indicated degree.
\end{remark}

Put $A=h_\Omega(\one)$.  By Lemma~\ref{found:rectification} we may
work with a strict commutative differential graded $D$-algebra model
of $A$.  Choose that model on the smaller grading
$\Omega/\langle w_0\rangle$ before pulling it back.  The resulting
model is strictly $w_0$-periodic.  Let
\[
 u_0\in A_{w_0},\qquad |u_0|_{\mathrm{chain}}=0,
\]
be the homogeneous periodic unit corresponding to $1\in A_0$.
We now show that finite algebraic constructions over $A$ describe
actual finite constructions from these sphere objects.

\subsection{Finite-cell comparison and the objects \texorpdfstring{$Y_j$}{Yj}}

For a graded module write $V\langle\eta\rangle_\theta
=V_{\theta-\eta}$ for weight shift.  A \emph{finite-cell} graded
$A$-module is obtained from finitely many homological suspensions of
the modules $A\langle\eta\rangle$ by finite sums and cofibers.  We
do not include retracts in this terminology.  Let $\cC_{\mathrm{cell}}$
be the corresponding full stable subcategory of $\cC$ generated by
the spheres $S(\eta)$.

\begin{proposition}\label{found:finite-cell-morita}
The functor $h_\Omega$ induces a symmetric monoidal equivalence
between $\cC_{\mathrm{cell}}$ and the category of finite-cell graded
$A$-modules.  For $X,Y\in\cC$, the natural comparison
\begin{equation}\label{eq:found-tensor-comparison}
 h_\Omega(X)\otimes_A h_\Omega(Y)
       \longrightarrow h_\Omega(X\otimes Y)
\end{equation}
is an equivalence whenever either input is in $\cC_{\mathrm{cell}}$.
These comparisons preserve units, compositions, and symmetric
multi-input products.  In particular, a commutative $A$-algebra whose
underlying module is finite-cell determines a commutative algebra
in $\cC$.
\end{proposition}

\begin{proof}
Invertibility gives
\[
 h_\Omega(S(\eta))_\theta
   \simeq\map_{\cC}(S(\theta-\eta),\one)
   =A_{\theta-\eta}.
\]
Thus a sphere generator maps to $A\langle\eta\rangle$.  For any
$X\in\cC$, mapping from this free graded module evaluates
$h_\Omega(X)$ at $\eta$, and therefore agrees with
$\map_{\cC}(S(\eta),X)$.  Exactness extends this comparison from
the generators to every finite-cell source.  It follows that
$h_\Omega$ is fully faithful on $\cC_{\mathrm{cell}}$.  A finite-cell
graded module is realized inductively: lift its attaching maps by
full faithfulness and take the corresponding cofibers in $\cC$.
This gives precisely the stated essential image.

The pairings of Lemma~\ref{found:mapping-pairings} are balanced over
$A$.  Their coherent balanced comparison is obtained from the tensor
bar construction: use the pairings with an arbitrary finite number
of inserted $A$-factors and then take its realization in graded
mapping spectra.  If $X=S(\eta)$,
Equation~\eqref{eq:found-tensor-comparison} is the weight-shift
equivalence supplied by invertibility.  Both sides are exact in
$X$, so the comparison remains an equivalence for every finite-cell
$X$, with $Y$ arbitrary.  Symmetry gives the other case.
The same bar pairings for several inputs and iterated relative
tensors retain compatibility with composition and symmetry.  Hence
the equivalence on the full tensor-closed finite-cell subcategories
is symmetric monoidal.  Commutative algebra objects in those full
subcategories transport across it, with all their coherence.
\end{proof}

The proposition holds with $\Lambda$ in place of $\Omega$, and with
the finite-cell subcategory generated by the spheres indexed by
$\Lambda$, by exactly the same generator and cofiber tests.  Only
finite-cell sources are involved; no conservativity assertion on
arbitrary objects is being made.

\begin{proposition}\label{found:koszul-realization}
Fix $0\leq r\leq n$. Suppose cycles $b_h$ of chain degree zero and
weight $w_h$, for $0\leq h\leq r$, have been chosen successively
in the strict algebras
\begin{equation}\label{eq:found-exterior-tower}
 \begin{gathered}
 A[e_0,\ldots,e_{j-1}],\qquad
 |e_h|_{\mathrm{chain}}=1,\quad \wt(e_h)=w_h,\\
 \partial e_h=b_h\in A[e_0,\ldots,e_{h-1}],
 \qquad b_0=\varpi u_0.
 \end{gathered}
\end{equation}
For $0\leq j\leq r+1$, these algebras determine commutative
algebras $Y_j\in\cC$, with $Y_0=\one$. For $0\leq j\leq r$,
the cycle $b_j$ represents a map
$S(w_j)\to Y_j$, and there is a cofiber sequence
\begin{equation}\label{eq:found-cofiber-tower}
 S(w_j)\otimes Y_j\xrightarrow{\,b_j\cdot-\,}Y_j
                    \longrightarrow Y_{j+1},
\end{equation}
whose second map is the algebra map adjoining the null $e_j$.
For $M=[(\MU_{(p)})_p]$ and $0\leq j\leq r+1$, there are
compatible equivalences
\begin{equation}\label{eq:found-mu-base-change}
 h_\Omega(M\otimes Y_j)
   \simeq h_\Omega(M)\otimes_A A[e_0,\ldots,e_{j-1}].
\end{equation}
\end{proposition}

\begin{proof}
Each variable $e_h$ is exterior.  The underlying $A$-module has the
finite basis of ordered exterior monomials $e_S$ for
$S\subseteq\{0,\ldots,j-1\}$.  Order these subsets by the largest
entry at which they differ.  In the differential of $e_S$, replacing
an index $h$ by indices strictly less than $h$ decreases this order;
terms with a repeated index vanish.  The differential therefore
provides a finite-cell filtration on this basis.  Proposition~\ref{found:finite-cell-morita}
transports the algebra and its unit to $Y_j$.

A degree-zero cycle at weight $w_j$ represents a class in
$\pi_0\map_{\cC}(S(w_j),Y_j)$.  Its product with the identity of
$Y_j$ corresponds to multiplication by $b_j$ on the algebraic module.
Lemma~\ref{found:free-null} identifies the extension by $e_j$, through
its specified null, with the cofiber of this multiplication map.
This proves Equation~\eqref{eq:found-cofiber-tower}, including its
unit and cofiber-map identifications.  Finally, $Y_j$ is finite-cell,
so Equation~\eqref{eq:found-tensor-comparison} with the other input
$M$ gives Equation~\eqref{eq:found-mu-base-change}, compatibly with
all the algebra maps and specified nulls.
\end{proof}

In particular, $Y_1$ is the cofiber of $\varpi$ on the unit.  The
remaining task is to construct the cycles $b_1,\ldots,b_n$ with the
required cobordism images.  The algebra structures above are available
in $\cC$ throughout that construction.  At the eventual passage to a
prime we will retain only finite diagrams of classes, products,
units, and cofiber sequences, as permitted by
Proposition~\ref{found:ultraproduct}.
==== END Finite-Smith-Toda-Complexes-at-Varying-Primes//01-foundations.tex ====

==== BEGIN Finite-Smith-Toda-Complexes-at-Varying-Primes//02-uniform-power.tex ====
\section{Uniform localization of permutation powers}
\label{sec:power}

The permutation power of a spectrum is multiplicative but is not exact.
We construct a localization in which it becomes exact, uniformly as the
prime varies. The resulting operation will take values in a graded
mapping algebra with an invertible Euler parameter. In
Section~\ref{sec:layer}, before inverting that parameter, we will identify
its cofiber with ordinary reduction modulo $\varpi$.

For each odd prime $p$, let
\[
 G=G_p=C_p\rtimes\F_p^\times
\]
act on the $p$ elements of $\F_p$ by affine transformations. Let $\rho$
be its real permutation representation and let $V=\rho-1$ be the
augmentation representation. We use the category of naive actions
\[
 \cG=\prod_{\U}\operatorname{Fun}(BG_p,\Sp_{(p)}).
\]
All equivariant spheres and spaces below denote their $p$-locally
stabilized sequences in this category. Put
\[
 T=S^V,\qquad J=S(V)_+,\qquad J_2=S(2V)_+.
\]
Here $S(V)$ is the unit sphere, whereas $S^V$ is the one-point
compactification. The two Euler cofiber sequences are
\begin{equation}
 \label{power:euler-triangles}
 J\longrightarrow\one\longrightarrow T,
 \qquad
 J_2\longrightarrow\one\xrightarrow{a_2}T^2.
\end{equation}
Tensor powers of invertible objects may have negative exponents. We
write $\Tr$ for the functor that gives a spectrum trivial action.

\begin{lemma}[The two localizations]
\label{power:localizations}
Let $\cE$ be the symmetric monoidal stable localization of
$\Ind(\cG)$ by the localizing tensor ideal generated by
$G_+\otimes X$, for all $X\in\cG$. For $b\geq1$, put
\[
 E_b=G^{*b},\qquad
 \widetilde E_b=\cofib(E_{b+}\longrightarrow\one).
\]
The image of every object of $\cG$ is compact in $\cE$, and, for
$X,Y\in\cG$,
\begin{equation}
 \label{power:join-formula}
 \map_{\cE}(X,Y)
 \simeq\colim_b\map_{\cG}(X,\widetilde E_b\otimes Y).
\end{equation}
There is a further symmetric monoidal localization $\cEp$ in which
$a_2$ is invertible. Its localization of an object $Y$ is represented
in $\cE$ by
\[
 \colim_{s\geq0}
 \bigl(Y\xrightarrow{a_2}T^2Y\xrightarrow{a_2}T^4Y
 \longrightarrow\cdots\bigr).
\]
Mapping out of a sequence object commutes with this telescope.
\end{lemma}

\begin{proof}
For fixed $b$, the free $G$-space $E_b$ has cells in dimensions
$0,\ldots,b-1$. Their multiplicities can depend on $p$, but a whole
prime-dependent wedge of free cells is still an induced sequence
object. Thus $E_{b+}\otimes Y$ belongs to the indicated ideal by a
bounded number of stable extensions. The cone of the coaugmentation
\[
 Y\longrightarrow\colim_b\widetilde E_b\otimes Y
\]
therefore belongs to that ideal.

After forgetting the action, the transition
$\widetilde E_b\to\widetilde E_{b+1}$ is null: the old join cones to
any chosen vertex in the new copy of $G$. Maps out of $G_+\otimes X$
are tested by forgetting the action and using induction adjunction.
Since these objects are compact in $\Ind(\cG)$, the telescope is
local against all the generators of the ideal. It is consequently the
local reflection. The generating objects are compact, so local
objects are closed under filtered colimits and the localized images
of sequence objects remain compact. This proves
Equation~\eqref{power:join-formula}. The ideal is a tensor ideal, which
gives the symmetric monoidal localization.

For the second localization, the cone of the displayed telescope
coaugmentation lies in the tensor ideal generated by the cones of
$a_2\otimes Z$. On the telescope, multiplication by $a_2$ is an
equivalence: it translates the telescope by one stage. The
interchanges of the $T^2$ factors may be taken coherently trivial by
Lemma~\ref{found:picard-grading}, since $T^2$ is a square. Equivalently,
one can verify the translation equivalence on maps from compact
objects using this coherent grading. This proves locality and the
second assertion. The cones being inverted are again generated by
maps between compact objects, so the same compactness argument
computes maps by the telescope.
\end{proof}

We will repeatedly use the following consequence of the invertibility
of an isotropy-group order.

\begin{lemma}[Prime-to-$p$ actions]
\label{power:prime-to-p}
If a finite group $H$ has order prime to $p$, every naive $H$-spectrum
in $\Sp_{(p)}$ is a retract of the induction of its underlying spectrum.
In particular, homotopy fixed points preserve underlying lower
connectivity bounds for such actions.
\end{lemma}

\begin{proof}
For finite groups, induction and coinduction agree. The unit into
coinduction, followed by this identification and the counit of
induction, has composite multiplication by $|H|$. On underlying
spectra this is an equivalence. Rescaling supplies the claimed
retraction. Homotopy fixed points of an induced object are its
underlying inducing spectrum, so the connectivity assertion follows
by taking the retract.
\end{proof}

At each prime, permuting the factors makes $X^{\otimes p}$ a naive
$G_p$-spectrum. These functors define a strong symmetric monoidal
functor
\[
 N:\cC\longrightarrow\cG.
\]
The next argument is the uniform replacement for the fixed-prime
exactness of the cyclic Tate tensor power
\cite[Proposition~III.1.1]{NS2018}.

\begin{lemma}[A single Euler map makes the power exact]
\label{power:exactness}
The composite $N:\cC\to\cEp$ is exact. More precisely, for a map
$f:A_0\to A_1$, form the canonical comparison in $\cG$
\[
 \kappa_f:\cofib(NA_0\longrightarrow NA_1)
             \longrightarrow N(\cofib f).
\]
Its fiber is annihilated by one multiplication by $a_2$,
simultaneously at all primes.
\end{lemma}

\begin{proof}
At a prime, index the tensor cube by subsets $L$ of the $p$ letters,
with value
\[
 P(L)=A_1^{\otimes L}\otimes A_0^{\otimes L^c}.
\]
The cofiber of the proper-face homotopy colimit mapping to
$P(\F_p)=NA_1$ is $N(\cofib f)$. For example, choose a cofibration
between cofibrant spectra representing $f$ in a monoidal model. The
pushout-product axiom identifies the proper-face union with the
homotopy colimit and its quotient with the smash of the cofibers.
This calculation is compatible with the permutation action and is
verified on underlying spectra.

First quotient by the empty-face value $NA_0$. The fiber of
$\kappa_f$ is the homotopy colimit of
\[
 \cofib\bigl(NA_0\longrightarrow P(L)\bigr)
\]
over nonempty proper subsets. Indeed, after taking these cofibers,
the empty face has value zero and can be omitted.

Let $\mathcal P$ be this nonempty-proper-subset poset, with its
$G$-action. Consider the natural transformation on its action category
from the constant sphere diagram to the diagram pulled back from
$T^2$, given by $a_2$. Its space of transformations is computed by
equivariant maps from the plus order complex $|\mathcal P|_+$ to
$T^2$: push the constant source diagram forward to $BG$.
The order complex has dimension at most $p-2$. Each simplex
stabilizer has order prime to $p$, because a subgroup with $p$-torsion
contains the transitive translation subgroup and cannot stabilize a
nonempty proper subset. On an orbit cell, the target mapping
spectrum is a homotopy fixed point spectrum of a sphere of underlying
degree $2(p-1)$. Lemma~\ref{power:prime-to-p} preserves that
connectivity. A cell of dimension $k$ shifts the lower bound to
$2(p-1)-k$. Since $k\leq p-2$, the cellular filtration gives the
whole mapping spectrum the lower connectivity bound
\[
 2(p-1)-(p-2)=p>0.
\]
Its $\pi_0$ therefore vanishes, so the Euler transformation has a
nullhomotopy.

Tensor this null transformation with the diagram of cofibers and
then push forward to $BG$. Tensoring with an object pulled back from
$BG$ commutes with this pushforward, by distribution over colimits.
Thus one copy of $a_2$ annihilates the entire intervening homotopy
colimit. Although the cube dimensions grow with $p$, the Euler
exponent is one at every prime. The chosen primewise nullhomotopies
define a null in the germ category. After passage to $\cEp$, the
comparison of cofibers is consequently an equivalence.

The functor preserves zero. Preservation of these cofiber sequences
implies preservation of finite pushouts in stable categories, and
hence exactness. In particular the exact strong symmetric monoidal
functor to $\cEp$ acts on stable mapping spectra, compatibly with
their composition and products. One can also obtain this enrichment
by extending the exact functor to a colimit-preserving functor on
Ind-categories, where it commutes with tensors by spectra.
\end{proof}

\subsection{The Euler-graded mapping algebra}

The exact power now gives maps on mapping spectra. To record the Euler
filtration before localization, use the index lattice
\[
 I=\Lambda\oplus\Lambda\oplus\Z,
 \qquad i=(\lambda,\delta,m),\qquad \tau=(0,0,1).
\]
Choose the coherent invertible grading
\begin{equation}
 \label{power:W}
 W_i=N(S(\lambda))\otimes\Tr S(\delta)\otimes T^{-2m}.
\end{equation}
The first two factors use the sphere gradings already chosen in
Section~\ref{sec:foundations}; the last uses
Lemma~\ref{found:picard-grading}. At the primes, the corresponding
representation sphere has virtual representation
\[
 d(\lambda)\rho+d(\delta)-2mV.
\]

Let $(X_p)_p$ be a sequence of $\Lambda$-graded spectra; an ungraded
input is taken constant at all weights. Define
\begin{equation}
 \label{power:B}
 B(X)_i=\map_{\cE}
 \left(W_i,\Tr\bigl(X_p(p\lambda+\delta)\bigr)_p\right).
\end{equation}
If the input consists of graded commutative algebras, tensoring maps
and then applying their multiplication makes $B(X)$ a rational
$I$-graded commutative algebra. The unit and the adjoint
$T^{-2}\to\one$ of $a_2$ give a homogeneous class
\[
 t\in\pi_0B(X)_\tau.
\]
Multiplication by $t$ is the Euler precomposition
$B(X)_{i-\tau}\to B(X)_i$. These polynomial actions are compatible
with algebra maps in $X$: take the class first for the graded sphere
unit supported at weight zero and then apply the algebra unit.

\begin{lemma}[Euler inversion and ordinary scalars]
\label{power:graded-localization}
Replacing $\cE$ by $\cEp$ in Equation~\eqref{power:B} gives the
algebra $B(X)[t^{-1}]$. At $\lambda=m=0$, trivial action gives a
natural algebra map on the $\delta$-grading
\[
 h_\Lambda(X)\longrightarrow B(X)_{0,*,0}.
\]
We call this the scalar action in the $\delta$ direction.
\end{lemma}

\begin{proof}
At an index $i$, algebraic inversion of $t$ takes the telescope of
$B(X)_{i+s\tau}$. Moving the factor $T^{-2s}$ from the source to the
target identifies it with the mapping telescope into $T^{2s}$ times
the target. Lemma~\ref{power:localizations} and compactness identify
this with the mapping spectrum in $\cEp$. These identifications
respect multiplication by the coherent tensor powers of $T^2$.
For the second assertion, at $(0,\delta,0)$ the source sphere is
$\Tr S(\delta)$, and the map is simply the one induced by the
symmetric monoidal trivial-action functor.
\end{proof}

\begin{proposition}[The unnormalized power]
\label{power:operation}
For a sequence of actual primewise $\Lambda$-graded commutative
algebras, there is a natural map of graded commutative algebras over
rational spectra
\begin{equation}
 \label{power:raw-operation}
 P_X:h_\Lambda(X)\longrightarrow B(X)[t^{-1}]_{*,0,0}.
\end{equation}
On a represented degree-zero class, its value is obtained by taking
the permuted $p$-fold smash and then multiplying the $p$ factors. In
particular, that value has this represented class in $B(X)$ before
Euler inversion. No $D$-linearity is asserted for $P_X$.
\end{proposition}

\begin{proof}
Apply the exact functor of Lemma~\ref{power:exactness} to mapping
spectra out of $S(\lambda)$, and then use
\[
 N\bigl(X_p(\lambda)\bigr)\longrightarrow
 \Tr X_p(p\lambda).
\]
The latter map is the $p$-ary multiplication. It is equivariant,
coherently permutation invariant, and compatible with multiplying
tuples, by the commutative algebra structure at each prime. It can
equivalently be read from the algebra codiagonal, since tensor product
is the coproduct of commutative algebras. The exact strong monoidal
mapping-spectrum construction and these coherent multiplications
give the asserted algebra map. Naturality holds for actual primewise
algebra maps. The description on represented classes follows before
localization from the same construction. Rational linearity follows
from stable enrichment and rationality of the mapping spectra; the
operation can change the action of the larger scalar ring $D$.
\end{proof}

Only primewise commutative algebra inputs will be used in
Proposition~\ref{power:operation}. In particular, it does not assert
primewise commutative algebra structures on the objects $Y_j$ of the
Koszul construction.
==== END Finite-Smith-Toda-Complexes-at-Varying-Primes//02-uniform-power.tex ====

==== BEGIN Finite-Smith-Toda-Complexes-at-Varying-Primes//03-layer.tex ====
\section{The Euler layer and ordinary reduction}
\label{sec:layer}

We identify the cofiber of the Euler parameter in the mapping algebra
of Section~\ref{sec:power}. The comparison will be multiplicative and
will respect both ordinary scalar classes and the particular null
used to form the cofiber. These compatibilities allow a later algebra
attachment to determine an actual class in a Koszul object.

Define the additive map $F:I\to\Omega$ by
\[
 F(w_h^{(\lambda)})=w_{h+1}-w_1,
 \qquad F(w_h^{(\delta)})=w_h,
 \qquad F(\tau)=-w_1.
\]
Recall that $q=d_1=2(p-1)$. The ordinary degrees satisfy
\begin{equation}
 \label{layer:degree}
 d(Fi)=p\,d(\lambda)+d(\delta)-qm
 \qquad (i=(\lambda,\delta,m)).
\end{equation}
Here $\lambda$ records the source of the permutation power,
$\delta$ the ordinary scalar shift, and $m$ the Euler shift. The
map $F$ records the resulting ordinary source sphere. In particular,
for $0\leq j<n$,
\[
 F(w_j,0,-1)=w_{j+1},\qquad
 d\bigl(F(w_j,0,-1)\bigr)=p d_j+q=d_{j+1}.
\]
This is the sphere degree required for the next attaching class.
At this index a graded target is still evaluated at $p w_j$, as in
$B(X)$; $F$ specifies the source sphere.

For a graded sequence $X$, set
\[
 \begin{split}
 Q(X)_i&=\map_{\cC}
 \left(S(Fi),\bigl(X_p(p\lambda+\delta)\bigr)_p\right),\\
 \Phi(X)_i&=\map_{\cE}
 \left(W_i,J^\vee\otimes
 \Tr\bigl(X_p(p\lambda+\delta)\bigr)_p\right).
 \end{split}
\]
The stable dual $J^\vee$ is a commutative algebra by the diagonal of
$S(V)$. Thus these are graded algebras when $X$ is. We will compare
$B(X)//t$ with $\Phi(X)$, and then compute $\Phi(X)$ as the reduction
of $Q(X)$ by $\varpi$.

\subsection{The sphere with affine action}

\begin{lemma}[Acyclicity of the Borel sphere]
\label{layer:acyclic-sphere}
For every odd prime $p$, the space
$Z_p^{\mathrm{orb}}=S(V)_{hG_p}$ has the $p$-local homology of a
point. Its stabilized augmentation to the $p$-local sphere is an
equivalence.
\end{lemma}

\begin{proof}
The two-periodic resolution of $C_p$ gives
\[
 H^*(BC_p;\F_p)=\Lambda(\eta)\otimes\F_p[\xi],
 \qquad |\eta|=1,\quad |\xi|=2.
\]
The class $\xi$ is the Bockstein of $\eta$, or equivalently the
reduction of a first Chern class. Under an automorphism of $C_p$,
both classes transform with multiplier weight one, up to replacing
the character by its inverse throughout. As a $C_p$-representation,
$V$ is a sum of $(p-1)/2$ nontrivial complex lines. Its Euler class
is a nonzero scalar times $\xi^{(p-1)/2}$. The sphere-bundle sequence
therefore gives
\[
 H^*(S(V)_{hC_p};\F_p)
 =\Lambda(\eta)\otimes
 \F_p[\xi]/\bigl(\xi^{(p-1)/2}\bigr).
\]
Every positive-degree monomial has multiplier weight strictly between
zero and $p-1$. Passage to $G$ takes invariants under
$\F_p^\times$, whose order is invertible in $\F_p$, so only the
degree-zero class remains.

The $p$-local homology is finitely generated in every degree, since
the group is finite and the sphere is finite dimensional. The
universal coefficient sequence and Nakayama's lemma now imply that
its reduced $\Z_{(p)}$-homology vanishes. The augmentation cofiber is
bounded below; the stable Hurewicz theorem applied to its first
possible nonzero homotopy group makes it contractible.
\end{proof}

\begin{lemma}[Even spherical trivializations]
\label{layer:even-trivializations}
For every even integer $r$, including negative integers, the local
$p$-local sphere of $rV$ over $Z_p^{\mathrm{orb}}$ is trivializable
with fiber degree $r(p-1)$. Consequently there are coherent
$J^\vee$-module equivalences
\begin{equation}
 \label{layer:oriented-W}
 J^\vee\otimes W_i\simeq
 J^\vee\otimes\Tr S(Fi)
 \qquad(i\in I).
\end{equation}
They may be chosen to agree with the given trivial identifications
on the $\delta$ summand, and they remain equivalences after
localization.
\end{lemma}

\begin{proof}
For even $r$, the determinant character of $rV$ is trivial. After
fixing the fiber degree, its classifying map to the classifying space
of sphere automorphisms lifts to the simply connected cover. All
higher homotopy groups of this cover are $p$-local abelian groups.
By Lemma~\ref{layer:acyclic-sphere}, reduced cohomology of
$Z_p^{\mathrm{orb}}$ with every such constant coefficient group
vanishes. Induction along the Eilenberg--Mac Lane fibers of the
Postnikov tower shows that the corresponding based mapping spaces
are contractible. Mapping into the limit gives a based null of the
classifying map. Dualization handles virtual negative multiples as
well.

Maps between free twisted rank-one $J^\vee$-modules are sections of
the corresponding sphere mapping local system over
$S(V)_{hG}$. Indeed, this follows from free-module adjunction and
duality for $S(V)_+$; composition of these maps is pointwise
composition of sections. The trivializations just constructed
therefore give the required module equivalences primewise on
generators. In $W_i$, the twisting exponent of $V$ is
$d(\lambda)-2m$, which is even, and its total underlying dimension
is Equation~\eqref{layer:degree}. Apply
Lemma~\ref{found:picard-grading} in the rational category of
$J^\vee$-modules in $\Ind(\cG)$ to extend the generator
equivalences coherently over $I$. All the required generator objects
are squares. The relative assertion in that lemma keeps the
$\delta$ identifications fixed. Finally apply the symmetric
monoidal localization.
\end{proof}

In particular, free-module adjunction allows the coherent replacement
of $W_i$ by $\Tr S(Fi)$ in the definition of $\Phi(X)$. It remains
to compute maps into $J^\vee$ with trivial coefficients. We retain a
bounded-join orbit term in this computation; replacing it by full
homotopy orbits before taking the ultraproduct would lose the required
uniform information.

\subsection{The partial norm calculation}

For $U=(U_p)_p\in\cG$, define spectra
\[
 \begin{split}
 \operatorname{Part}(U)
 &=\colim_b\prod_{\U}(E_{b+}\otimes U_p)_{hG_p},\\
 \operatorname{Full}(U)
 &=\prod_{\U}U_p^{hG_p}.
 \end{split}
\]
Ultraproducts of spectra in these formulas mean filtered colimits of
products. The first construction uses the fixed finite join stages;
the second takes the full homotopy fixed points at each prime.

\begin{lemma}[The partial and full norm model]
\label{layer:norm-model}
There is a natural cofiber sequence
\begin{equation}
 \label{eq:norm-triangle}
 \operatorname{Part}(U)\longrightarrow
 \operatorname{Full}(U)\longrightarrow
 \map_{\cE}(\one,U).
\end{equation}
The first map is projection from a join stage followed by the norm.
The norm is an equivalence on induced spectra and on their finite
stable extensions. On underlying fibers, fiber inclusion followed
by norm and evaluation is the sum of the group actions.
\end{lemma}

\begin{proof}
For a finite group, the norm is the natural map from homotopy orbits
to homotopy fixed points; see
\cite[Definition~I.1.10 and Example~I.1.11]{NS2018}. One concrete
description uses the group ring spectrum $\mathcal H=S_{(p)}[G]$.
To specify its derived construction, regard $\mathcal H$ as a
$G\times G$-spectrum with action $(g,h)\cdot k=gkh^{-1}$.
It is induced from the trivial sphere along the diagonal subgroup
$\Delta G$. Finite induction agrees with coinduction, so the unit
of restriction--coinduction gives a map from the trivial
$G\times G$-sphere to $\mathcal H$. On underlying spectra this
is the diagonal into the finite product of spheres. Equivalently,
it is the $\mathcal H$-bimodule norm-element map that sums all
group elements.

Take its derived tensor over the right $\mathcal H$-action with
$U_p$. The remaining left action makes the resulting map
from the trivially acted-on spectrum $S_{(p)}\otimes_{\mathcal H}U_p$
to $U_p$ equivariant. Its adjoint is
\[
 S_{(p)}\otimes_{\mathcal H}U_p
 \longrightarrow\map_{\mathcal H}(S_{(p)},U_p).
\]
In particular, composing the inclusion of an underlying fiber with
this map and then evaluating at a point sums the group actions.
On an induced object $\mathcal H\otimes Z$, the orbit and
fixed-point adjunctions identify both sides with $Z$; the diagonal
entry at the identity element identifies the norm with
$\id_Z$. Both functors are exact, so the same
equivalence holds on finite stable extensions of induced objects;
compare \cite[Lemma~I.3.8]{NS2018}.

Apply this to $E_{b+}\otimes U_p$, whose equivariant cellular
filtration consists of free cells. The join triangle, followed by
homotopy fixed points, thus identifies its first term with homotopy
orbits. Take the ultraproduct and then the colimit in $b$.
Equation~\eqref{power:join-formula} identifies the last term, giving
Equation~\eqref{eq:norm-triangle}.
\end{proof}

\begin{lemma}[Partial Thom spectra with arbitrary coefficients]
\label{layer:partial-thom}
Let $Z=(Z_p)_p$ be any sequence of $p$-local spectra, without
connectivity or boundedness assumptions. Let $r=(r_p)_p$ be a
sequence of integers of fixed parity. If $r$ is odd, then
\[
 \operatorname{Part}(T^r\Tr Z)=0.
\]
If $r$ is even, inclusion of a fiber point at the identity vertex of
$E_b$ induces an equivalence
\[
 \prod_{\U}(T_p^{r_p}\otimes Z_p)\xrightarrow{\simeq}
 \operatorname{Part}(T^r\Tr Z),
\]
where the source has forgotten the group action.
\end{lemma}

\begin{proof}
First omit $Z_p$. The partial Thom spectrum
$(E_{b+}\otimes T_p^{r_p})_{hG}$ has cells in degrees
$0,\ldots,b-1$ relative to $r_p(p-1)$. This assertion also applies
to a virtual representation, after shifting by its virtual
dimension. Since $E_b$ is $(b-2)$-connected, its Thom homology in
relative degrees less than $b-1$ agrees with group homology with
coefficient $\F_p(\det^{r_p})$.

For fixed $b$ and sufficiently large $p$, these group homology
groups consist only of degree zero when $r_p$ is even, and vanish
in this range when $r_p$ is odd. To see this, use the cyclic-group
generators $\eta,\xi$ from Lemma~\ref{layer:acyclic-sphere} and take
multiplier invariants; the corresponding homology calculation is
its degreewise dual. The sign of the multiplier permutation by
$\alpha$ is $\alpha^{(p-1)/2}$ modulo $p$, as follows by applying
the permutation to the Vandermonde product. Thus the odd
determinant twist shifts the required multiplier weight by
$(p-1)/2$. For a fixed degree range this weight, and every positive
untwisted invariant weight, eventually lies beyond that range.

In the even case, choose the identity vertex in the first join
factor at every stage. The fiber-point inclusions then commute with
the maps from stage $b$ to stage $b+4$, and hence induce maps of
their cofibers. Take these cofibers as remainders; in the odd case,
keep the entire partial Thom spectrum. For $b\geq2$ each remainder
is a finite $p$-local cell spectrum with relative cells in degrees
at most $b-1$, and it has no $p$-local homology below degree $b-1$.
Here mod-$p$ vanishing gives $p$-local vanishing because the homology
groups are finitely generated. Stable Hurewicz after desuspension
therefore gives the same lower connectivity bound. The remainder
at $b+4$ is connective in relative degree $b+3$, so the transition
from the remainder at $b$ to the remainder at $b+4$ is null by
comparison with the source cell dimensions. This null holds at
almost all primes for each fixed $b$.

The common virtual dimension shift does not affect the argument,
even if $r_p$ varies without bound. Moreover the transition remains
null after smashing with any $Z_p$. Since $Z_p$ has trivial action,
this smash commutes with the homotopy orbit construction. The
remainder telescope is consequently zero after the ultraproduct.
The compatible fiber-point maps in the even case give the asserted
equivalence.
\end{proof}

Write $\mathcal H(Z)=\map_{\cC}(\one,Z)$ for an ordinary sequence.
We next compute the whole $J^\vee$ coefficient object, with its
multiplication rather than only its homotopy groups.

\begin{lemma}[The coefficient algebra of the layer]
\label{layer:coefficient-algebra}
For every ordinary sequence $Z$, external multiplication is an
equivalence
\begin{equation}
 \label{layer:external-product}
 \map_{\cE}(\one,J^\vee)\otimes_D\mathcal H(Z)
 \xrightarrow{\simeq}
 \map_{\cE}(\one,J^\vee\otimes\Tr Z).
\end{equation}
The unit-input algebra is equivalent, as a commutative $D$-algebra,
to $D//\varpi$. Under the point and unit identifications of the
partial and full terms, the first map of
Equation~\eqref{eq:norm-triangle} for $J^\vee\otimes\Tr Z$ is
multiplication by $(p(p-1))_p$ on $\mathcal H(Z)$.
\end{lemma}

\begin{proof}
Dualizing the first Euler triangle gives
\[
 T^{-1}\longrightarrow\one\longrightarrow J^\vee.
\]
Lemma~\ref{layer:partial-thom} and exactness identify
$\operatorname{Part}(J^\vee\Tr Z)$ with $\mathcal H(Z)$, using the
fiber point and the unit of $J^\vee$. On the other hand,
\[
 (J_p^\vee\otimes\Tr Z_p)^{hG_p}
 \simeq\map\bigl(S(V)_{hG_p+},Z_p\bigr)
 \simeq Z_p
\]
by Lemma~\ref{layer:acyclic-sphere}. Evaluation at a point is the
inverse of the constant-class equivalence. By
Lemma~\ref{layer:norm-model}, norm followed by this evaluation on
the constant-unit copy sums all group actions. It is therefore
multiplication by $|G_p|=p(p-1)$.

To check Equation~\eqref{layer:external-product}, pair the norm
triangle for $J^\vee$ with $\mathcal H(Z)$ by tensoring trivial
maps. Do this first on the partial homotopy fixed point terms,
before using their norm identifications with homotopy orbits. These
pairings are balanced over $D$, acting by trivial sphere scalars,
and give a map of the entire join triangles. Passage to the
telescope gives the localization pairing on their cofibers.

The comparisons on the first two terms are equivalences. For the
partial term, the point-and-unit class at $Z=\one$, followed by
the norm, multiplies a class from $\mathcal H(Z)$ to the
corresponding point-and-unit-norm class for $Z$, by naturality at
each prime. Lemma~\ref{layer:partial-thom}, also applied to
suspended sphere classes, identifies this generator with the free
rank-one $D$-module and identifies its external product with
$\mathcal H(Z)$. For the full term the same assertion follows from
the constant generator. Exactness gives the equivalence on
cofibers.

At $Z=\one$, both initial terms are $D$ in degree zero, and their
map is multiplication by $\varpi(p-1)_p$. The factor $(p-1)_p$ is a
unit and $\varpi$ is a nonzerodivisor. The cofiber algebra thus has
homotopy only in degree zero, equal to $D/\varpi$, and its unit
induces the quotient map. Choose a null of $\varpi$ in this
algebra. The resulting algebra map from $D//\varpi$ is an
equivalence on underlying modules, hence an algebra equivalence.
\end{proof}

The coefficient computation identifies the proposed layer after
trivializing its source spheres. To compare it with the Euler
cofiber, we must still show that the restriction from $S(2V)$ to
$S(V)$ loses nothing. The following odd-twist vanishing supplies
that step.

\begin{lemma}[Odd twists vanish in the layer]
\label{layer:odd-vanishing}
For every ordinary sequence $Z$ and every sequence of odd integers
$r=(r_p)_p$,
\[
 \map_{\cE}(\one,T^rJ^\vee\Tr Z)=0.
\]
\end{lemma}

\begin{proof}
The dual Euler triangles give
$J^\vee\simeq\Sigma T^{-1}J$. It suffices to apply
Equation~\eqref{eq:norm-triangle} to $T^{r-1}J\Tr Z$ and show that
its first map is an equivalence.

At each prime, $S(V)$ has a finite equivariant cell decomposition
with prime-to-$p$ stabilizers. One may use the boundary of the
permutation simplex: a subgroup with $p$-torsion has no fixed point
in $S(V)$. Induction from each such stabilizer is $p$-locally a
retract of free induction by Lemma~\ref{power:prime-to-p}. The
ordinary full orbit-to-fixed-point norm on
$T^{r-1}J\Tr Z$ is therefore an equivalence, by its finite
equivariant cellular filtration.

Its full homotopy orbit spectrum is identified with the underlying
fiber by inclusion of a fiber point: $r-1$ is even, so use the
trivialization of Lemma~\ref{layer:even-trivializations}, followed
by the acyclic augmentation of
Lemma~\ref{layer:acyclic-sphere}. Its partial orbit spectrum is
identified by the same point. Indeed projection $J\to\one$ and
the Euler triangle reduce that assertion to the even calculation
for $T^{r-1}\Tr Z$ and the odd vanishing for $T^r\Tr Z$ in
Lemma~\ref{layer:partial-thom}. The projection from partial to
full orbits respects these fiber-point maps, so it is an
equivalence. Composing with the full norm proves the assertion.
\end{proof}

\subsection{The multiplicative comparison and completion}

\begin{proposition}[The multiplicative Euler layer]
\label{layer:comparison}
For every sequence of graded commutative algebras $X$, there are
natural product-compatible equivalences
\begin{equation}
 \label{eq:layer}
 B(X)//t\xrightarrow{\simeq}\Phi(X)
 \xleftarrow{\simeq}Q(X)\otimes_D(D//\varpi).
\end{equation}
The right-hand comparison applies to arbitrary spectrum inputs as
well. The scalar action in the $\delta$ direction agrees with
ordinary evaluation on $Q(X)$ followed by reduction. The
comparisons are compatible with algebra units and maps of inputs.
The first map uses the tautological Euler null on $S(2V)$, restricted
to $S(V)$.
\end{proposition}

\begin{proof}
For the right-hand equivalence, replace $W_i$ coherently by
$\Tr S(Fi)$ using Lemma~\ref{layer:even-trivializations} and
free-module adjunction. Apply
Equation~\eqref{layer:external-product} with the target shifted by
$S(-Fi)$. Multiplication of the coefficient algebra at index zero
with ordinary maps from these spheres gives the comparison in
every degree and weight. The same construction respects all
products, since the sphere trivializations are coherent and the
external pairing is multiplication in the mapping algebra. The
chosen agreement on the $\delta$ summand makes its scalar action
ordinary evaluation. Naturality in targets is built into all of
these maps. In particular no algebra structure on the target is
needed for the underlying coefficient comparison.

For the left-hand equivalence, first replace $J^\vee$ by
$J_2^\vee$ in the definition of $\Phi(X)$. In that algebra, null
$t$ by the tautological Euler null on $S(2V)$, multiplied by the
target unit. Concretely the tautological nonzero vector gives a
radial path from the zero section of the disk bundle to its
boundary, which becomes the point at infinity in the representation
sphere. This is the null in the dual Euler triangle
\[
 T^{-2}\longrightarrow\one\longrightarrow J_2^\vee.
\]
The universal property of the algebra quotient gives an algebra
map from $B(X)//t$ to this $J_2^\vee$ mapping algebra.

We verify its underlying map with the null specified, since the
existence of some module equivalence would not suffice. By
Lemma~\ref{found:free-null}, its source is the cofiber of
multiplication by $t$, and the map on the added module cell is the
chosen null multiplied by the input from $B(X)_{i-\tau}$. The
coherent identifications of $W_\tau$, $W_{i-\tau}$, and $W_i$
identify multiplication by $t$ with the map obtained from
$T^{-2}\to\one$ and the target unit. The multiplied null is,
by the same unitality, the geometric null in the dual Euler
triangle tensored with the target. The original input maps are
constant along $S(2V)$. Exactness consequently identifies this
particular map with the cofiber equivalence
\begin{equation}
 \label{layer:chosen-null-cofiber}
 \cofib\bigl(B(X)_{i-\tau}\xrightarrow{t}B(X)_i\bigr)
 \simeq\map_{\cE}
 \left(W_i,J_2^\vee\otimes
 \Tr\bigl(X_p(p\lambda+\delta)\bigr)_p\right).
\end{equation}

Now restrict along the inclusion of the first copy
$S(V)\to S(2V)$. Its quotient is stably $T\otimes J$:
the open complement with nonzero second component is an open
$V$-ball times $S(V)$, and its one-point compactification gives
this description. After dualizing, the fiber of restriction
therefore has an extra factor $T^{-1}J^\vee$. Every $W_i$ has
even twisting exponent in $V$, so moving it to the target leaves
an odd twist. Lemma~\ref{layer:odd-vanishing} makes this fiber
zero, including any ordinary sphere shift. Restriction is thus an
equivalence. Its composite with
Equation~\eqref{layer:chosen-null-cofiber} proves the first map
of Equation~\eqref{eq:layer}. All choices of nulls have been
multiplied with units and transported by the stated maps, so the
construction has the claimed unit and target compatibilities.
\end{proof}

\begin{corollary}[The first Koszul layer]
\label{layer:first-koszul}
For constant input $\one$, the comparison identifies
\[
 B(\one)//t\simeq F^*h_\Omega(Y_1)
\]
as graded algebras. For $M=(\MU_{(p)})_p$ there is a compatible
identification
\[
 B(M)//t\simeq F^*h_\Omega(M\otimes Y_1).
\]
These identifications preserve the algebra maps induced by the
unit and the scalar action in the $\delta$ direction.
\end{corollary}

\begin{proof}
For constant input $X$, $Q(X)=F^*h_\Omega(X)$. The first Koszul
attachment kills $b_0=\varpi u_0$. Its quotient is base change by
$D//\varpi$: send the null of $b_0$ to $u_0$ times the null of
$\varpi$. The chosen-null cofiber description and
Proposition~\ref{found:finite-cell-morita} give this identification also
after tensoring with $M$. Scalar base change supported in weight
zero is componentwise tensor on the other input, so it commutes
with the pullback of weights by $F$. Apply
Proposition~\ref{layer:comparison} and its unit compatibility.
\end{proof}

\begin{lemma}[Derived Euler completion]
\label{layer:completion}
For a rational graded commutative algebra $C$ with a homogeneous
chain-degree-zero polynomial class $t$, define
\[
 \widehat C=\lim_{s\geq1}C//t^s,
\]
using the natural derived polynomial quotient maps. For every
$r\geq1$, the natural map induces an equivalence
\[
 \widehat C//t^r\simeq C//t^r.
\]
Thus $\widehat C$ is complete for derived reduction by powers of
$t$. Any algebra extension whose underlying $\widehat C$-module
is perfect is complete as well. In particular these statements
apply to
$\widehat B(X)=\lim_s B(X)//t^s$.
\end{lemma}

\begin{proof}
Derived reduction modulo $t^r$ is tensoring over the polynomial
algebra with its two-term perfect quotient. It commutes with
limits. After this reduction, the cones of the maps from the
initial module $C$ to its quotients modulo $t^s$ form a pro-zero
tower: their transitions are multiplications by $t$ on the
corresponding shifts, and any $r$ consecutive transitions are
null because $t^r$ has its quotient nullhomotopy. Their inverse
limit is zero. Taking the limit of the reduced quotient maps
therefore gives the asserted equivalence. Applying this for all
$r$ proves completeness of $\widehat C$. A perfect
$\widehat C$-module is dualizable; tensoring it commutes with
this limit and with the finite quotient cofibers. The same
calculation proves the final assertion.
\end{proof}
==== END Finite-Smith-Toda-Complexes-at-Varying-Primes//03-layer.tex ====

==== BEGIN Finite-Smith-Toda-Complexes-at-Varying-Primes//04-universal.tex ====
\section{Universal coefficients for successive cycles}
\label{sec:universal}

The power construction applies to commutative algebras defined at the
individual primes. Our successive Koszul algebras, however, need only
exist in the rational mapping category. We connect these settings by
constructing a universal algebra for the coefficients of finitely many
cycles. Its generators have positive formal mass. This permits both a
rational comparison at fixed weights and a calculation of the same
primewise models at weights that increase with the prime.

Fix $0\leq a<n$, and suppose that cycles $b_j$, for $0\leq j\leq a$,
have been chosen in the successive strict extensions
$A[e_0,\ldots,e_{j-1}]$ of Section~\ref{sec:foundations}.
Here $|e_j|_{\mathrm{chain}}=1$, $\wt(e_j)=w_j$, and
$\partial e_j=b_j$. In particular, $b_0=\varpi u_0$.
We write $[j)=\{0,\ldots,j-1\}$ and take every exterior monomial
$e_S$ in increasing order of its indices. All signs below are the
signs of the homological grading; the formal lattice contributes no
additional sign.

\subsection{The coefficient algebra and its filtration}

\begin{definition}\label{universal:coefficients}
The graded algebra underlying $R^a$ is the free graded-commutative
$D$-algebra on variables
\[
 z_{j,S},\qquad 0\leq j\leq a,\quad S\subseteq[j),
 \qquad |z_{j,S}|_{\mathrm{chain}}=-|S|,
 \qquad \wt(z_{j,S})=\gamma_{j,S}
       :=w_j-\sum_{h\in S}w_h.
\]
Set
\[
 b_j^u=\sum_{S\subseteq[j)}z_{j,S}e_S,
 \qquad \partial e_j=b_j^u.
\]
The differential on $R^a$ is determined by
$\partial b_j^u=0$, with coefficients compared in the exterior basis.
Define
\[
 K_-^a=R^a[e_j:j<a],\qquad K^a=R^a[e_j:j\leq a].
\]
The superscript $u$ indicates these universal cycles.
\end{definition}

For example, when $n\geq2$ the stage $a=1$ has universal cycles
\[
 b_0^u=z_{0,\emptyset},\qquad
 b_1^u=z_{1,\emptyset}+z_{1,\{0\}}e_0.
\]
The coefficient $z_{1,\{0\}}$ has chain degree $-1$, whereas the
two empty-subset coefficients have chain degree zero. The cycle
equations require
\[
 \partial z_{0,\emptyset}=\partial z_{1,\{0\}}=0,
 \qquad
 \partial z_{1,\emptyset}=z_{1,\{0\}}z_{0,\emptyset}.
\]
Indeed, differentiating $z_{1,\{0\}}e_0$ introduces a minus sign,
so these identities give $\partial b_1^u=0$. The differential on
the coefficient algebra records the condition that the prescribed
expression be a cycle after the earlier null $e_0$ has been adjoined.
An actual evaluation may send some of these coefficients to zero.

\begin{lemma}\label{universal:triangular}
Definition~\ref{universal:coefficients} determines dg algebras
$R^a$, $K_-^a$, and $K^a$. The differential of each $z_{j,S}$ is
decomposable and uses only earlier generators when the generators
are ordered by increasing $j$ and, for a fixed $j$, by decreasing
binary order on subsets of $[j)$. There is a dg algebra evaluation
\begin{equation}\label{universal:evaluation}
 R^a\longrightarrow A|_\Lambda=h_\Lambda(\one)
\end{equation}
that sends $z_{j,S}$ to the coefficient of $e_S$ in $b_j$.
The constructions and evaluations extend those at every earlier stage.
\end{lemma}

\begin{proof}
Binary order compares two subsets at their largest differing index.
In a term of $\partial e_S$, one replaces an index $h\in S$ by a
subset of $[h)$. If the resulting exterior monomial is nonzero, its
index set is strictly smaller than $S$ in binary order. Therefore the
coefficient equation for $e_U$ in $\partial b_j^u$ expresses
$\partial z_{j,U}$ in terms of coefficients $z_{j,S}$ with $S>U$
and coefficients $z_{h,T}$ with $h<j$. The signs are fixed by
\[
 \partial(z_{j,S}e_S)
 = (\partial z_{j,S})e_S
    +(-1)^{|S|}z_{j,S}\partial e_S.
\]
Each nonzero summand in the expression for $\partial z_{j,U}$
contains one coefficient of each of these two kinds. It is thus
decomposable, with the required triangular ordering.

Inductively suppose that $\partial^2 e_h=0$ for $h<j$.
The square of an odd derivation is a derivation, since its mixed
terms cancel. Applying $\partial$ to $\partial b_j^u=0$ consequently
gives
\[
 0=\partial^2 b_j^u
   =\sum_{S\subseteq[j)}(\partial^2 z_{j,S})e_S.
\]
Independence of the exterior basis gives
$\partial^2z_{j,S}=0$ for every $S$, and the defining cycle equation
gives $\partial^2e_j=0$. This proves the assertion by induction.
The actual coefficients of $b_j$ satisfy precisely these equations,
so they define Equation~\eqref{universal:evaluation}; in particular
$z_{0,\emptyset}$ maps to $\varpi u_0$. All equations for an earlier
stage are retained when a new value of $j$ is added.
\end{proof}

Define an additive mass function on $\Lambda$ by
$\sigma(w_i)=3^i$. Every displayed generator has strictly positive
integral mass, since
\begin{equation}\label{universal:positive-mass}
 \sigma(\gamma_{j,S})
 \geq 3^j-\sum_{h<j}3^h=\frac{3^j+1}{2}>0,
 \qquad \sigma(w_j)=3^j>0.
\end{equation}
Give each generator $z_{j,S}$ and $e_j$ length one. We filter
$R^a$, $K_-^a$, and $K^a$ decreasingly by length: $\mathcal F_s$
is spanned by monomials of length at least $s$. At a fixed weight
$\gamma$, the filtration vanishes for $s>\sigma(\gamma)$ and is
constant for $s\leq0$. In particular it is bounded at each weight.

\begin{lemma}\label{universal:filtered-cells}
The length filtrations are dg filtrations. The associated graded
algebra $\gr R^a$ has zero differential, and in $\gr K^a$ one has
\[
 \partial e_j=z_{j,\emptyset}.
\]
As a filtered $R^a$-module, $K^a$ is built by finitely many free
cells indexed by the subsets $S\subseteq\{0,\ldots,a\}$, in
increasing binary order. The cell $e_S$ has chain degree $|S|$,
weight $\sum_{h\in S}w_h$, and filtration order $|S|$.
In particular, $K^a$ and $K_-^a$ are perfect $R^a$-modules.
\end{lemma}

\begin{proof}
The differential of a coefficient generator increases length,
by Lemma~\ref{universal:triangular}. The summand
$z_{j,S}e_S$ of $\partial e_j$ has length $1+|S|$, so only
$S=\emptyset$ contributes at the original length one.
For the module cell $e_S$, differentiating one factor $e_h$
and using the term indexed by $T\subseteq[h)$ gives total length
$|S|+|T|$. This is at least $|S|$, and the resulting exterior
basis vector is earlier in binary order. Thus the differential
is a filtered attachment to earlier module cells. There are
$2^{a+1}$ basis vectors, including the initial one indexed by
the empty set.
\end{proof}

\subsection{Filtered models at the primes}

The primewise models have two distinct roles. The coefficient algebras
$R_p^a$ will be actual commutative algebras, so their permutation powers
are defined. The objects $K_p^a$ will be finite-cell modules over them;
their role is to test vanishing at weights growing with $p$. They need
no primewise algebra structure. The rational Koszul algebra $K^a$
already has a commutative algebra structure, which will supply the
Amitsur construction in Section~\ref{sec:normalization}.

We retain the positive-mass filtration in both primewise models.
A filtered object is a diagram
$\mathcal F_s\to\mathcal F_{s-1}$ indexed by $s\in\Z$, with
convolution tensor product along addition. A step at order $s$
is constant at indices at most $s$ and zero at larger indices.
The tensor product of steps at orders $s$ and $t$ is a step at
order $s+t$.

We use filtrations supported in the weight monoid generated by
the relevant generator weights, constant below order zero,
and zero at orders greater than $\sigma(\gamma)$ in weight
$\gamma$. These conditions are preserved by colimits and tensor
products. For tensor products, one may decompose a bounded
filtration in each weight into finitely many extensions of
steps, starting at its highest nonzero order. On steps the
assertion follows from additivity of mass and of filtration order.

\begin{lemma}\label{universal:associated-graded}
On filtered spectra, the functor
\[
 (\gr\mathcal F)_s
   =\cofib(\mathcal F_{s+1}\longrightarrow\mathcal F_s)
\]
preserves colimits and is symmetric monoidal. It therefore preserves
free commutative algebras and algebra pushouts. The comparison
\[
 h_\Lambda(\gr X_p)\simeq\gr h_\Lambda(X_p)
\]
respects the algebra and module pairings whenever they are present.
\end{lemma}

\begin{proof}
The associated graded pairing is obtained by taking the cofiber
of the pushout-product of the filtration arrows. On steps this
gives the claimed tensor equivalence. Steps generate the filtered
category under colimits, and both constructions preserve colimits,
so the equivalence holds in general, with its symmetric monoidal
compatibilities. The assertions for free algebras and their
pushouts follow.

For the last assertion, fix a finite tuple of weights and
filtration orders. The tensor cube for the filtration arrows
maps to the arrow at the sum of those orders: every vertex
with at least one order increased maps into filtration at
least one higher than that sum. Taking the cube's
pushout-product and cofiber constructs the associated graded
pairing. The exact lax pairings defining $h_\Lambda$ commute
with these finite cofibers and produce the same construction.
This verifies the assertion for all finite tuples, together
with the compatibility under composition of pairings. It
does not require $h_\Lambda$ to commute with infinite
filtration colimits.
\end{proof}

\begin{proposition}\label{universal:primewise-models}
For the chosen cycles through $b_a$, there exist the following data
as germs along $\U$:
\begin{enumerate}
\item Filtered $\Lambda$-graded commutative ring spectra $R_p^a$,
with an equivalence
$R^a\simeq h_\Lambda(R_p^a)$ at every filtration order, compatible
with the filtered algebra structures.
\item Maps of whole unfiltered graded commutative algebras
$R_p^a\to S_{(p)}$, where the sphere is constant in the weight
grading, together with a homotopy identifying their normalized
evaluation with Equation~\eqref{universal:evaluation}.
\item Filtered finite-cell $R_p^a$-modules $K_p^a$, with an
equivalence $K^a\simeq h_\Lambda(K_p^a)$ of filtered modules.
\end{enumerate}
The algebra models, their comparisons, and the evaluation
homotopies may be chosen relative to the preceding stage.
Thus the maps $R^{a-1}\to R^a$ and $R_p^{a-1}\to R_p^a$
and their evaluations form compatible diagrams of algebras.
No algebra structure is required on $K_p^a$.
\end{proposition}

\begin{proof}
We first construct the algebras in the triangular generator order
of Lemma~\ref{universal:triangular}. Start at the unit, in weight
and filtration order zero. Suppose that the preceding generators
have been constructed, and let $z=z_{j,S}$, $r=|S|$, and
$\gamma=\gamma_{j,S}$. Its boundary $\partial z$ is a cycle of
chain degree $-r-1$, weight $\gamma$, and filtration at least two.
The comparison already established at this fixed weight and
filtration lifts the boundary to maps
\begin{equation}\label{universal:attaching-map}
 S^{d(\gamma)-r-1}
    \longrightarrow(\mathcal F_2R_p^{\mathrm{old}})(\gamma),
\end{equation}
together with a path matching the normalized class with
$\partial z$. Compose with the map to $\mathcal F_1$ and attach
a free commutative algebra cell that nulls this map. Explicitly,
push out the free algebra on the indicated sphere, placed as
a step at order one, along its augmentation to the unit.
The support and filtration bounds are preserved.

After taking associated graded, the attaching map in order one
is null by its specified factorization through $\mathcal F_2$.
Lemma~\ref{universal:associated-graded} identifies the result
with free adjunction of the sphere
$S^{d(\gamma)-r}$ in weight $\gamma$ and order one. Repeating
this construction gives a primewise free algebra on all the
generator spheres as the associated graded of $R_p^a$.

We verify carefully that the comparison with the filtered dg
algebra extends at each step. Over the $\Q$-algebra $D$, free
algebras on free step generators are computed by ordinary
graded symmetric powers, with their step orders added. For
a cycle generator $x$, the free disk algebra with a generator
$z$ satisfying $\partial z=x$, both placed at order one,
resolves the filtered unit over the free cycle algebra.
Indeed every component of positive word length is
contractible. If the null generator is even, the needed
contraction divides by its positive multiplicity, which is
invertible in $D$. Moreover, filtering the disk algebra by
the number of null generators gives successive free step
modules over the cycle algebra. The disk therefore computes
the derived relative tensor, and its base change is exactly
the extension by $z$ with the specified differential and
length filtration.

These computations use derived tensor products. To see
directly that the strict monomial models compute them,
order monomials at each weight by their highest differing
generator multiplicity in the triangular list. There are
finitely many such monomials by positive mass, and their
differentials use earlier ones. For $K^a$, place the $e_j$
after all coefficient generators, in increasing $j$.
The resulting filtered complexes are built from free
$D$-module steps. Their strict tensor pairings thus
represent the derived pairings as well.

The old comparison and the tautological null in the spectral
attachment now induce a map from the dg pushout to the
normalized spectral pushout. We use the chosen matching
path from Equation~\eqref{universal:attaching-map} inside
$\mathcal F_2$. In order one modulo order two, the new
generator is the difference of two nulls: the attached null
and the null obtained by projecting the boundary from
$\mathcal F_2$. The matching path respects the latter null.
Consequently the comparison sends the new associated graded
generator to the spectral generator supplied by the attachment.
Taking cofibers and projecting the filtration commute with
normalized Hom, so this is also an identification of the
chosen generator classes, not merely of the objects.

At any fixed weight, the associated graded spectral algebra
is a finite sum of extended powers of the generator spheres,
with arities bounded by the mass of that weight independently
of $p$. For sufficiently large $p$, the symmetric groups
in these terms have invertible order. An even sphere
repeated $k$ times contributes a sphere of the summed degree:
the fiber inclusion into its homotopy orbits is an equivalence
after $p$-localization. An odd sphere repeated at least twice
contributes zero. To justify both assertions, its extended
power is the Thom spectrum of the permutation sphere over
$B\Sigma_k$, and its ordinary homology is the group homology
with the permutation orientation character. When
$p\nmid k!$, higher group homology vanishes. The degree-zero
coinvariants are $\Z_{(p)}$ in the even case and zero in the
odd case, since a transposition acts by $-1$ and $2$ is a
unit. The spectra are bounded below after translating the
sphere degree, so the homology statements imply the claimed
equivalences by stable Hurewicz.

Products of distinct generator factors are treated separately.
Multiplication of their generator classes uses the fiber
inclusions. The coherent sphere grading and
$\map_{\cC}(\one,\one)=HD$ identify their normalized Hom
with the expected free graded-commutative basis over $D$.
The extra chain shifts are fixed at each fixed weight, so
the fixed-stem calculation applies. Our comparison is
therefore an equivalence on associated gradeds, and hence
on the bounded filtration at every weight. This completes
the induction constructing the filtered algebra comparison.

It remains to realize the evaluation, including its specified
nulls. If $(X_p)$ is any sequence of graded commutative
algebras, there is a comparison
\begin{equation}\label{universal:algebra-mapping}
 \prod_{\U}\Map_{\CAlg^\Lambda}(R_p^a,X_p)
 \longrightarrow
 \Map_{\CAlg_D^\Lambda}(R^a,h_\Lambda(X_p)),
\end{equation}
where the source uses maps of whole graded systems, after
forgetting filtration. This is an equivalence by induction
over the finite list of free algebra cells. For one cell,
extending a map means choosing a null of the image of its
attaching map. Its mapping space is the corresponding
homotopy pullback of mapping spaces from the generator
sphere at its specified weight. Those sphere mapping
spaces agree by the definition of $h_\Lambda$. The initial
unit case agrees as well, and finite homotopy pullbacks
commute with the ultraproduct. The comparison constructed
above sends attached-null data to the attached-null data
in these pullbacks, since it used both the tautological
null and the specified matching path.

Forgetting filtration in this argument does preserve the
free-cell pushouts: it is colimit towards decreasing
filtration order, a colimit-preserving symmetric monoidal
functor, as one sees on steps. Here it takes the constant
value at orders at most zero. Apply
Equation~\eqref{universal:algebra-mapping} to the constant
graded sphere target and the evaluation in
Equation~\eqref{universal:evaluation}. It supplies the
required primewise evaluations and the matching homotopy.
The same comparison commutes with restriction to earlier
cells. It is therefore an equivalence on the homotopy fibers
over already chosen maps, including their comparison paths.
The new evaluation can thus extend the preceding evaluation
with its chosen compatibility, which proves the assertion
about diagrams across stages.

Finally, construct $K_p^a$ using the finite filtered module
cells from Lemma~\ref{universal:filtered-cells}. A free module
cell over $R_p^a$ with generator of weight $\gamma$, order
$s$, and ordinary degree $d(\gamma)+u$, for fixed $u\in\Z$,
normalizes to the corresponding free module shift over
$h_\Lambda(R_p^a)$ with chain degree $u$. This uses the
coherent sphere grading. A boundary is, by adjunction, a
map from one sphere into the specified weight and
filtration of the preceding module. Lift that map and
choose a path matching the boundary. Taking its cofiber
extends the module comparison. Exactness proves the
comparison at each stage and gives the compatible
associated graded comparison.
\end{proof}

\subsection{A comparison of whole primewise modules}

The fixed-weight equivalences just proved do not, by themselves,
control a weight that depends on $p$. We now obtain a stronger
statement using the finite number of module cells. This is the
point at which the whole graded systems in
Proposition~\ref{universal:primewise-models} are essential.

\begin{proposition}\label{universal:whole-module}
On one $\U$-large set of primes there is an equivalence of whole
$(\text{weight},\text{order})$-graded $\gr R_p^a$-modules
\begin{equation}\label{eq:whole-module}
 \gr K_p^a\simeq
 \bigotimes_{j\leq a}^{\gr R_p^a}
 \cofib\bigl(\text{multiplication by }z_{j,\emptyset}
                         \text{ on }\gr R_p^a\bigr).
\end{equation}
Each multiplication map has source the free module whose
generator has weight $w_j$, order one, and ordinary degree
$d_j$. On the right, $\gr R_p^a$ is the free graded algebra
of all the generator spheres, with the splittings constructed
in Proposition~\ref{universal:primewise-models}.
\end{proposition}

\begin{proof}
After applying $h_\Lambda$, the right side becomes the tensor
of two-term cofibers for the classes $z_{j,\emptyset}$ in
$\gr R^a$. The tensor comparison is an equivalence for
these finite free module cells: it is the free-shift
identification for one cell and extends by finite cofibers.
This is the Koszul module with differential
$\partial e_j=z_{j,\emptyset}$, hence agrees with the
normalized left side by
Lemma~\ref{universal:filtered-cells}. A change in a sphere
identification can multiply a chosen basis element by a
unit and does not change this equivalence.

To lift that equivalence, consider the ultraproduct of
categories of whole $(\text{weight},\text{order})$-graded
modules over $\gr R_p^a$. There is a natural comparison
from its mapping spectra to mapping spectra of graded
modules over $h_\Lambda(\gr R_p^a)$. For a free source
at one fixed weight, order, and extra integer suspension,
both mapping spectra are exactly the same normalized
evaluation of the target at that weight and order.
Induction through cofiber sequences proves full faithfulness
for sources built from finitely many such free shifts,
against any whole-system target.

Both modules in Equation~\eqref{eq:whole-module} satisfy
this finiteness condition. For the left side, take
associated graded of the finite filtered cell construction.
For the right side, expand the tensor of the finitely many
two-term cofibers. Their formal weight and order shifts and
their extra integer suspensions are fixed independently
of $p$. Full faithfulness lifts the normalized equivalence,
its inverse, and the two homotopies identifying their
composites with the respective identities. These are
finitely many maps and homotopies of whole systems. After
restricting to one $\U$-large prime set, they are inverse
equivalences at each prime as whole graded module maps.
Thus the same prime set works for all weights; no
intersection over the lattice of weights is taken.
\end{proof}

Equation~\eqref{eq:whole-module} lets us examine the same modules
at $p$-dependent weights. The next section estimates these
weights by analyzing the extended powers of each generator
sphere and the multiplication cofibers of the empty-subset
generators.
==== END Finite-Smith-Toda-Complexes-at-Varying-Primes//04-universal.tex ====

==== BEGIN Finite-Smith-Toda-Complexes-at-Varying-Primes//05-connectivity.tex ====
\section{Connectivity at weights growing with the prime}
\label{sec:connectivity}

Fix a stage $0\leq a<n$ and the universal coefficient algebra and
finite-cell module constructed in Section~\ref{sec:universal}.
Equation~\eqref{eq:whole-module} describes $\gr K_p^a$ as a whole
graded module on a set of primes in $\U$. We will use that description
at the moving weight $p\lambda+\delta$, where $\lambda,\delta\in\Lambda$
are fixed. The goal is a lower degree bound with explicit linear
dependence on $p$ after subtracting the ordinary sphere degree
$d(p\lambda+\delta)$.
The resulting vanishing, after tensoring with $K^a$, is the input to
the normalization construction in Section~\ref{sec:normalization}.

We say that a spectrum belongs to $\Sp_{\geq L}$ if its homotopy
groups vanish in degrees strictly below $L$. All spectra in the
primewise calculations below are $p$-local. Sphere degrees may be
negative; the arguments apply to virtual Thom spectra after an
ordinary integer suspension.

\subsection{Symmetric groups and multiplication cofibers}

For an integer $d$, write
\[
 \Sym^k(S^d)=\big((S^d)^{\otimes k}\big)_{h\Sigma_k}.
\]
This is the Thom spectrum of the virtual bundle $d\rho_k$ over
$B\Sigma_k$, where $\rho_k$ is the real permutation representation.
Its virtual rank is $kd$, and its mod-$p$ orientation character is
$\operatorname{sgn}^d$. Thus its mod-$p$ cohomology, with degrees
translated by $kd$, is group cohomology with trivial coefficients
when $d$ is even and with sign coefficients when $d$ is odd.

\begin{lemma}\label{conn:stable-elements}
Let $p$ be odd, let $0\leq k<p^2$, and write $k=lp+b$ with
$0\leq b<p$. Let $H=(C_p)^l\subseteq\Sigma_k$ act by translations
on $l$ disjoint blocks of $p$ letters, and let $N$ be its normalizer.
For either the trivial or the sign module $M$ over $\F_p$, restriction
identifies
\[
 H^*(\Sigma_k;M)
 \xrightarrow{\ \cong\ }
 H^*(H;M)^{N/H}.
\]
Here the normalizer acts both by conjugation on $H$ and by its given
action on $M$. Moreover,
\[
 N/H\cong
 \big((\F_p^\times)^l\rtimes\Sigma_l\big)\times\Sigma_b.
\]
\end{lemma}

\begin{proof}
The exponent of $p$ in $k!$ is $l$, since $k<p^2$.
Consequently $H$ is a Sylow subgroup. Its nontrivial orbits are
exactly the $p$-element blocks, so a normalizing permutation permutes
these blocks and the $b$ remaining letters separately. On each
block the normalizer of translations consists of affine maps
$x\mapsto ux+v$. Dividing out the translations gives the displayed
normalizer quotient. Its order is $(p-1)^l l!b!$, which is prime
to $p$ because $l,b<p$.

We recall the transfer identities needed here, including their
effect on coefficients. Cohomological transfer is obtained by
summing a cochain over representatives of the relevant finite
coset set, using the group action to identify the coefficient
fibers. It follows that transfer after restriction from a group to
a subgroup is multiplication by the subgroup index. For the
opposite composite, decomposing that coset set into subgroup orbits
gives the double-coset formula: each double coset contributes
restriction to the corresponding intersection, conjugation with
its coefficient action, and transfer from that intersection.
These descriptions also give the projection formula for transfer
and cup products.

Since $[\Sigma_k:H]$ is prime to $p$, transfer after restriction
proves that restriction to $H$ is injective. Its image is invariant
under the normalizer: conjugation together with the coefficient
action acts trivially on cohomology of the full group, and
restriction respects this action.

To prove that every invariant is in the image, consider the other
transfer composite. The restriction of either $M$ to $H$ is trivial:
each $p$-cycle is even. If $L$ is a proper subgroup of $H$, it is a
proper linear subspace of the elementary abelian group $H$. A
linear retraction $H\to L$ shows that
$H^*(H;\F_p)\to H^*(L;\F_p)$ is surjective. Choose a basis of the
one-dimensional, trivial $H$-module $M$. Every class with
coefficients in $M$ over $L$ is then the restriction of a class
over $H$. The projection formula gives
\[
 \operatorname{tr}_L^H\operatorname{res}_L^H(x)
   =x\,[H:L]=0.
\]
Hence transfer from every proper intersection subgroup in the
double-coset formula vanishes, in all degrees. An intersection
$H\cap gHg^{-1}$ equals $H$ exactly when $g$ normalizes $H$.
The remaining terms are therefore
\[
 \operatorname{res}_H^{\Sigma_k}
 \operatorname{tr}_H^{\Sigma_k}(x)
   =\sum_{\bar g\in N/H}\bar g\cdot x.
\]
For an invariant $x$ this is $|N/H|x$. The scalar is invertible in
$\F_p$, proving surjectivity onto the invariants.
\end{proof}

For even $d$ define the multiplication cofibers
\begin{equation}\label{conn:cofiber-definition}
 T_{p,0}(d)=S^0,\qquad
 T_{p,k}(d)=\cofib\left(
 S^d\otimes\Sym^{k-1}(S^d)
 \longrightarrow\Sym^k(S^d)\right)\quad(k\geq1).
\end{equation}
The arrow is multiplication in the free commutative algebra on
$S^d$. In the homotopy-orbit description of its symmetric powers,
this multiplication comes from the block inclusion
$\Sigma_1\times\Sigma_{k-1}\subseteq\Sigma_k$. Thus the arrow in
Equation~\eqref{conn:cofiber-definition} is induced by
$B\Sigma_{k-1}\to B\Sigma_k$, with the compatible Thom bundles.
In particular its map on cohomology is restriction. This
identification specifies the map itself and its bottom-degree
normalization; no transfer or factor of $k$ is inserted.

\begin{proposition}\label{conn:symmetric}
Fix a positive integer $C$. For every odd prime $p>C$, every
$0\leq k\leq Cp$, and every integer $d$, one has
\[
 \Sym^k(S^d)\in\Sp_{\geq kd}.
\]
If $d$ is odd, there is the additional bound
\begin{equation}\label{conn:odd-bound}
 \Sym^k(S^d)\in\Sp_{\geq kd+k-(2C+1)}.
\end{equation}
If $d$ is even, the objects in
Equation~\eqref{conn:cofiber-definition} satisfy
\begin{equation}\label{conn:cofiber-bound}
 T_{p,k}(d)\in\Sp_{\geq kd+2k-3C}.
\end{equation}
More precisely, writing $k=lp+b$ with $0\leq b<p$, an odd-degree
extended power is contractible if $b\geq2$, and otherwise its
cohomology above its rank begins no earlier than $l(p-2)$.
For even $d$ and $k>0$, the multiplication cofiber is contractible
if $b>0$, and if $b=0$ its cohomology above its rank begins no
earlier than $l(2p-3)$.
\end{proposition}

\begin{proof}
The Thom spectrum has a cellular filtration with its cells in
degrees $kd+j$, where $j\geq0$ is a base-cell degree. This proves
the first bound, including for virtual bundles. Since $p>C$ and
$k\leq Cp$, we have $k<p^2$ and $l\leq C<p$, so
Lemma~\ref{conn:stable-elements} applies.

The two-periodic cyclic-group resolution, together with its
periodicity class and Bockstein, gives
\[
 H^*(C_p;\F_p)=\Lambda(\eta)\otimes\F_p[\xi],
 \qquad |\eta|=1,\quad |\xi|=2.
\]
Both $\eta$ and $\xi$ transform with multiplier weight one under
$\F_p^\times$, or both with its inverse if the conjugation
convention is reversed. The periodic resolution gives the
polynomial degree-two class, the Bockstein identifies its
multiplier action with that of the degree-one class, and
$\eta^2=0$ follows from graded commutativity since $p$ is odd.

The sign of multiplication by $u\in\F_p^\times$ on the $p$
letters of a block is $u^{(p-1)/2}$, viewed in $\F_p$.
Indeed, applying the permutation to the Vandermonde product on
the $p$ field elements multiplies that product both by its sign
and by $u^{p(p-1)/2}=u^{(p-1)/2}$. Translations have sign one.
A block transposition has sign $(-1)^p=-1$, and permutations of
the remaining $b$ letters have their ordinary sign.

Suppose first that $d$ is odd. If $b\geq2$, a transposition of
the remaining letters acts trivially on $H$ and by $-1$ on the
coefficient line. The normalizer invariants are consequently zero
in every degree. If $b=0$ or $1$, invariance under the independent
multipliers on each block requires multiplier weight
$(p-1)/2$ modulo $p-1$ in each factor of
$H^*(H;\F_p)$. The least degree with this weight is $p-2$,
represented by $\eta\xi^{(p-3)/2}$; the least even degree is
$p-1$. Reversing the multiplier convention gives the same
condition. Therefore the normalizer invariants have no classes
below degree $l(p-2)$. The further block-permutation condition
can only remove classes. Since $b\leq1$ in this case,
\[
 l(p-2)=k-(2l+b)\geq k-(2C+1).
\]
This gives the cohomological vanishing required for
Equation~\eqref{conn:odd-bound}.

For even $d$, put
\[
 U=H^*(C_p;\F_p)^{\F_p^\times},\qquad
 V=\ker(U\longrightarrow\F_p).
\]
The least positive degree in $U$ is $2p-3$, represented by
$\eta\xi^{p-2}$. By Lemma~\ref{conn:stable-elements}, trivial
coefficients give
\[
 H^*(\Sigma_k;\F_p)\cong(U^{\otimes l})^{\Sigma_l}.
\]
The permutations of the tensor factors here include the Koszul
signs of their cohomological degrees. The leftover $\Sigma_b$
acts trivially and imposes no further condition.

We now compute restriction to $\Sigma_{k-1}$. If $b>0$, choose
the fixed letter outside the $p$-blocks. Both symmetric groups
then have the same Sylow subgroup $H$, and their normalizer
invariants are the same: the only change is from $\Sigma_b$ to
$\Sigma_{b-1}$ on the unused letters. The restriction is an
isomorphism, since its further restriction to $H$ is the identity.

If $b=0$ and $k>0$, choose the fixed letter in the last block.
A Sylow subgroup of $\Sigma_{k-1}$ consists of the first $l-1$
blocks. Naturality of restriction and its injectivity on these
Sylow subgroups identify the restriction map with
\begin{equation}\label{conn:restriction}
 (U^{\otimes l})^{\Sigma_l}
 \longrightarrow (U^{\otimes(l-1)})^{\Sigma_{l-1}},
 \qquad \id^{\otimes(l-1)}\otimes\epsilon,
\end{equation}
where $\epsilon:U\to\F_p$ is augmentation.
To determine this map without a choice of normalization, decompose
$U=\F_p\oplus V$. A tensor with exactly $r$ entries from $V$
has its nonunit entries in an $r$-element subset of $\{1,\ldots,l\}$.
An invariant on all such subsets is uniquely determined by its
component on one subset, which belongs to
$(V^{\otimes r})^{\Sigma_r}$. Conversely, place such a component
in each $r$-element subset and sum the resulting tensors, without
dividing the sum by its number of terms. This gives all the
invariants and respects the graded signs. Applying augmentation
to the last factor kills exactly the summands whose subset
contains that factor. For $r<l$, the remaining sum is precisely
the corresponding sum over the $r$-element subsets of
$\{1,\ldots,l-1\}$. For $r=l$, every summand is killed.
Consequently Equation~\eqref{conn:restriction} is surjective and
its kernel is $(V^{\otimes l})^{\Sigma_l}$. Every class in this
kernel has degree at least $l(2p-3)$.

The Thom isomorphism is compatible with the block inclusion
defining $T_{p,k}(d)$: on $\Sigma_{k-1}$ the bundle $d\rho_k$
restricts to $d\rho_{k-1}$ plus a trivial bundle of rank $d$.
Both Thom spectra therefore have rank $kd$. The preceding
restriction maps are surjective in every degree. The cohomology
long exact sequence of their cofiber identifies its cohomology
with the kernel of restriction in the same degree, without an
additional degree shift. Thus it vanishes entirely if $b>0$;
if $b=0$ its least possible degree above $kd$ is
\[
 l(2p-3)=2k-3l\geq2k-3C.
\]
The case $k=0$ of Equation~\eqref{conn:cofiber-bound} follows
directly from $T_{p,0}(d)=S^0$.

For completeness, these mod-$p$ calculations imply the stated
homotopy bounds, not just cohomology bounds. Classifying spaces
of the finite groups used above have finitely many cells in
each degree, and the virtual Thom spectra are bounded below
after suspension. Their $p$-local integral homology groups,
and those of the indicated cofibers, are finitely generated in
each degree. Cohomology vanishing over the field $\F_p$ gives
homology vanishing in the same range. The universal coefficient
sequence then shows that each integral homology group in that
range has zero quotient modulo $p$. Finite generation over
$\Z_{(p)}$ forces that group to vanish. Finally, for a
bounded-below spectrum, the first nonzero homotopy group, if
there is one below the claimed bound, maps isomorphically to
the first nonzero integral homology group by the stable
Hurewicz theorem. This would contradict the integral homology
vanishing. The same argument in every degree proves
contractibility in the cases where all mod-$p$ cohomology
vanishes.
\end{proof}

\subsection{The bound for the universal Koszul module}

The estimates in Proposition~\ref{conn:symmetric} now apply
factor by factor to Equation~\eqref{eq:whole-module}. The crucial
point is that the number of variable types is fixed, although
their multiplicities may grow with the prime.

\begin{proposition}\label{conn:estimate}
Fix $0\leq a<n$ and $\lambda,\delta\in\Lambda$,
and set
\begin{equation}\label{conn:constants}
 C=\max\{0,\sigma(\lambda)\}+|\sigma(\delta)|+1,
 \qquad N_a=2^{a+1}-1,
 \qquad E=N_a(5C+1).
\end{equation}
On a set of primes in $\U$, and for all its sufficiently large
members, the underlying unfiltered module satisfies
\begin{equation}\label{conn:module-bound}
 K_p^a(p\lambda+\delta)
 \in\Sp_{\geq d(p\lambda+\delta)
                  +2\nu(p\lambda+\delta)-E}.
\end{equation}
The constants are independent of the individual multiplicities
occurring at this weight and of the prime.
\end{proposition}

\begin{proof}
Write $\gamma=p\lambda+\delta$. Restrict to the set of primes
where Equation~\eqref{eq:whole-module} holds for whole graded
modules. If $\gamma$ is outside the support monoid of the
universal model, its component is zero and there is nothing to
prove. Otherwise decompose the free algebra $\gr R_p^a$ as the
tensor product of the free algebras on the individual generator
spheres. There are
$\sum_{j=0}^a2^j=N_a$ such generators.

For a nonempty subset $S\subseteq[j)$, set $r=|S|$ as before.
The factor with multiplicity $k=k_{j,S}$ is
\[
 \Sym^k\big(S^{d(\gamma_{j,S})-r}\big).
\]
For $S=\varnothing$, the corresponding quotient factor in
Equation~\eqref{eq:whole-module} has multiplicity-$k$ component
$T_{p,k}(d_j)$. Here multiplicity includes the generator in
the source of the multiplication map, or equivalently the
possible Koszul cell, as well as the ordinary polynomial
copies. Its filtration order is $k$ on both sides of that
map. Distinct factors tensor together, and the weight-$\gamma$
component is the finite sum of the terms satisfying
\begin{equation}\label{conn:weight-identity}
 \sum_{j,S}k_{j,S}\gamma_{j,S}=\gamma.
\end{equation}

Every generator weight has positive integral mass. Taking mass
in Equation~\eqref{conn:weight-identity} therefore gives
\[
 0\leq k_{j,S}\leq\sigma(\gamma)
     =p\sigma(\lambda)+\sigma(\delta)\leq Cp.
\]
This also proves finiteness of the sum at each prime. For
$p>C$, Proposition~\ref{conn:symmetric} applies to every
factor. Since $d(\gamma_{j,S})$ is even, the odd-degree
improvement applies exactly when $r$ is odd. For $r=0$, the
multiplication-cofiber improvement applies instead.
Tensoring lower degree bounds adds them: after desuspending
each factor to a connective spectrum, their tensor product is
connective. The bound for each tensor term, relative to
$d(\gamma)$, is consequently at least
\begin{equation}\label{conn:sum-bound}
 -\sum_{j,S}r k_{j,S}
 +\sum_{r\text{ odd}} k_{j,S}
 +2\sum_{r=0}k_{j,S}-E.
\end{equation}
Indeed the odd factors lose at most $2C+1$ each, the
multiplication cofibers lose at most $3C$ each, and there are
at most $N_a$ factors; the displayed value of $E$ bounds their
total loss, including factors of multiplicity zero.

The elementary inequality
\[
 -r+\mathbf 1_{\{r\text{ odd}\}}
       +2\mathbf 1_{\{r=0\}}\geq2(1-r)
 \qquad(r\geq0)
\]
and the identity $\nu(\gamma_{j,S})=1-r$ turn
Equation~\eqref{conn:sum-bound} into the lower bound
$2\nu(\gamma)-E$. Thus every associated graded piece at
weight $\gamma$ lies in the range asserted in
Equation~\eqref{conn:module-bound}.

At this weight the filtration is zero above
$\sigma(\gamma)$ and constant in filtration degrees at most
zero. Starting at its zero top term, the cofiber sequences
relating successive filtration terms and associated graded
pieces prove the same bound for the unfiltered component.
This uses only closure of $\Sp_{\geq L}$ under extensions.
The filtration length is finite at each prime; no uniform
bound on that length is required.
\end{proof}

\subsection{Vanishing after the Koszul-module test}

Recall that $i=(\lambda,\delta,m)\in I$, that $q=2(p-1)$,
and that the layer comparison uses
\[
 d(Fi)=d(p\lambda+\delta)-qm.
\]
The following consequence retains the strict inequality in
$m$ that will determine the normalization range.

\begin{lemma}\label{conn:degree-gap}
For every fixed $i=(\lambda,\delta,m)$ with
$m>-\nu(\lambda)$, one has
\[
 Q(K_p^a)_i=0,\qquad \Phi(K_p^a)_i=0.
\]
Here $Q$ and $\Phi$ are applied to the primewise underlying
graded modules, as allowed by the arbitrary-object part of
the layer comparison.
\end{lemma}

\begin{proof}
Let $s\in\Z$ be any fixed homotopy degree. The $s$th homotopy
group of $Q(K_p^a)_i$ is computed by maps from the sphere of
ordinary degree $d(Fi)+s$ to $K_p^a(p\lambda+\delta)$.
Subtracting this requested degree from the lower bound of
Proposition~\ref{conn:estimate} gives
\begin{align*}
 &d(p\lambda+\delta)+2\nu(p\lambda+\delta)-E
       -\big(d(Fi)+s\big)\\
 &\hspace{1cm}=
 2p\big(\nu(\lambda)+m\big)
       +2\nu(\delta)-2m-E-s.
\end{align*}
Because $\nu(\lambda)+m$ is a positive integer, this expression
tends to $+\infty$ with $p$. The requested primewise homotopy
group therefore vanishes on a set in $\U$. It follows that
$\pi_s Q(K_p^a)_i=0$. This argument applies separately to
every fixed $s$, so the mapping spectrum is zero. The
arbitrary-object base-change comparison of
Equation~\eqref{eq:layer} identifies
\[
 \Phi(K_p^a)_i\simeq
 Q(K_p^a)_i\otimes_D(D//\varpi),
\]
which proves the second assertion.
\end{proof}

\begin{proposition}\label{conn:k-test}
For every $\lambda\in\Lambda$ and every integer
$m>-\nu(\lambda)$, there is a vanishing of
$\delta$-graded modules
\begin{equation}\label{eq:k-test}
 K^a\otimes_{R^a}
       \big(B(R_p^a)/t\big)_{\lambda,*,m}=0.
\end{equation}
The tensor product uses the $R^a$-action in the
$\delta$-grading. The notation $/t$ denotes the underlying
module cofiber of multiplication by $t$.
\end{proposition}

\begin{proof}
Fix $\lambda$ and $m$. Trivial-action maps in the
$\delta$-direction, followed by the primewise module action,
give the balanced pairing
\begin{equation}\label{conn:module-pairing}
 K^a\otimes_{R^a}\Phi(R_p^a)_{\lambda,*,m}
 \longrightarrow \Phi(K_p^a)_{\lambda,*,m}.
\end{equation}
In constructing it we use the chosen comparisons
$h_\Lambda(R_p^a)\simeq R^a$ and
$h_\Lambda(K_p^a)\simeq K^a$ from
Section~\ref{sec:universal}. The compatibility of the module
action with these comparisons makes the pairing balanced over
the indicated $\delta$-scalar action.

We verify that Equation~\eqref{conn:module-pairing} is an
equivalence using the finite unfiltered module cells of $K_p^a$.
For a free cell generated in fixed weight $\alpha$ and ordinary
sphere degree $d(\alpha)+u$, with $u\in\Z$ fixed, the pairing
identifies both sides with the shift of
$\Phi(R_p^a)_{\lambda,*,m}$ by weight $\alpha$ and chain degree
$u$. Concretely, the primewise module input at $\delta$ on the
right is
$S^{d(\alpha)+u}\otimes R_p^a(p\lambda+\delta-\alpha)$.
After applying $J^\vee\otimes\Tr$, the Picard grading moves
the trivially acting $S(\alpha)$ into the source, changing its index to
$(\lambda,\delta-\alpha,m)$ and leaving the fixed suspension
$u$. This is exactly the free-cell shift on the left.
Both sides of Equation~\eqref{conn:module-pairing} are exact in
the module variable: tensor products preserve finite cofibers,
and the mapping constructions are exact componentwise.
Induction over the finitely many cells now proves the
equivalence for $K_p^a$.

If $m>-\nu(\lambda)$, Lemma~\ref{conn:degree-gap} shows that
the right side is zero at every fixed $\delta$. Finally,
Equation~\eqref{eq:layer} identifies
$B(R_p^a)//t$ with $\Phi(R_p^a)$ compatibly with the scalar
action. Since $t$ has chain degree zero, its algebra quotient
has the multiplication cofiber $B(R_p^a)/t$ as underlying
module, with the identification specified by its universal
null. Substituting this comparison into
Equation~\eqref{conn:module-pairing} proves
Equation~\eqref{eq:k-test}.
\end{proof}

The bounds in this section are asserted separately at each
fixed lattice index and each fixed integer suspension. Their
prime thresholds may depend on those indices. This is enough
to prove the vanishing of the corresponding mapping objects;
it does not require intersecting infinitely many large sets
of primes. In contrast, the use of
Equation~\eqref{eq:whole-module} at the moving weight
$p\lambda+\delta$ was justified by its equivalence of whole
primewise modules on one large set.
==== END Finite-Smith-Toda-Complexes-at-Varying-Primes//05-connectivity.tex ====

==== BEGIN Finite-Smith-Toda-Complexes-at-Varying-Primes//06-normalization.tex ====
\section{Normalization and successive attaching classes}
\label{sec:normalization}

We now use the vanishing in Equation~\eqref{eq:k-test} to construct
the cycles $b_1,\ldots,b_n$.  The vanishing is obtained only after
tensoring with a Koszul algebra.  Accordingly, we first pass to a
localization in which those tests detect equivalences, and then show
that the actual completed targets already belong to that localization.
This produces algebra maps, together with the boundary choices needed
to attach the next cell.  No identification of the cobordism Hurewicz
images is used in this construction.

\subsection{The inductive data}

Retain the lattices $\Lambda$, $\Omega$, and
$I=\Lambda\oplus\Lambda\oplus\Z$, the homomorphism $F:I\to\Omega$,
and the functions $\nu$ and $d$ from the preceding sections.  Set
\[
 i_j=(w_j,0,-1)\in I \qquad (0\leq j<n),
 \qquad \tau=(0,0,1).
\]
In particular, $F(i_j)=w_{j+1}$.  Let
$M=(\MU_{(p)})_p\in\cC$ and put
\[
 L_0=\widehat B(\one),\qquad L_0^M=\widehat B(M).
\]
Here the hats denote derived completion with respect to $t$, as in
Lemma~\ref{layer:completion}.  Starting with
$b_0=\varpi u_0$, we have the algebra
$h_\Omega(Y_1)=A[e_0]$, with $\partial e_0=b_0$.

At stage $0\leq a\leq n$, the inductive data will consist of the
strict cycle models through $b_a$ and $Y_{a+1}$, and a tower of
$I$-graded commutative algebras
\[
 L_a=L_0[e'_0,\ldots,e'_{a-1}],\qquad
 \wt(e'_j)=i_j,\quad |e'_j|_{\mathrm{chain}}=1,
 \quad \partial e'_j=g_j.
\]
This notation means successive algebra attachments with their
specified nulls.  In particular, it does not require the maps used to
construct the $g_j$ to have been prescribed as strict dg maps.  Define
the parallel tower by base change:
\[
 L_a^M=L_0^M\otimes_{L_0}L_a.
\]
Its attached classes and nulls are the images of those in $L_a$.
All tensor products of algebras in this section are derived.

The invariant is a pair of equivalences of $I$-graded commutative
algebras
\begin{equation}
\label{eq:target-layer}
 \begin{aligned}
 L_a//t&\simeq F^*h_\Omega(Y_{a+1}),\\
 L_a^M//t&\simeq F^*h_\Omega(M\otimes Y_{a+1}).
 \end{aligned}
\end{equation}
These equivalences are compatible with the sphere-to-$M$ unit, with
the maps from the initial layers, and with the algebra maps in the
$Y$-tower.  In the $\delta$ direction, their scalar action is ordinary
evaluation followed by the indicated algebra unit.  The case $a=0$
is supplied by Corollary~\ref{layer:first-koszul} and the equality of the
finite reductions of $B$ and $\widehat B$.

Each null attachment has a two-term underlying module by
Lemma~\ref{found:free-null}.  Thus $L_a$ is perfect over $L_0$, and
$L_a^M$ is perfect over $L_0^M$.  Lemma~\ref{layer:completion} shows
that both towers remain derived $t$-complete.

\subsection{Localization by the Koszul algebra}

Fix a stage $a<n$ for which these data have been constructed.
At a weight $\lambda\in\Lambda$, the power on the universal
coefficients takes values in
\[
 \bigl(B(R_p^a)[t^{-1}]\bigr)_{(\lambda,0,0)}.
\]
We seek an algebra map from $R^a$ into the components
\[
 \bigl\{(L_a)_{(\lambda,0,-\nu(\lambda))}\bigr\}_{\lambda\in\Lambda}.
\]
These components form a graded algebra because their index map is
additive. After inverting $t$ and multiplying its weight-$\lambda$
component by $t^{\nu(\lambda)}$, the new map must recover the power
followed by evaluation $R_p^a\to S_{(p)}$ and the map
$B(\one)\to L_a$. At weight $w_a$, the new target index is $i_a$.
Extending this coefficient map across the older Koszul nulls will
then let us apply it to $b_a^u$ and reduce modulo $t$ to construct
$b_{a+1}$. Thus the normalization must preserve algebra maps and
their specified nulls, rather than only divide individual classes
after inversion.

For $m>-\nu(\lambda)$, Equation~\eqref{eq:k-test} makes multiplication
by $t$ into the row $(\lambda,*,m)$ an equivalence after tensoring
with $K^a$. The following completion construction makes those
comparisons invertible; we will then show that the target $L_a$ is
unchanged by it.

For a commutative graded algebra $R$ and a commutative
$R$-algebra $K$, call an $R$-module $V$ \emph{$K$-acyclic} if
$K\otimes_R V=0$.  An $R$-module $U$ is \emph{$K$-local} if
$\map_R(V,U)=0$ for every $K$-acyclic $V$.  These definitions refer
to graded modules and their graded module category; any additional
indices can be kept as parameters.

\begin{lemma}[Perfect algebra completion]
\label{norm:amitsur}
Let $R$ be a commutative graded rational algebra, and let $K$ be a
commutative $R$-algebra whose underlying $R$-module is perfect.  Define
\[
 \mathcal T(U)=
 \Tot\bigl(K^{\otimes_R(\bullet+1)}\otimes_R U\bigr),
 \qquad c_U:U\longrightarrow\mathcal T(U),
\]
using the Amitsur coaugmentation.  Then $\mathcal T$ is exact,
$\mathcal T(U)$ is $K$-local, and $c_U$ becomes an equivalence after
tensoring with $K$.  Consequently $c_U$ is an equivalence whenever
$U$ is $K$-local, and $\mathcal T$ sends every $K$-acyclic module to
zero.  On commutative $R$-algebras the construction and its
coaugmentation are maps of commutative algebras.
\end{lemma}

\begin{proof}
The cosimplicial structure inserts the unit of $K$ in the coface maps
and multiplies adjacent $K$ factors in the codegeneracy maps.
Tensoring with $K$ commutes with totalization because a perfect module
is dualizable.  After that tensor, the augmented cosimplicial object
is split: multiplication into the extra, distinguished $K$ factor
gives the extra codegeneracy.  It follows that
\[
 K\otimes_R U\ \xrightarrow{\simeq}\
 K\otimes_R\mathcal T(U).
\]

Every nonaugmented term is a $K$-module.  If $W$ is a $K$-module and
$V$ is $K$-acyclic, extension and restriction of scalars give
\[
 \map_R(V,W)\simeq\map_K(K\otimes_R V,W)=0.
\]
Thus those terms are $K$-local.  Local objects are closed under
limits, so their totalization is local as well.  The fiber of $c_U$
is $K$-acyclic by the preceding split calculation.  If $U$ is local,
that fiber is also local, and hence is zero: its endomorphism spectrum
vanishes by applying the definition of locality to the fiber itself.
This proves the assertion about local inputs.  If $U$ is acyclic,
each term in its totalization is zero.

Tensoring over $R$ is exact in the stable module category, and limits
preserve finite limits.  Therefore $\mathcal T$ is exact.  Finally,
for algebra input the cosimplicial terms are commutative algebras and
all structure maps are algebra maps.  Their limit has the asserted
algebra structure and coaugmentation.  This is the usual
dualizable-algebra completion argument; compare
\cite[Example~2.18 and Proposition~2.21]{MNN2017}.
\end{proof}

Use the universal algebras and primewise models of
Proposition~\ref{universal:primewise-models}, and abbreviate
\[
 R=R^a,\qquad K=K^a,\qquad B=B(R_p^a).
\]
The small-weight comparison identifies $R$ with $h_\Lambda(R_p^a)$.
It acts on $B$ in the $\delta$ direction.  Evaluation of the
primewise universal algebra gives
\[
 v:B\longrightarrow B(\one)\longrightarrow L_0
       \longrightarrow L_a.
\]
The $R$-module $K$ is perfect by Lemma~\ref{universal:filtered-cells}.
We apply Lemma~\ref{norm:amitsur} with $R$ and $K$ supported at
indices $(0,\delta,0)$ of $I$.  Write the resulting functor as
$\mathcal T_a$.  Its module statements apply separately to each
row with fixed $(\lambda,m)$, while its algebra construction retains
all the $I$-graded products.

\begin{lemma}[Locality of the completed target]
\label{norm:target-locality}
Every $(\lambda,m)$ row of $L_a$ is $K^a$-local for the evaluated
$R^a$-action.  Thus the coaugmentation
$c_{L_a}:L_a\to\mathcal T_a L_a$ is an equivalence of
$I$-graded algebras.
\end{lemma}

\begin{proof}
Under Equation~\eqref{eq:target-layer}, restriction of the right-hand
algebra to $(0,\delta,0)$ is
\[
 h_\Lambda(Y_{a+1})
   =\bigl(A[e_0,\ldots,e_a]\bigr)|_\Lambda.
\]
There is an algebra map from $K^a$ to this restriction: send the
universal coefficients to the chosen coefficients in $A|_\Lambda$,
and send each $e_j$ to the actual null cell $e_j$.  Its differential
identities are exactly the universal-cycle identities after
evaluation.  The scalar compatibility in
Equation~\eqref{eq:layer} and the chosen path matching primewise and
actual coefficient evaluations identify its restriction to $R^a$
with the $R^a$-action on $L_a//t$.  Transporting along
Equation~\eqref{eq:target-layer} therefore makes every row of that
layer a $K^a$-module, and hence a local module.

The cofiber of a composite of $s$ multiplications by $t$ is a finite
extension of shifts of the cofiber of one multiplication by $t$.
Applied row by row, this shows that $L_a/t^s$ is local for every
$s\geq1$.  Here $/$ denotes the module cofiber; its underlying
module agrees with that of the corresponding algebra quotient.
Local modules are closed under finite extensions and inverse limits.
Since $L_a$ is derived $t$-complete, it is the inverse limit of these
local finite quotients and is itself local.  The conclusion follows
from Lemma~\ref{norm:amitsur}.  This proof uses the already attached
cells, and makes no assumption about any Hurewicz image.
\end{proof}

\subsection{The normalized algebra map}

For an $I$-graded algebra $U$ equipped with the parameter $t$, write
\[
 U^\#_\lambda=U_{(\lambda,0,-\nu(\lambda))},
 \qquad
 U^\flat_\lambda=U_{(\lambda,0,0)}.
\]
Both restrictions are $\Lambda$-graded algebras because their index
maps are additive.  There is a natural algebra map
\begin{equation}
\label{norm:regrading-map}
 s_U:U^\#\longrightarrow\bigl(U[t^{-1}]\bigr)^\flat,
 \qquad (s_U)_\lambda=t^{\nu(\lambda)}.
\end{equation}
In this formula one first maps to the localization and then
multiplies by the indicated Laurent power.  The powers, including
negative powers, form a coherent multiplicative family: pull back
the homogeneous units of the rational Laurent polynomial algebra
along $\lambda\mapsto\nu(\lambda)\tau$ and use the specified
algebra map into $U[t^{-1}]$.  Additivity of $\nu$ and
commutativity of multiplication prove that $s_U$ is an algebra map.

\begin{lemma}[The boundary row]
\label{norm:normalization}
For $B=B(R_p^a)$, the map
\[
 s_{\mathcal T_a B}:
 (\mathcal T_a B)^\#\longrightarrow
 \bigl((\mathcal T_a B)[t^{-1}]\bigr)^\flat
\]
is an equivalence of $\Lambda$-graded rational algebras.
\end{lemma}

\begin{proof}
By Equation~\eqref{eq:k-test}, the cofiber of multiplication by $t$
into a row $(\lambda,*,m)$ is $K$-acyclic whenever
$m> -\nu(\lambda)$.  Exactness of $\mathcal T_a$ sends this
cofiber to zero.  Multiplication by the coaugmented $t$ on
$\mathcal T_a B$ is the image under $\mathcal T_a$ of multiplication
by $t$ on $B$: this holds termwise in its cosimplicial construction
and therefore after totalization.  Consequently the transitions
\[
 (\mathcal T_a B)_{(\lambda,\delta,m-1)}
   \xrightarrow{t}
 (\mathcal T_a B)_{(\lambda,\delta,m)}
\]
are equivalences whenever $m> -\nu(\lambda)$.

The telescope for localization is thus constant up to equivalence
starting at the boundary row $m=-\nu(\lambda)$.  In particular,
that row maps equivalently to the same row after localization.
Multiplication by $t^{\nu(\lambda)}$ identifies the latter row
with the localized row at $m=0$.  This proves the assertion in every
weight.  When $\nu(\lambda)<0$, the last multiplication is simply
a negative Laurent power; the telescope still reaches the same
invertible tail.  Multiplicativity follows from
Equation~\eqref{norm:regrading-map}.
\end{proof}

We can now define the normalized operation.  The power map of
Proposition~\ref{power:operation}, the coaugmentation, and the two
equivalences just proved give
\begin{equation}
\label{norm:normalized-operation}
\begin{aligned}
 P^a:R^a\simeq h_\Lambda(R_p^a)
 &\xrightarrow{P_{R_p^a}}\bigl(B[t^{-1}]\bigr)^\flat
 \xrightarrow{(c_B[t^{-1}])^\flat}
       \bigl((\mathcal T_a B)[t^{-1}]\bigr)^\flat\\
 &\xrightarrow{s_{\mathcal T_a B}^{-1}}
       (\mathcal T_a B)^\#
 \xrightarrow{(\mathcal T_a v)^\#}
       (\mathcal T_a L_a)^\#
 \xrightarrow{(c_{L_a}^\#)^{-1}} L_a^\#.
\end{aligned}
\end{equation}
This is a map of graded algebras over rational spectra.  The power
map can change the action of the varying scalar ring $D$, and
$P^a$ is not asserted to be $D$-linear.

\begin{lemma}[Recovery and naturality]
\label{norm:naturality}
The normalized map recovers the evaluated power after inversion:
\begin{equation}
\label{norm:raw-recovery}
 s_{L_a}P^a\simeq (v[t^{-1}])^\flat P_{R_p^a}.
\end{equation}
For $a>0$, its restriction to $R^{a-1}$ agrees, as a rational
algebra map, with $P^{a-1}$ followed by the map $L_{a-1}^\#\to
L_a^\#$.
\end{lemma}

\begin{proof}
The first assertion says that the following square commutes:
\[
\begin{array}{ccc}
 R^a&\xrightarrow{\ P^a\ }&L_a^\#\\[2pt]
 {\scriptstyle P_{R_p^a}}\downarrow&&
       \downarrow{\scriptstyle s_{L_a}}\\[2pt]
 (B[t^{-1}])^\flat&\xrightarrow{\ (v[t^{-1}])^\flat\ }&
       (L_a[t^{-1}])^\flat .
\end{array}
\]
To verify it, naturality of $s$ for $c_{L_a}$ and $\mathcal T_a v$,
followed by naturality of the coaugmentation for $v$, gives
\[
\begin{aligned}
 &s_{L_a}(c_{L_a}^\#)^{-1}(\mathcal T_a v)^\#
        s_{\mathcal T_a B}^{-1}(c_B[t^{-1}])^\flat\\
 &\qquad=\bigl((c_{L_a}[t^{-1}])^\flat\bigr)^{-1}
       \bigl((\mathcal T_a v)[t^{-1}]\bigr)^\flat
       (c_B[t^{-1}])^\flat
   \simeq (v[t^{-1}])^\flat.
\end{aligned}
\]
The inverse in this display is the localization of an already
invertible coaugmentation on $L_a$.

For the second assertion, the compatible universal models and
evaluations give a map from the preceding diagram
$K^{a-1}\leftarrow R^{a-1}\to B(R_p^{a-1})\to L_{a-1}$
to the current diagram $K^a\leftarrow R^a\to B(R_p^a)\to L_a$.
In cosimplicial degree $r$, it induces an algebra map
\[
 (K^{a-1})^{\otimes_{R^{a-1}}(r+1)}
       \otimes_{R^{a-1}}B(R_p^{a-1})
 \longrightarrow
 (K^a)^{\otimes_{R^a}(r+1)}\otimes_{R^a}B(R_p^a).
\]
The same construction applies with $L_{a-1}$ and $L_a$ in the last
positions.  These maps commute with the units and multiplications
defining the cosimplicial structure, so they induce maps of
coaugmented totalizations.  They also preserve $t$.

The maps $s$ are natural for these maps of totalizations and for their
localizations.  Both normalization maps being inverted in
Equation~\eqref{norm:normalized-operation} are equivalences, so
their inverses respect these naturality squares.  The same is true
of the two target coaugmentations.  Finally, $P_{R_p^a}$ is natural
for the map of actual primewise algebras, and the chosen small-weight
comparisons and evaluation paths extend those of the preceding
stage.  Comparing all the arrows in
Equation~\eqref{norm:normalized-operation} therefore proves the
claimed homotopy of algebra maps.  This comparison uses actual maps
between the two Amitsur diagrams.  It does not interchange
totalization with $t$-inversion or identify a totalization with its
extension of scalars.
\end{proof}

\subsection{Extending across nulls and preserving the layer}

The map $P^a$ acts on universal coefficients.  To apply it to the
universal cycle $b_a^u$, we must also specify images of its older
Koszul nulls.  We construct
\[
 f^a:K_-^a\longrightarrow L_a^\#
\]
extending $P^a$.  For $a=0$, set $f^0=P^0$, since $K_-^0=R^0$.
For $a>0$, there is a pushout description
\[
 K_-^a\simeq R^a\otimes_{R^{a-1}}K^{a-1}.
\]
By induction, $f^{a-1}$ gives the map on $K_-^{a-1}$, followed by
$L_{a-1}^\#\to L_a^\#$.  The attached cell $e'_{a-1}$ in $L_a$
is a specified null of
$g_{a-1}=f^{a-1}(b_{a-1}^u)$.  Its index
$(w_{a-1},0,-1)$ is exactly the index selected by the $\#$ restriction
at $w_{a-1}$.  The universal property of the null attachment extends
the map to $K^{a-1}$.  Its restriction to $R^{a-1}$ agrees with the
restriction of $P^a$ by Lemma~\ref{norm:naturality}.  The pushout
therefore supplies $f^a$, including the required boundary data.
Although the displayed pushout consists of $D$-algebras, it is also
the pushout of the same span of rational algebras: the common middle
algebra already carries the specified scalar maps.  Thus this
construction does not impose $D$-linearity on $P^a$.

Define the homogeneous cycle class and the next algebra by
\begin{equation}
\label{norm:next-cell}
 g_a=f^a(b_a^u)\in\pi_0(L_a)_{i_a},
 \qquad L_{a+1}=L_a//g_a.
\end{equation}
Use the attached null as $e'_a$, and put
$L_{a+1}^M=L_0^M\otimes_{L_0}L_{a+1}$.
The first equivalence in Equation~\eqref{eq:target-layer} carries
the reduction of $g_a$ to a class of
$h_\Omega(Y_{a+1})$ at weight $F(i_a)=w_{a+1}$.  Represent this
class by a strict cycle $b_{a+1}$ in $A[e_0,\ldots,e_a]$, choosing
a path matching its class with that reduction.  Attach the next
null $e_{a+1}$ and realize this finite-module algebra extension as
$Y_{a+2}$ by Proposition~\ref{found:koszul-realization}.

\begin{lemma}[Compatibility of the new attachment]
\label{norm:attachment}
The preceding choices extend Equation~\eqref{eq:target-layer} to
stage $a+1$, with its unit, initial-layer, and tower compatibilities.
Both new algebras are derived $t$-complete.
\end{lemma}

\begin{proof}
Algebra pushouts commute with one another, so reducing the first
attachment modulo $t$ gives
\[
 L_{a+1}//t\simeq (L_a//t)//\overline g_a.
\]
The old layer equivalence, the chosen path matching attaching
classes, and the actual null $e_{a+1}$ define an algebra map from
this pushout to $F^*h_\Omega(Y_{a+2})$.  We check this particular
map on its underlying module.

At an index $i\in I$, the source module is the cofiber of
multiplication by $\overline g_a$ from
$(L_a//t)_{i-i_a}$ to $(L_a//t)_i$.  Under the old equivalence these
terms become
\[
 h_\Omega(Y_{a+1})_{F(i)-w_{a+1}}
 \xrightarrow{\ b_{a+1}\ }
 h_\Omega(Y_{a+1})_{F(i)}.
\]
Their cofiber is $h_\Omega(Y_{a+2})_{F(i)}$.  The map on the new
cell is the chosen null $e_{a+1}$ multiplied by the base input.
Lemma~\ref{found:free-null} therefore identifies the constructed
map with this cofiber equivalence.  This argument uses only the
additivity of $F$ and a two-term cofiber at each index.  It does not
require $F$ to be injective or its pullback to preserve arbitrary
relative algebra tensors.

For the $M$-tower, base change gives
\[
 L_{a+1}^M//t\simeq
 (L_a^M//t)\otimes_{L_a//t}(L_{a+1}//t).
\]
Map its first input into $F^*h_\Omega(M\otimes Y_{a+2})$ by the
old $M$-comparison followed by the actual $Y$-attachment.  Map its
second input there by the new sphere comparison followed by the
$M$-unit.  Their agreement on $L_a//t$ is the previous unit
compatibility together with the same attaching-class path.
The resulting pushout map is again the underlying cofiber
comparison: the new null is the image of $e_{a+1}$ under the
$M$-unit, and the finite-module tensor comparison identifies the
target with the cofiber of multiplication by this image of
$b_{a+1}$.  This proves the second equivalence and its compatibility
with the first.  The construction uses the canonical maps from the
old algebras, so it also preserves the initial-layer and tower maps.

The new attachment is a finite perfect-module extension of its
base.  Completeness follows from Lemma~\ref{layer:completion}, for
both the sphere tower and its $M$-base change.
\end{proof}

Figure~\ref{fig:towers} displays the induction just proved. The
auxiliary tower is one attachment ahead after reduction: its $a$th
stage records the ordinary object $Y_{a+1}$. The comparison is
compatible with the specified nulls, which is essential for the
next cycle to belong to that ordinary object.
\input{figures/towers}

\begin{proposition}[The compatible towers]
\label{norm:construction}
There exist cycles $b_0,\ldots,b_n$ and their realizations
$Y_0,\ldots,Y_{n+1}$, together with the algebras $L_a,L_a^M$ for
$0\leq a\leq n$, satisfying Equation~\eqref{eq:target-layer} and
its sphere-to-$M$, initial-layer, tower-map, and ordinary
$\delta$-scalar compatibilities. Each $b_j$ has chain degree zero
and weight $w_j$. For each $0\leq a<n$
there are rational algebra maps $P^a$ and $f^a$ as constructed above,
and $g_a=f^a(b_a^u)$ has chain degree zero and weight $i_a$.
It defines the next tower attachment and reduces modulo $t$ to
$b_{a+1}$ under Equation~\eqref{eq:target-layer}.
\end{proposition}

\begin{proof}
The initial layer was established at the beginning of the section.
Given stage $a<n$, Propositions~\ref{universal:primewise-models} and
\ref{conn:k-test} supply the universal models and
Equation~\eqref{eq:k-test}.  Lemmas~\ref{norm:target-locality} and
\ref{norm:normalization} define $P^a$; Lemma~\ref{norm:naturality}
and the older attached nulls extend it to $f^a$.
Equation~\eqref{norm:next-cell} defines $g_a$, and
Lemma~\ref{norm:attachment} constructs the next cycle and preserves
the invariant.  There are exactly $n$ such steps.
\end{proof}

\subsection{The comparison used for cobordism detection}

The construction has produced the next cycle before identifying
its Hurewicz image.  We finish this section with the comparison that
will perform that identification.  It relates the normalized
operation on the universal cycle to the original power on $M$.

\begin{proposition}[Recovery on the preceding Koszul object]
\label{norm:recovery}
For $0\leq a<n$ there is a map of $\Lambda$-graded rational algebras
\begin{equation}
\label{eq:recovery}
 h_\Lambda(M\otimes Y_a)
 \simeq h_\Lambda(M)\otimes_{R^a}K_-^a
 \longrightarrow \bigl(L_a^M[t^{-1}]\bigr)^\flat.
\end{equation}
On $h_\Lambda(M)$ it is the unnormalized power $P_M$ followed by
the map to the completed tower and inversion.  The $M$-Hurewicz
image of $b_a$ maps to $t g_a^M$, where $g_a^M$ is the image of
$g_a$ in $L_a^M$.
\end{proposition}

\begin{proof}
The actual evaluation $R^a\to A|_\Lambda$ sends the universal
coefficients to those of the chosen cycles.  Consequently
\[
 \bigl(A[e_0,\ldots,e_{a-1}]\bigr)|_\Lambda
 \simeq A|_\Lambda\otimes_{R^a}K_-^a,
\]
with each universal null sent to its actual counterpart.  Tensoring
with $h_\Lambda(M)$ over $A|_\Lambda$ gives the equivalence on the
left of Equation~\eqref{eq:recovery}.  Its identification with
$h_\Lambda(M\otimes Y_a)$ follows from the finite-module tensor
comparison of Proposition~\ref{found:finite-cell-morita}.  Only the finite
Koszul module is used here, so the same comparison applies after
restriction from $\Omega$ to $\Lambda$.

Define a map on the first pushout input $h_\Lambda(M)$ using
$P_M$ and the maps
$B(M)[t^{-1}]\to L_0^M[t^{-1}]\to L_a^M[t^{-1}]$.
On the other input use
\[
 K_-^a\xrightarrow{f^a}L_a^\#\longrightarrow(L_a^M)^\#
        \xrightarrow{s_{L_a^M}}
                    (L_a^M[t^{-1}])^\flat.
\]
These maps agree on $R^a$.  Indeed, restriction of the second map
to $R^a$ recovers the evaluated, unnormalized power by
Equation~\eqref{norm:raw-recovery}.  Naturality of the primewise
power for $R_p^a\to S_{(p)}\to\MU_{(p)}$ identifies this with
$P_M$ applied to the evaluated coefficients.  The chosen path
between spectral evaluation and the actual coefficient evaluation
is part of this comparison.  It gives agreement as algebra maps,
including the scalar action and the boundary data.

The universal property of the pushout now gives the arrow in
Equation~\eqref{eq:recovery}.  Although its source was described
using $D$-algebras, the same span is a pushout of rational algebras:
the common algebra $R^a$ supplies the scalar identifications.  No
additional $D$-linearity condition on the map into the powered
target is required.

Finally, the Hurewicz image of $b_a$ corresponds in that pushout to
the universal cycle $b_a^u\in K_-^a$.  Its image under $f^a$ is
$g_a$ by definition.  Since $\nu(w_a)=1$, the map
$s_{L_a^M}$ multiplies this class by $t$, changing its index from
$(w_a,0,-1)$ to $(w_a,0,0)$.  This proves the asserted value
$t g_a^M$.
\end{proof}
==== END Finite-Smith-Toda-Complexes-at-Varying-Primes//06-normalization.tex ====

==== BEGIN Finite-Smith-Toda-Complexes-at-Varying-Primes//07-cobordism.tex ====
\section{The cobordism calculation}
\label{sec:cobordism}

The preceding construction gives candidate attaching classes and recovers
their unnormalized powers after inverting $t$.  To identify their
Hurewicz images, we now calculate these powers in complex cobordism.
Everything in this section takes place at one fixed odd prime $p$.
The relation we obtain will be applied, prime by prime, to the completed
towers in Section~\ref{sec:realization}.

Write $M_p=\MU_{(p)}$, and use cohomological grading for spaces, so that
$M_p^{-s}(\mathrm{pt})=(M_p)_s$.  We use the commutative multiplication
on the complex Thom spectrum and its complex orientation, supplied by
the coherent Thom construction
\cite[Chapter~IV, Section~2, Construction~2.5 and Lemma~2.2]{May1977}.
Localization at $p$ carries this structure to $M_p$: the smashing
localization $X\mapsto S_{(p)}\otimes X$ is idempotent, and its
equivalences are preserved by tensoring, so it is symmetric monoidal
\cite[Proposition~2.2.1.9]{LurieHA}. In particular,
complex bundles have multiplicative Thom classes, and tensor product of
line bundles defines a graded one-dimensional commutative formal group
law.  The coefficient calculation is
\[
 (M_p)_*=\Z_{(p)}[a_s:s\geq 1],\qquad |a_s|=2s;
\]
see \cite[Chapter~2, Section~2.4, Theorem~4 of the English translation]{Novikov1962}.
We also use Quillen's multiplicative projection
\[
 M_p\longrightarrow\BP
\]
and the standard $p$-typical orientation of $\BP$
\cite[Section~5, Theorem~4]{Quillen1969}.  A map of ring spectra in the
homotopy category is sufficient here.  We do not require this projection
to preserve power operations, and we do not place a commutative
ring-spectrum structure on $\BP$.

\subsection{Coefficients of the \texorpdfstring{$p$}{p}-series}

Let $Z$ be the Euler coordinate of the universal complex line, of
cohomological degree two.  Define homogeneous coefficients and ideals by
\begin{equation}
 \label{cob:p-series}
 [p](Z)=\sum_{u\geq 1}k_uZ^u,
 \qquad k_u\in(M_p)_{2(u-1)},
 \qquad x_h=k_{p^h},
 \qquad I_h=(x_0,\ldots,x_{h-1}).
\end{equation}
Here $I_0=0$, $x_0=p$, and $|x_h|=d_h=2(p^h-1)$.

\begin{lemma}
 \label{cob:regularity}
 The sequence $x_0,x_1,\ldots$ is regular in $(M_p)_*$.
 Under the multiplicative projection to $\BP_*$, the ideal $I_h$ has
 image $(p,v_1,\ldots,v_{h-1})$ for $h\geq 1$, and the image of $x_h$
 modulo this ideal is a scalar unit times $v_h$.  Moreover,
 \[
  k_u\in I_h\qquad(1\leq u<p^h).
 \]
\end{lemma}

\begin{proof}
 In the standard $p$-typical coordinate, the logarithm of the
 Brown--Peterson formal group is
 \[
  \log(Z)=\sum_{h\geq 0}\ell_hZ^{p^h},\qquad \ell_0=1,
  \qquad
  p\ell_h=\sum_{0\leq s<h}\ell_s v_{h-s}^{p^s}.
 \]
 This is the Hazewinkel recursion
 \cite[Section~3.1, Equations~(3.1.1)--(3.1.3)]{HazewinkelIII}.
 Over the rationals, the $p$-series is
 $\log^{-1}(p\log(Z))$.  Its indecomposable term at $Z^{p^h}$ is
 \[
  (p-p^{p^h})\ell_h
   =(1-p^{p^h-1})v_h
       \pmod{\text{lower generators}},\qquad h>0.
 \]
 By homogeneity no generator $v_j$ with $j>h$ occurs in this
 coefficient; every term other than the displayed multiple of $v_h$
 uses lower generators.  The coefficient itself is integral, so these
 remaining terms lie in $\Z_{(p)}[v_1,\ldots,v_{h-1}]$.
 The scalar $1-p^{p^h-1}$ is a $p$-local unit.  Also, modulo
 $(p,v_1,\ldots,v_{h-1})$, degree considerations exclude every
 coefficient of the $p$-series below $Z^{p^h}$.

 These leading-coefficient assertions are unchanged, up to a scalar
 unit, by a change of Euler coordinate.  Indeed, the projective-space
 formula identifies a second coordinate with an integral power series
 in the first, with invertible linear coefficient.  Conjugating a
 series whose terms below order $p^h$ vanish preserves that order and
 multiplies its leading coefficient by a power of this invertible
 linear coefficient.  For normalized complex orientations the change
 is strict, and the leading coefficient is unchanged.  Induction on
 $h$ therefore gives the asserted images of the ideals $I_h$ and of
 their next generators, also for the Euler coordinate induced from
 $M_p$.

 We next prove regularity in $(M_p)_*$ itself. For $h\geq1$, homogeneity in the
 polynomial coefficient ring gives
 \[
  x_h=c_h a_{p^h-1}
       +D_h(a_1,\ldots,a_{p^h-2}),\qquad c_h\in\Z_{(p)},
 \]
 where $D_h$ is decomposable.  Under the projection to $\BP_*$, the
 image of a generator of smaller degree cannot involve $v_h$.
 Consequently the coefficient of $v_h$ in the image of $x_h$ is the
 product of $c_h$ and the coefficient of $v_h$ in the image of
 $a_{p^h-1}$.  The preceding calculation says that this product is a
 unit modulo $p$.  Thus $c_h$ is a $p$-local unit.  Replacing
 $a_{p^h-1}$ successively by $x_h$ is a triangular change of polynomial
 generators.  Since $p$ is a nonzerodivisor, this proves regularity of
 $p,x_1,x_2,\ldots$.

 Finally, work modulo $I_h$, with $h>0$.  This coefficient ring has
 characteristic $p$.  Suppose that the $p$-series has a first nonzero
 term $cZ^s$.  Since it is a formal group endomorphism, comparison of
 total degree $s$ in its compatibility with formal addition gives
 \[
  c(X+Y)^s=cX^s+cY^s.
 \]
 Hence $\binom{s}{i}c=0$ for $0<i<s$.  If $s=p^rm$ with
 $p\nmid m$ and $m>1$, then
 $\binom{s}{p^r}\equiv m\pmod p$, which is a unit in any
 $\F_p$-algebra.  This is impossible for nonzero $c$.
 Thus $s$ must be a $p$-power.  All possible such coefficients below
 $p^h$ are among $x_0,\ldots,x_{h-1}$ and have already been killed.
 This proves $k_u\in I_h$ in the stated range.  The argument uses no
 assumption that the quotient ring is a domain.
\end{proof}

\subsection{Borel cohomology and Euler divisibility}

Retain $G=C_p\rtimes\F_p^\times$, its real permutation representation
$\rho$, and $V=\rho-1$.  For a subgroup $K\leq G$, set
$H_K^*=M_p^*(BK)$.  These are ordinary Borel cohomology groups; no
Euler inversion or categorical localization is being performed in this
section.

\begin{lemma}
 \label{cob:cohomology}
 For the Euler coordinate $x$ of a faithful complex character of $C_p$,
 restriction gives
 \begin{equation}
  \label{cob:borel-rings}
  H_{C_p}^*=(M_p)^*(\mathrm{pt})[[x]]/([p](x)),
  \qquad
  H_G^*\xrightarrow{\ \simeq\ }
     (H_{C_p}^*)^{\F_p^\times}.
 \end{equation}
 The projective-space formulas are
 \[
  M_p^*(BK\times\mathbb CP^r)=H_K^*[Z]/(Z^{r+1}),
  \qquad
  M_p^*(BK\times\mathbb CP^\infty)=H_K^*[[Z]],
 \]
 with graded power series in the infinite-dimensional case.
 Evaluation at a point of $BG$ detects units in $H_G^0$.
\end{lemma}

\begin{proof}
 The classes $1,Z,\ldots,Z^r$ form a free basis for
 $M_p^*(\mathbb CP^r)$.  To see this at the module level, the cellular
 spectral sequence collapses because $M_p$ has even coefficients.
 The class $Z$ is a generator in filtration two, and its powers
 generate the successive even filtrations by the ordinary cup-product
 calculation.  The resulting map
 \[
  \bigoplus_{j=0}^r\Sigma^{-2j}M_p
     \longrightarrow\map(\mathbb CP^r_+,M_p)
 \]
 of $M_p$-modules is an equivalence.  The class $Z^{r+1}$ vanishes
 because its filtration exceeds the dimension of $\mathbb CP^r$.
 Applying cochains from $BK$ to this finite splitting proves the
 product formula at finite $r$.  Passage to the inverse limit gives
 the power-series formula; the truncation maps are surjective, so
 there is no additional derived-limit term.

 The unit circle bundle of the $p$-th tensor power of the universal
 line over $\mathbb CP^\infty$ models $BC_p$.  Its sphere-bundle
 triangle, together with the Thom isomorphism, identifies the relevant
 cochain map with multiplication by $[p](x)$.  This power series is a
 nonzerodivisor in $(M_p)^*(\mathrm{pt})[[x]]$, since the coefficient
 ring is a polynomial domain.  The resulting exact sequence gives
 the first formula in Equation~\eqref{cob:borel-rings}.

 For the second formula, compute homotopy fixed points first by $C_p$
 and then by its complement $\F_p^\times$.  If a finite group $K$ has
 order invertible on a spectrum, homotopy fixed points calculate the
 invariants on homotopy groups without a boundedness hypothesis.
 Indeed, the map to the coinduction of the underlying spectrum and
 the sum map back have composite $|K|$.  Finite induction and
 coinduction agree, so dividing this composite by $|K|$ gives a
 retraction.  After taking homotopy fixed points, the coinduced term
 is the underlying spectrum, and the opposite composite is the
 averaging operator.  Apply this to $K=\F_p^\times$ and to the
 $C_p$-homotopy fixed points of the constant spectrum $M_p$.

 Finally, a degree-zero Borel class acts as an $M_p$-linear
 endomorphism of the constant rank-one $M_p$-module over $BG$.
 If its value at a point is a unit, this endomorphism is a fiberwise
 equivalence, hence an equivalence of local systems.  Its inverse is
 again such an endomorphism, proving the assertion about units.
\end{proof}

Define
\begin{equation}
 \label{cob:epsilon}
 \epsilon=e_{M_p}(V\otimes_{\mathbb R}\mathbb C)\in H_G^q,
 \qquad q=2(p-1).
\end{equation}
The restriction of this Euler class to $C_p$ is
\[
 \epsilon|_{C_p}=\prod_{\alpha=1}^{p-1}[\alpha](x)
                  =x^{p-1}\cdot\text{a unit}.
\]
Here $[\alpha](x)$ is the formal multiple for the character indexed by
$\alpha$, and its linear coefficient $\alpha$ is a $p$-local unit.

\begin{lemma}
 \label{cob:euler-divisibility}
 If a homogeneous class $y\in H_G^*$ restricts to an element of
 $(x^p)\subset H_{C_p}^*$, then $y$ is divisible by $\epsilon^2$ in
 $H_G^*$.
\end{lemma}

\begin{proof}
 For every $s>0$, the quotient $(x^s)/(x^{s+1})$ in $H_{C_p}^*$ is
 annihilated by $p$: multiply the relation $[p](x)=0$ by $x^{s-1}$.
 A multiplier $\alpha\in\F_p^\times$ acts on this quotient by
 $\alpha^s$, or its inverse if the opposite action convention is used.
 For
 \[
  p\leq s<2(p-1),
 \]
 this character is nontrivial.  An invariant element in $(x^s)$
 therefore maps to zero in $(x^s)/(x^{s+1})$, since some
 $\alpha^s-1$ is a unit modulo $p$.  Iterating over this finite range
 shows that $y|_{C_p}$ lies in $(x^{2(p-1)})$.

 Since $\epsilon|_{C_p}$ is $x^{p-1}$ times a unit, there is a
 homogeneous $z\in H_{C_p}^*$ with
 $y|_{C_p}=\epsilon^2z$.  Average $z$ over $\F_p^\times$.
 Both $y|_{C_p}$ and $\epsilon$ are invariant, so the averaged divisor
 still has product $y|_{C_p}$ with $\epsilon^2$.  By
 Lemma~\ref{cob:cohomology}, this invariant divisor comes from $H_G^*$.
 Notice that this argument neither asserts that $\epsilon$ is a
 nonzerodivisor nor cancels a factor of $\epsilon$.
\end{proof}

\subsection{Thom-corrected permutation powers}

We next define the ordinary primewise operation used in the
calculation. Its Thom construction belongs to the cobordism power
methods of tom Dieck and Quillen
\cite{tomDieck1968,Quillen1971}; the Euler factors will be retained
explicitly here. For a space $X$ and $f\in M_p^{2d}(X)$, represent $f$ by
a map
\[
 S^{-2d}\wedge X_+\longrightarrow M_p.
\]
Apply the permuted $p$-fold smash, pull back along the diagonal of
$X^p$, and use the commutative multiplication on $M_p$.  This gives a
map of spectra with naive $G$-action
\[
 S^{-2d\rho}\wedge X_+\longrightarrow\Tr M_p.
\]
The virtual real representation $2d(\rho-p)$ is complex.  Choose its
complex Thom trivialization
\[
 S^{2d(\rho-p)}\longrightarrow\Tr M_p.
\]
Tensoring the two maps and multiplying in $M_p$ replaces the source
representation by $-2pd$.  We obtain
\begin{equation}
 \label{cob:pi-definition}
 \Pi_p(f)\in M_p^{2pd}(BG\times X).
\end{equation}
The choices of Thom trivialization depend only on the integer $d$,
not on $X$ or $f$.  Such a Thom map induces an equivalence after
extension to $M_p$-modules because it does so on every fiber.  Negative
values of $d$ cause no difficulty: the corresponding trivializations
are obtained by dualizing those of actual complex bundles.  This
construction agrees with the even-degree Thom construction of
\cite[Section~3.2]{JohnsonNoel2010}, with allowance for the chosen
orientation units.

For a space $X$, let $\mathfrak t_G(X)$ denote the additive subgroup of
$M_p^*(BG\times X)$ generated by transfers from the subgroups of $G$
whose orders are prime to $p$.  It is a graded ideal by the projection
formula.  Write $\mathfrak t_G=\mathfrak t_G(\mathrm{pt})$.

\begin{lemma}
 \label{cob:power-rules}
 The operation $\Pi_p$ is natural in $X$.  For homogeneous even-degree
 classes $f$ and $g$, its product rule has the form
 \[
  \Pi_p(fg)=\alpha\,\Pi_p(f)\Pi_p(g),
  \qquad \alpha\in(H_G^0)^\times,
 \]
 where $\alpha$ depends on their degrees and the Thom choices and is
 pulled back from $BG$.  On any finite sum of classes in the same
 even degree, $\Pi_p$ is additive modulo $\mathfrak t_G(X)$.
 For $X=\mathbb CP^r$, extracting a coefficient of $Z^s$ sends
 $\mathfrak t_G(X)$ into $\mathfrak t_G$.
\end{lemma}

\begin{proof}
 Naturality follows from the diagonal construction.  Before the Thom
 correction, the product rule follows from the symmetric monoidal
 permuted smash and the commutative multiplication of $M_p$.
 Compare the product of the two chosen Thom trivializations with the
 chosen trivialization for the sum of the degrees.  Their ratio is
 an automorphism of a constant suspended rank-one $M_p$-module over
 $BG$, hence a unit in $H_G^0$.  This comparison is made before
 introducing $X$, so the factor has no dependence on its
 cohomology classes.

 To examine a finite sum of classes in one degree, factor the sum
 map through a finite biproduct and distribute the $p$-fold smash
 over that biproduct.  The summands are indexed by label tuples on
 the $p$ letters.  A constant label tuple gives exactly the power
 of the corresponding summand.  The stabilizer of a nonconstant
 tuple has order prime to $p$: otherwise it contains the unique
 subgroup $C_p$ of order $p$, whose transitive action on the letters
 would force the labels to be constant.  The other orbits therefore
 factor through inductions from prime-to-$p$ subgroups.

 A composite between trivially acting endpoints that factors through
 an induction from a subgroup is the corresponding additive transfer.
 This follows by the induction--restriction adjunction and the
 coinduction--restriction adjunction, using equality of finite
 induction and coinduction; their unit and counit give the transfer
 of the composite at the subgroup.  Tensoring with a Thom
 trivialization and multiplying do not change the property of
 factoring through such an induction.  This proves additivity
 modulo $\mathfrak t_G(X)$ and also explains directly its stability
 under multiplication by classes from the larger group.

 For the last assertion, use the finite $M_p$-module splitting of
 the cochains of $\mathbb CP^r$ from Lemma~\ref{cob:cohomology}.
 Coefficient extraction is the projection onto one of its summands.
 Taking cochains from a space with trivial group action commutes
 with finite induction, since that induction is a finite sum.
 Consequently these projections commute with transfers and take
 a transfer error to a transfer error on the coefficient spectrum.
\end{proof}

Let $P$ be the universal line bundle on $\mathbb CP^\infty$, and put
$h(Z)=\Pi_p(Z)$.  The next formula both identifies its first
coefficient and controls the remaining coefficients by the
Euler filtration.

\begin{lemma}
 \label{cob:euler-power}
 There is a unit
 $U(Z)\in M_p^0(BG\times\mathbb CP^\infty)^\times$ such that
 \begin{equation}
  \label{cob:euler-product}
  h(Z)=U(Z)\,e_{M_p}\bigl(P\otimes
                      (\rho\otimes_{\mathbb R}\mathbb C)\bigr).
 \end{equation}
 In particular, $h(Z)$ has no constant term and
 \begin{equation}
  \label{cob:linear-coefficient}
  [Z]h(Z)=U(0)\epsilon,
  \qquad
  h(Z)|_{C_p}=U(Z)|_{C_p}
          \prod_{\alpha=0}^{p-1}\bigl(Z+_{\MU}[\alpha](x)\bigr).
 \end{equation}
\end{lemma}

\begin{proof}
 Represent $Z$ by the zero section of $P$ followed by its Thom map.
 Fiberwise, the normalized Thom map is a map from the local sphere
 $S^{P-\mathbb C}$ to $M_p$ that becomes an equivalence after
 extending to $M_p$-modules.  Thom spectra can be formed as colimits
 of these sphere local systems.  Thus the external permuted smash
 of the Thom maps, which commutes with their colimits, is the
 corresponding fiberwise construction on the external sum of the
 bundles.  On pulling back to the diagonal and making the Thom
 correction in Equation~\eqref{cob:pi-definition}, the zero section
 becomes that of
 $P\otimes(\rho\otimes_{\mathbb R}\mathbb C)$.

 The subsequent map of its sphere local system, normalized by
 $\mathbb C^p$, is an $M_p$-linear equivalence after extension: on
 each fiber it is the product of the chosen units.  Its ratio to
 the ordinary Thom equivalence of this bundle is therefore a unit
 over $BG\times\mathbb CP^\infty$.  This is the class $U(Z)$ in
 Equation~\eqref{cob:euler-product}.

 Restricted to $C_p$, the complex permutation representation splits
 into the characters indexed by $\alpha\in\F_p$.  Multiplicativity
 of Euler classes and the formal group law give the product in
 Equation~\eqref{cob:linear-coefficient}.  The trivial summand gives
 the factor $Z$, so the constant term vanishes; evaluating every
 other factor at $Z=0$ gives the linear coefficient $U(0)\epsilon$.
 With compatible canonical Thom choices this is the usual exact
 Euler-product formula
 \cite[Equations~(4.11) and~(4.14)]{JohnsonNoel2010}; the unit-factor
 version suffices here.
\end{proof}

\subsection{Extraction of the next coefficient}

Fix an integer $a\geq0$. We now compare the power of the $p$-series
with the $p$-series substituted into $h$. The relation below has leading terms
$\Pi_p(x_a)$ and $\epsilon x_{a+1}$, each with a unit coefficient.
It separates four kinds of error: multiples of $\epsilon^2$,
products $x_jx_k$ with $j,k\leq a$, earlier powers $\Pi_p(x_j)$
with $j<a$, and additive transfers. In the Hurewicz induction of
Section~\ref{sec:realization}, the first two kinds will become
divisible by $t^2$, and the last two will vanish. This leaves the
comparison of leading terms from which one factor of $t$ can be
canceled before reduction modulo $t$.

\begin{proposition}
 \label{cob:coefficient-relation}
 For each integer $a\geq 0$ there are homogeneous coefficients in
 $H_G^*$ and units $C,C'\in(H_G^0)^\times$ such that
 \begin{equation}
  \label{eq:cobordism}
  \begin{split}
   C\Pi_p(x_a)={}&C'\epsilon x_{a+1}+\epsilon^2D_*
      +\sum_{0\leq j,k\leq a}D_{jk}x_jx_k\\
      &+\sum_{0\leq j<a}H_j\Pi_p(x_j)+T_*,
      \qquad T_*\in\mathfrak t_G.
  \end{split}
 \end{equation}
 All terms have cohomological degree $-p d_a$; coefficients in the
 two sums and $D_*$ are in the complementary degrees.
\end{proposition}

\begin{proof}
 Set $r_*=p^{a+1}$, and work on $\mathbb CP^{r_*}$.  Tensoring the
 universal line $p$ times gives a classifying map to
 $\mathbb CP^\infty$ that pulls $Z$ back to $[p](Z)$.  Naturality
 of $\Pi_p$ therefore gives $h([p](Z))$ for the power of this
 Euler class.  All substitutions can be made at finite truncation:
 by cellular approximation, the classifying map from
 $\mathbb CP^{r_*}$ factors up to homotopy through its
 $2r_*$-dimensional skeleton, and $Z^{r_*+1}=0$ on this space.

 Alternatively, expand $[p](Z)$ as in Equation~\eqref{cob:p-series}.
 Each summand $k_uZ^u$ has cohomological degree two, so the
 additivity assertion of Lemma~\ref{cob:power-rules} applies.
 Repeated use of its product rule gives
 \begin{equation}
  \label{cob:series-comparison}
  h([p](Z))\equiv
     \sum_{1\leq u\leq r_*}
          \alpha_u\Pi_p(k_u)h(Z)^u
       \pmod{Z^{r_*+1},\ \mathfrak t_G(\mathbb CP^{r_*})},
 \end{equation}
 where every $\alpha_u$ is a unit in $H_G^0$ pulled back from
 $BG$.  These factors may depend on $u$ and on the Thom choices;
 no compatibility between them is required below.

 Extract the coefficient of $Z^{r_*}$.  Since both sides of
 Equation~\eqref{cob:series-comparison} have degree $2p$, this
 coefficient has degree
 \[
  2p-2r_*=-p d_a.
 \]
 The transfer error remains an element of $\mathfrak t_G$ by
 Lemma~\ref{cob:power-rules}.

 First consider the term $u=p^a$ on the right.  Put
 $A_u=[Z^{r_*}]h(Z)^u$.  Its degree at $u=p^a$ is zero.
 At a point of $BG$, the permutation bundle is trivial and
 Equation~\eqref{cob:euler-product} becomes
 $h(Z)=U(Z)Z^p$.  Consequently
 \[
  A_{p^a}|_{\mathrm{pt}}=U(0)|_{\mathrm{pt}}^{\,p^a},
 \]
 a scalar unit.  Lemma~\ref{cob:cohomology} shows that
 $A_{p^a}$ is a unit of $H_G^0$.  Multiplication by
 $\alpha_{p^a}$ gives the coefficient $C$ of $\Pi_p(x_a)$.

 If $u>p^a$, restrict to $C_p$.  Each of the $pu$ formal-addition
 factors in the product expression for $h(Z)^u$ lies in the
 ideal $(Z,x)$.  The additional factor $U(Z)^u$ is an ordinary
 power series in these variables, and hence cannot decrease this
 ideal order.  Therefore
 \[
  A_u|_{C_p}\in(x^{pu-r_*}),\qquad pu-r_*\geq p.
 \]
 This assertion can be checked using any power-series lifts before
 quotienting by $[p](x)$; all exponents of $x$ and $Z$ are
 nonnegative.  Since $A_u$ comes from $BG$,
 Lemma~\ref{cob:euler-divisibility} gives
 $A_u\in\epsilon^2H_G^*$.  After multiplication by
 $\alpha_u\Pi_p(k_u)$ and summation, all such terms contribute
 to $\epsilon^2D_*$.

 For $1\leq u<p^a$, Lemma~\ref{cob:regularity} gives a finite
 homogeneous expression
 \[
  k_u=\sum_{0\leq j<a}c_{u,j}x_j
       \quad\text{in }(M_p)_*.
 \]
 All summands have the same degree as $k_u$.  Applying the sum
 and product rules of Lemma~\ref{cob:power-rules} expresses
 $\Pi_p(k_u)$, modulo transfers, as a sum of multiples of
 $\Pi_p(x_j)$ for $j<a$.  Multiplying by $\alpha_uA_u$ and
 collecting terms yields the last ideal sum in
 Equation~\eqref{eq:cobordism}.  When $a=0$ this range is empty.

 It remains to describe the coefficient on the left of
 Equation~\eqref{cob:series-comparison}.  Write
 $h(Z)=\sum_{s\geq 1}h_sZ^s$.  Its linear term contributes
 \[
  h_1[Z^{r_*}][p](Z)
      =U(0)\epsilon x_{a+1}.
 \]
 This is the displayed main term, with a unit coefficient.
 Every contribution from $h_s[p](Z)^s$ for $s\geq 2$ is a
 product of at least two coefficients $k_u$ with
 $1\leq u<r_*$.  Each is in $I_{a+1}$ by
 Lemma~\ref{cob:regularity}.  Thus these contributions belong to
 $I_{a+1}^2H_G^*$ and can be written as the displayed sum of
 $x_jx_k$ for $j,k\leq a$.

 Move the other right-hand terms to the left or absorb their signs
 into their coefficients.  The unit multiplying the main term
 remains a unit, giving Equation~\eqref{eq:cobordism}.  Every sum
 used above is finite at this fixed prime.  After collection there
 are only the displayed finitely many ideal factors, although
 their coefficients and the expression for $T_*$ may vary with
 $p$ without a uniform bound on the number of intermediate terms.
\end{proof}
==== END Finite-Smith-Toda-Complexes-at-Varying-Primes//07-cobordism.tex ====

==== BEGIN Finite-Smith-Toda-Complexes-at-Varying-Primes//08-realization.tex ====
\section{Hurewicz detection and finite realization}
\label{sec:realization}

Fix \(n\geq 1\).  Proposition~\ref{norm:construction} constructed the
classes \(b_0,\ldots,b_n\), the objects \(Y_0,\ldots,Y_{n+1}\), and
the auxiliary algebras \(L_a\).  We now identify the cobordism Hurewicz
image of each \(b_j\).  This identification will show that the successive
cofibers kill the first powers of the coefficients \(x_j\), and hence
the first powers of the Hazewinkel generators after passage to \(\BP\).

Write \(M=(M_p)_p\), where \(M_p=\MU_{(p)}\), and retain the ideals
\(I_j=(x_0,\ldots,x_{j-1})\subseteq (M_p)_*\) of
Section~\ref{sec:cobordism}.  We set \(I_0=0\).  For each prime, let
\(\overline I_j\subseteq\BP_*\) denote the ideal generated by the
images of these coefficients.  Lemma~\ref{cob:regularity} identifies
\(\overline I_j=(p,v_1,\ldots,v_{j-1})\) for \(j\geq 1\), with
\(\overline I_0=0\).

\subsection{The finite diagram and its effect on homology}

Let \(u_j:\one\to Y_j\) be the unit and let
\[
 \mu_j:S(w_j)\otimes Y_j\longrightarrow Y_j,
 \qquad c_j:Y_j\longrightarrow Y_{j+1}
\]
be, respectively, multiplication by \(b_j:S(w_j)\to Y_j\) and its
cofiber map.  The construction supplies the homotopies
\begin{equation}
 \mu_j\circ(\id\otimes u_j)\simeq b_j,
 \qquad c_j\circ u_j\simeq u_{j+1},
 \qquad \cofib(\mu_j)\simeq Y_{j+1},
 \label{real:finite-identities}
\end{equation}
where the last equivalence respects the cofiber map.  Only these maps,
homotopies, and cofiber identifications, for \(0\leq j\leq n\),
will be needed at individual primes.

They form finite diagram data in \(\cC\).  By the description of
\(\cC\) as a filtered colimit of product categories, they have
representatives on a common \(\U\)-large set of primes.  To be
explicit, choose representatives of the finitely many objects and maps,
then representatives of the finitely many homotopies in
Equation~\eqref{real:finite-identities}.  An equivalence of cofibers is
represented together with an inverse and the two inverse homotopies.
Shrinking the prime set finitely many times makes all these choices
valid together.  Calibrate each of the finitely many source spheres
\(S(w_j)\) with \(S^{d_j}\) at the primes.  We obtain spectra
\(Y_{j,p}\), maps \(u_{j,p},b_{j,p},\mu_{j,p},c_{j,p}\), and
cofiber sequences
\begin{equation}
 \Sigma^{d_j}Y_{j,p}\xrightarrow{\mu_{j,p}}Y_{j,p}
 \xrightarrow{c_{j,p}}Y_{j+1,p}.
 \label{real:primewise-cofibers}
\end{equation}
No descent of the full commutative algebra structures on the \(Y_j\)
is involved in this assertion.

The following elementary observation explains why this finite diagram
is enough to compute homology in every degree.

\begin{lemma}\label{real:cofiber-quotient}
Let \(E\) be a ring spectrum whose coefficient ring \(R=\pi_*E\)
is graded commutative and concentrated in even degrees. Let \(J\)
be a homogeneous ideal of \(R\), and let \(u:S\to Y\) induce
the quotient identification \(E_*Y\cong R/J\). Suppose that
\(b:S^d\to Y\) and \(\mu:\Sigma^dY\to Y\), with \(d\) even, satisfy
\(\mu\circ\Sigma^du\simeq b\).  If the \(E\)-Hurewicz image
of \(b\) is \(a\) times the image of a homogeneous element
\(x\in R_d\), where \(a\in R_0^\times\) and multiplication
by \(x\) is injective on \(R/J\), then the
composite \(S\xrightarrow{u}Y\to\cofib(\mu)\) induces
\[
 E_*\cofib(\mu)\cong R/(J,x).
\]
This is an identification of the entire graded module.
\end{lemma}

\begin{proof}
The map on \(E\)-homology induced by any spectrum map is
\(R\)-linear.  Its source here is the shifted cyclic module
\(\Sigma^dR/J\), and the given homotopy says that it takes its
generator to \(a x\).  It is therefore multiplication by \(a x\)
on all homogeneous elements.  This map is injective.  The cofiber
long exact sequence consequently gives, for every integer \(k\),
\[
 0\longrightarrow (R/J)_{k-d}
 \xrightarrow{\,a x\,}(R/J)_k
 \longrightarrow E_k\cofib(\mu)\longrightarrow 0.
\]
The quotient map in this sequence is induced by the cofiber map, so
the resulting identification is induced by the stated map from the
sphere.  Since \(a\) is a unit, its cokernel is \(R/(J,x)\).
\end{proof}

\subsection{Detection before Euler inversion}

For \(0\leq a<n\), abbreviate
\[
 s_a=(w_a,0,0),\qquad i_a=(w_a,0,-1)=s_a-\tau.
\]
In the arguments below, a quotient by \(t\) denotes the underlying
cofiber of multiplication by \(t\).  It agrees with the underlying
module of the algebra quotient by \(t\).  Thus the \(M\)-version
of Equation~\eqref{eq:target-layer} reads
\begin{equation}
 (L_a^M/t)_i\simeq
 h_\Omega(M\otimes Y_{a+1})_{F(i)}
 \qquad(i\in I).
 \label{real:target-layer-M}
\end{equation}
These equivalences respect the unit, the maps from the base layer,
and the reduction of \(g_a\) used to define \(b_{a+1}\).

\begin{lemma}\label{real:t-injectivity}
Suppose that, on a \(\U\)-large set of primes,
\[
 (M_p)_*Y_{a+1,p}\cong (M_p)_*/I_{a+1}
\]
via the unit.  Then multiplication by \(t\) is injective on
\(\pi_0L_a^M\) at every index.  In particular,
\[
 \pi_0(L_a^M)_i\longrightarrow
 \pi_0(L_a^M[t^{-1}])_i
\]
is injective for every \(i\in I\).
\end{lemma}

\begin{proof}
The coefficient ring of \(M_p\) and its quotient by \(I_{a+1}\)
are concentrated in even degrees.  Every degree \(d(F(i))\) is
even.  Equation~\eqref{real:target-layer-M} therefore gives
\(\pi_1(L_a^M/t)_i=0\) for every fixed \(i\).  In the exact
sequence of the cofiber
\[
 (L_a^M)_{i-\tau}\xrightarrow{t}(L_a^M)_i
 \longrightarrow (L_a^M/t)_i,
\]
this vanishing proves the asserted injectivity.  Localization is the
componentwise telescope along multiplication by \(t\).  Homotopy
groups commute with that filtered colimit, and every transition on
\(\pi_0\) is injective, which proves the last assertion.  Notice
that this argument uses only the evenness of the layer; it makes no
evenness assertion about the completed algebra \(L_a^M\).
\end{proof}

For a coefficient \(x_j\), let \(\mathcal P(x_j)\) denote its
raw powered class in \(\pi_0(L_a^M)_{s_j}\): represent \(x_j\)
at each prime by a map \(S^{d_j}\to M_p\), take its permuted
\(p\)-fold smash, multiply in \(M_p\), and map the resulting
class from \(B(M)\) to \(L_a^M\).  This definition is made
before inverting \(t\).  For \(u\in D^\times\), define
\(\mathcal P(u)\in\pi_0(L_a^M)_0\) in the same way using
scalar maps.  We do not assert additivity of these raw classes
before localization.

\begin{lemma}\label{real:raw-powers}
Assume the hypothesis of Lemma~\ref{real:t-injectivity}.  Suppose
also that \((M_p)_*Y_{a,p}\cong (M_p)_*/I_a\) via the unit on
a \(\U\)-large set, and that the \(M\)-Hurewicz image of
\(b_a\) equals \(u x_a\) for some \(u\in D^\times\).
Then, before inverting \(t\),
\begin{equation}
 \begin{aligned}
  \mathcal P(x_j)&=0
     &&\text{in }\pi_0(L_a^M)_{s_j}\quad(0\leq j<a),\\
  t g_a&=\mathcal P(u)\mathcal P(x_a)
     &&\text{in }\pi_0(L_a^M)_{s_a}.
 \end{aligned}
 \label{eq:before-inversion}
\end{equation}
Here \(g_a\) also denotes its image under base change to \(M\),
and \(\mathcal P(u)\) is a unit at index zero.
\end{lemma}

\begin{proof}
Apply Proposition~\ref{norm:recovery} and
Equation~\eqref{eq:recovery}.  The classes \(x_j\) for
\(j<a\) vanish in its source
\(h_\Lambda(M\otimes Y_a)\), and their images in the target are
the localizations of \(\mathcal P(x_j)\).  The same comparison
takes the Hurewicz image of \(b_a\) to \(t g_a\).  Since it is
a map of rational graded algebras and that Hurewicz image is
\(u x_a\), its value is the localization of
\(\mathcal P(u)\mathcal P(x_a)\).  Lemma~\ref{real:t-injectivity}
now proves both equalities before inversion.  No linearity over
\(D\) is required of the powered comparison.

Raw powering is multiplicative on the represented scalar maps:
the product of the permuted powers of two maps is the permuted
power of their product, followed by the commutative multiplication
of \(M_p\).  Apply this to primewise representatives of \(u\)
and \(u^{-1}\).  It gives
\(\mathcal P(u)\mathcal P(u^{-1})=1\) already in \(B(M)\),
and hence in \(L_a^M\).
\end{proof}

\subsection{Putting the cobordism relation at fixed indices}

Equation~\eqref{eq:cobordism} is an identity of Borel cohomology
classes at each prime.  Its ordinary degree varies with the prime,
whereas \(L_a^M\) is indexed by the fixed lattice \(I\).  We
describe the comparison, including its unit ambiguities and its
effect on transfers.

For \(i=(\lambda,\delta,m)\), the representation underlying
\(W_i\) at the prime \(p\) is
\[
 d(\lambda)\rho+d(\delta)-2mV,
 \qquad
 d(F(i))=p\,d(\lambda)+d(\delta)-qm.
\]
This is a virtual complex representation: \(d(\lambda)\) and
\(d(\delta)\) are even, and \(2V\) is the underlying real
representation of \(V\otimes_{\mathbb R}\mathbb C\).
Complex Thom equivalences thus identify
\(W_i\otimes\Tr M_p\) with
\(\Tr S^{d(F(i))}\otimes\Tr M_p\), as \(M_p\)-modules with
naive \(G\)-action.  The equivalences for virtual negative
summands are obtained by dualizing.  For each fixed index these
equivalences give additive maps
\begin{equation}
 \theta_i:
 \prod_{\U}M_p^{-d(F(i))}(BG)
 \longrightarrow\pi_0B(M)_i
 \longrightarrow\pi_0(L_a^M)_i.
 \label{real:borel-comparison}
\end{equation}
At index zero choose the unit identification, so \(\theta_0\)
is a ring map.  The other \(\theta_i\) respect its scalar action.

If two such Thom equivalences are multiplied and compared with the
equivalence at the sum of their indices, the discrepancy is an
automorphism of a suspended free rank-one \(M_p\)-module over
\(BG\).  It is therefore multiplication by a unit of
\(M_p^0(BG)\).  Consequently products under the maps
\(\theta_i\) agree up to images of units under \(\theta_0\).
All comparisons we will use involve a fixed finite list of indices.
Their representatives and the tensor identifications can be chosen
on a common large prime set; a simultaneous primewise realization
of the entire Picard grading is unnecessary.

\begin{lemma}\label{real:thom-comparisons}
Fix \(0\leq a<n\), and suppose that
\((M_p)_*Y_{a+1,p}\cong(M_p)_*/I_{a+1}\) via the unit on a
\(\U\)-large set.  The maps in
Equation~\eqref{real:borel-comparison} have the following properties.
\begin{enumerate}
\item The image \(\theta_\tau(\epsilon)\) is a unit at index
zero times \(t\).
\item For \(j\leq a\), the image
\(\theta_{s_j}(\Pi_p(x_j))\) is a unit at index zero times
\(\mathcal P(x_j)\).
\item Put \(\delta_j=(0,w_j,0)\).  For \(j\leq a\), the
class \(\theta_{\delta_j}(x_j)\) lies in the image of
\[
 t:\pi_0(L_a^M)_{\delta_j-\tau}
 \longrightarrow\pi_0(L_a^M)_{\delta_j}.
\]
\item Set \(X_{a+1}=\theta_{i_a}(x_{a+1})\).  Under
Equation~\eqref{real:target-layer-M}, its reduction modulo \(t\)
is a unit at index zero times the image of the ordinary coefficient
\(x_{a+1}\) in \((M_p)_*Y_{a+1,p}\), at weight \(w_{a+1}\).
\item Every sequence of finite sums of additive transfers from
prime-to-\(p\) subgroups, in the degree assigned to any fixed
index, has zero image under \(\theta_i\).  The number of
summands may depend on \(p\).
\end{enumerate}
\end{lemma}

\begin{proof}
For the first assertion, recall that \(\epsilon\) is the Euler
class of the complex representation
\(V\otimes_{\mathbb R}\mathbb C\), whose underlying real
representation is \(2V\).  Its zero-section map, after the Thom
identification, is the map \(\one\to T^2\) defining \(t\).
Changing the Thom equivalence changes it by a degree-zero unit.
The second assertion follows by undoing the Thom conversion in the
definition of \(\Pi_p\): the source \(W_{s_j}\) is exactly
the permuted sphere \(N(S(w_j))\).  Both constructions then use
the same permuted smash of the coefficient map and multiplication
in \(M_p\).

At \(\delta_j\) the source is the untwisted sphere
\(\Tr S(w_j)\).  The scalar compatibility of
Equation~\eqref{eq:layer}, followed by its extension to
Equation~\eqref{eq:target-layer}, identifies the reduction of
\(\theta_{\delta_j}(x_j)\) with the usual coefficient action
on \(M\otimes Y_{a+1}\), up to the allowed unit.  This action
is zero because \(x_j\in I_{a+1}\).  Exactness of the
cofiber sequence for \(t\) proves the third assertion.

For the fourth assertion, first note that
\[
 F(i_a)=w_{a+1},\qquad
 d(F(i_a))=p d_a+q=d_{a+1}.
\]
It remains to check the coefficient, since equality of these degrees
alone does not identify the two maps.  Recall that
Equation~\eqref{eq:layer} is constructed through \(\Phi(M)\)
using the chosen \(J^\vee\)-linear sphere comparison
\[
 J^\vee\otimes W_i\simeq
 J^\vee\otimes\Tr S(F(i)).
\]
Extend this comparison to \(J^\vee\otimes\Tr M\).  Also
extend the Borel Thom comparison defining \(\theta_i\) along
the unit of \(J^\vee\).  The two are trivializations of the
same free twisted rank-one module.  Their ratio is an invertible
endomorphism, hence a unit in \(\pi_0\Phi(M)_0\).
With the first trivialization, the constant coefficient map
\(S^{d(F(i))}\to M_p\), followed by the \(J^\vee\)-unit,
is precisely the coefficient supplied by the right-hand comparison
of Equation~\eqref{eq:layer}.  The second trivialization therefore
gives that coefficient times the stated unit.  Extending from this
base layer to Equation~\eqref{real:target-layer-M} uses the actual
unit and tower maps, so it preserves this comparison.

Finally, an additive transfer from \(H\leq G\) factors through
an induced object \(\operatorname{Ind}_H^G Z\).  If
\(|H|\) is prime to \(p\), then \(Z\) is a retract of
\(\operatorname{Ind}_{\{1\}}^H\Res_{\{1\}}^H Z\): the
diagonal and sum maps have composite \(|H|\), which is
invertible \(p\)-locally.  Inducing to \(G\) makes the
original induction a retract of a free \(G\)-induction.  At
each prime, factor a finite sum of transfers through the direct sum
of these induced objects.  That entire sum is a retract of one free
induction on the direct sum of the underlying spectra.  Although its
size may grow with \(p\), these sums form a single sequence
object, and their free inductions are among the objects killed by
the defining localization to \(\cE\).  Tensoring with a
representation sphere preserves induction by the projection
formula, and composing with the Thom maps preserves the
factorization.  Thus the whole transfer error has zero image
already in \(B(M)\), before Euler inversion or completion.
\end{proof}

\subsection{The Hurewicz induction}

\begin{proposition}\label{real:hurewicz}
There is a \(\U\)-large set of primes on which the unit maps
induce, simultaneously for \(0\leq j\leq n+1\),
\begin{equation}
 (M_p)_*Y_{j,p}\cong (M_p)_*/I_j,
 \qquad
 \BP_*Y_{j,p}\cong\BP_*/\overline I_j.
 \label{real:homology-quotients}
\end{equation}
For every \(0\leq j\leq n\), there is a scalar
\(u^{(j)}\in D^\times\) such that the \(M\)-Hurewicz
image of \(b_j\) equals \(u^{(j)}x_j\) in
\(\pi_0h_\Omega(M\otimes Y_j)_{w_j}\).
\end{proposition}

\begin{proof}
At \(j=0\), the quotient assertions hold because \(Y_0=\one\),
and the Hurewicz assertion holds because
\(b_0=\varpi u_0\) represents the sequence \(p=x_0\) at
weight \(w_0\).  Lemmas~\ref{cob:regularity} and
\ref{real:cofiber-quotient}, applied to
Equation~\eqref{real:primewise-cofibers}, give both quotient
assertions for \(Y_1\).

Suppose now that \(0\leq a<n\), that the Hurewicz assertions
hold through \(a\), and that the quotient assertions hold
through \(a+1\).  Lemmas~\ref{real:t-injectivity} and
\ref{real:raw-powers} apply.  We will use
Proposition~\ref{cob:coefficient-relation} to identify the reduction
of \(g_a\), which by construction is the Hurewicz image of
\(b_{a+1}\).

Map Equation~\eqref{eq:cobordism} to \(L_a^M\) at total
index \(s_a=i_a+\tau\).  Its total cohomological degree is
\(-p d_a\).  The index assignments for its factors and error
coefficients are
\[
\begin{array}{c|c@{\qquad}c|c}
 \text{class}&\text{index}&\text{class}&\text{index}\\ \hline
 C,C'&0&\Pi_p(x_j)&s_j\\
 \epsilon&\tau&x_j\ (j\leq a)&\delta_j\\
 x_{a+1}&i_a&D_*&s_a-2\tau\\
 D_{jk}&s_a-\delta_j-\delta_k&H_j&s_a-s_j.
\end{array}
\]
These are fixed indices in \(I\).  They have exactly the required
ordinary degrees: for example,
\(d(F(i_a))=d_{a+1}\),
\(d(F(s_a-2\tau))=p d_a+2q\), and
\(d(F(s_a-\delta_j-\delta_k))=p d_a-d_j-d_k\).
Thus each coefficient in the primewise relation defines a sequence
in the source of the appropriate \(\theta_i\).  There is no
requirement that these coefficients themselves have formulas
independent of \(p\).

Lemma~\ref{real:thom-comparisons} and
Equation~\eqref{eq:before-inversion} now account for every term.
The earlier powered coefficients \(\mathcal P(x_j)\),
\(j<a\), vanish.  The \(\epsilon^2D_*\) term is divisible
by \(t^2\).  Each \(x_j\), \(j\leq a\), assigned to
\(\delta_j\), is divisible by \(t\), so every
\(D_{jk}x_jx_k\) term is also divisible by \(t^2\).
The complete transfer error vanishes, including when the number of
its primewise summands grows.  Products under the Thom comparisons
introduce only index-zero units.  Since \(C,C'\) and
\(\mathcal P(u^{(a)})\) are units, the two remaining leading
terms give
\begin{equation}
 t\bigl(U_1g_a-U_2X_{a+1}\bigr)=t^2z
 \quad\text{in }\pi_0(L_a^M)_{s_a},
 \label{real:cancel-equation}
\end{equation}
where \(U_1,U_2\in\pi_0(L_a^M)_0^\times\) and
\(z\in\pi_0(L_a^M)_{s_a-2\tau}\).  After the error
coefficients in Equation~\eqref{eq:cobordism} have been collected,
the list of terms just used has size bounded in terms of \(a\).

Multiplication by \(t\) from index \(i_a\) to index \(s_a\)
is injective by Lemma~\ref{real:t-injectivity}.  Canceling it in
Equation~\eqref{real:cancel-equation} gives
\[
 U_1g_a-U_2X_{a+1}=tz.
\]
Reduce modulo \(t\) and use
Equation~\eqref{real:target-layer-M}.  The reduction of \(g_a\)
is the \(M\)-Hurewicz image of \(b_{a+1}\), because the
construction of the new cycle and its chosen attaching-map path
commutes with the unit from the sphere to \(M\).
Lemma~\ref{real:thom-comparisons} identifies the reduction of
\(X_{a+1}\) with a unit times the ordinary coefficient
\(x_{a+1}\).  We have therefore proved that the Hurewicz image
of \(b_{a+1}\) is a unit at index zero times that coefficient,
at weight \(w_{a+1}\).

This unit can be replaced by a scalar in \(D^\times\).
Indeed, at index zero the actual layer has
\begin{equation}
 \pi_0(L_a^M/t)_0
 \cong\prod_{\U}(M_p)_0Y_{a+1,p}
 \cong\prod_{\U}\F_p
 \cong D/\varpi.
 \label{real:scalar-layer}
\end{equation}
The identifications are ring identifications via the unit: the
quotient assertion identifies the underlying degree-zero module,
and the unit map is already a ring map in the ultraproduct.
Here \(I_{a+1}\) contains \(p\), and its other generators have
positive ordinary degree.  A unit in \(D/\varpi\) has a
representative whose residue at \(p\) is nonzero on a
\(\U\)-large set.  Lift those residues to elements of
\(\Z_{(p)}^\times\), and choose arbitrary units at the
remaining primes.  This gives the required lift in \(D^\times\).
Its action agrees with that of the original layer unit.  Denote the
resulting scalar by \(u^{(a+1)}\).

The resulting Hurewicz equality is an equality in the one specified
mapping group
\[
 \pi_0\map_{\cC}(S(w_{a+1}),M\otimes Y_{a+1})
 \cong
 \prod_{\U}(M_p)_{d_{a+1}}Y_{a+1,p}.
\]
It consequently holds on a \(\U\)-large set at the actual
primes, with scalar representatives in \(\Z_{(p)}^\times\).
Naturality under the multiplicative projection \(M_p\to\BP\)
gives the analogous Hurewicz equality in \(\BP\)-homology.
Lemma~\ref{cob:regularity} makes both coefficient classes
nonzerodivisors in the current quotients.  Applying
Lemma~\ref{real:cofiber-quotient} gives the two quotient assertions
for \(Y_{a+2,p}\), in all degrees at once.

This completes the induction.  At each step the Hurewicz equality
requires only one new large-set restriction, together with the
finitely many comparisons used at that step.  The calculation of
the entire homology module then holds at every prime of that set
by coefficient linearity and the cofiber exact sequence.  Since
\(n\) is fixed, intersecting the finitely many resulting sets
proves the simultaneous assertion.  In particular, no intersection
over all homological degrees, weights, or infinitely many stages
is taken.
\end{proof}

\subsection{Finiteness and proof of the main theorem}

\begin{lemma}\label{real:finite-descent}
The \(p\)-localizations of finite CW spectra are closed under the
finite suspensions and cofibers in
Equation~\eqref{real:primewise-cofibers}.
\end{lemma}

\begin{proof}
Suspensions preserve this class.  For finite CW spectra \(F_0,F_1\),
compactness of \(F_0\), and the expression of localization as a
filtered colimit of multiplication maps, give
\[
 [ (F_0)_{(p)},(F_1)_{(p)} ]
 \cong [F_0,F_1]\otimes\Z_{(p)}.
\]
Thus a map between these localizations has the form \(f/s\),
where \(f:F_0\to F_1\) is a stable map and \(p\nmid s\).
Precomposing or postcomposing with multiplication by \(s\), an
equivalence after localization, identifies its cofiber with
\(\cofib(f)_{(p)}\).  The cofiber of a stable map of finite
CW spectra is again a finite CW spectrum: represent the map after
a finite suspension by a cellular map between finite CW models
and take its finite mapping cone, then desuspend.  This proves the
claim and applies inductively starting from the sphere.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:main}]
For \(n=0\), choose any prime \(p\) and take the cofiber of
\(p:S_{(p)}\to S_{(p)}\).  Multiplication by \(p\) is
injective on \(\BP_*\), so the cofiber exact sequence gives
\(\BP_*X\cong\BP_*/(p)\), via the map from the sphere.
This is a finite \(p\)-local spectrum.

For \(n\geq1\), fix the construction above and choose a prime
in the common \(\U\)-large set of
Proposition~\ref{real:hurewicz}.  Set \(X=Y_{n+1,p}\).
Equations~\eqref{real:primewise-cofibers} and
Lemma~\ref{real:finite-descent} show that \(X\) is the
\(p\)-localization of a finite CW spectrum, and hence is finite
\(p\)-local in the sense of the theorem.  By
Equation~\eqref{real:homology-quotients} and
Lemma~\ref{cob:regularity}, its actual unit map from the
\(p\)-local sphere induces a surjection
\[
 \BP_*\longrightarrow\BP_*X
 \quad\text{with kernel }(p,v_1,\ldots,v_n).
\]

For both cases, the polynomial cooperation algebra
\(\BP_*\BP=\BP_*[t_1,t_2,\ldots]\) is flat over the
coefficient ring \cite[Section~4.4]{HazewinkelIII}. Thus
\(\BP\)-homology of a spectrum carries its natural
\(\BP_*\BP\)-coaction, and the map from the sphere is
a comodule map. Since it is surjective on homology, naturality
determines the coaction on every element of the quotient: it is
exactly the coaction induced from the coefficient comodule
\(\BP_*\).  Thus the displayed isomorphism has the canonical
quotient comodule structure, and its generator is the image of
\(1\in\BP_0\), in degree zero.  This proves the assertion
for every \(n\), with the prime allowed to depend on \(n\).
\end{proof}
==== END Finite-Smith-Toda-Complexes-at-Varying-Primes//08-realization.tex ====

==== BEGIN Finite-Smith-Toda-Complexes-at-Varying-Primes//main.tex ====
\documentclass[11pt]{article}
\input{preamble}
\title{Finite Smith--Toda Complexes at Varying Primes}
\author{OpenAI}
\date{September 23, 2026}
\begin{document}
\maketitle
\begin{abstract}
For every nonnegative integer $n$, we construct a Smith--Toda complex
$V(n)$ at some prime $p$ depending on $n$. It is a finite $p$-local
spectrum whose Brown--Peterson homology is
$\BP_*/(p,v_1,\ldots,v_n)$ with its canonical comodule structure and
generator in degree zero. Each listed generator is killed to its first
power.
\end{abstract}
\tableofcontents
\input{sections/00-introduction}
\input{sections/01-foundations}
\input{sections/02-uniform-power}
\input{sections/03-layer}
\input{sections/04-universal}
\input{sections/05-connectivity}
\input{sections/06-normalization}
\input{sections/07-cobordism}
\input{sections/08-realization}
\bibliographystyle{plain}
\bibliography{references}
\end{document}
==== END Finite-Smith-Toda-Complexes-at-Varying-Primes//main.tex ====

==== BEGIN Finite-Smith-Toda-Complexes-at-Varying-Primes//preamble.tex ====
\usepackage[T1]{fontenc}
\usepackage{lmodern}
\usepackage{amsmath,amssymb,amsthm,mathtools}
\usepackage[margin=1.1in]{geometry}
\usepackage{microtype}
\usepackage{enumitem}
\usepackage{booktabs,longtable}
\usepackage[colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue]{hyperref}
\usepackage{url}
\hypersetup{pdftitle={Finite Smith--Toda Complexes at Varying Primes},pdfauthor={OpenAI},bookmarksnumbered=true}
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}
\newcommand{\Z}{\mathbb Z}
\newcommand{\Q}{\mathbb Q}
\newcommand{\F}{\mathbb F}
\newcommand{\U}{\mathcal U}
\newcommand{\cC}{\mathcal C}
\newcommand{\cG}{\mathcal C_G}
\newcommand{\cE}{\mathcal E}
\newcommand{\cEp}{\mathcal E'}
\newcommand{\one}{\mathbf 1}
\newcommand{\MU}{\mathrm{MU}}
\newcommand{\BP}{\mathrm{BP}}
\newcommand{\HZ}{H\Z}
\newcommand{\HQ}{H\Q}
\newcommand{\Sp}{\mathrm{Sp}}
\newcommand{\CAlg}{\mathrm{CAlg}}
\newcommand{\Mod}{\mathrm{Mod}}
\newcommand{\Ind}{\mathrm{Ind}}
\newcommand{\Pic}{\mathrm{Pic}}
\DeclareMathOperator{\map}{map}
\DeclareMathOperator{\Map}{Map}
\DeclareMathOperator{\Sym}{Sym}
\DeclareMathOperator{\Tot}{Tot}
\DeclareMathOperator{\Tr}{Tr}
\DeclareMathOperator{\cofib}{cofib}
\DeclareMathOperator{\fib}{fib}
\DeclareMathOperator{\id}{id}
\DeclareMathOperator{\gr}{gr}
\DeclareMathOperator{\wt}{wt}
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==== END Finite-Smith-Toda-Complexes-at-Varying-Primes//preamble.tex ====

==== BEGIN Finite-Smith-Toda-Complexes-at-Varying-Primes//towers.tex ====
\begin{figure}[htbp]
\centering
\[
\begin{array}{ccc}
 L_a & \xrightarrow{\ \text{adjoin }e'_a\ } & L_{a+1} \\[5pt]
 \big\downarrow\mathrlap{\scriptstyle\,//t} &&
 \big\downarrow\mathrlap{\scriptstyle\,//t} \\[5pt]
 F^*h_\Omega(Y_{a+1}) & \longrightarrow & F^*h_\Omega(Y_{a+2})
\end{array}
\]
\caption{Reduction of the auxiliary tower constructs the next ordinary
attaching class. The upper arrow nulls $g_a$; under the vertical layer
equivalences its reduction is $b_{a+1}$, and the lower arrow adjoins
the specified null $e_{a+1}$. The same square holds after the
compatible base change to complex cobordism.}
\label{fig:towers}
\end{figure}
==== END Finite-Smith-Toda-Complexes-at-Varying-Primes//towers.tex ====

==== BEGIN Finite-Smith-Toda-Complexes-at-Varying-Primes//references.bib ====
@article{Serre1951,
  author = {Serre, Jean-Pierre},
  title = {Homologie singuli{\`e}re des espaces fibr{\'e}s. Applications},
  journal = {Annals of Mathematics},
  series = {2},
  volume = {54},
  number = {3},
  year = {1951},
  pages = {425--505},
  doi = {10.2307/1969485},
  url = {https://webhomes.maths.ed.ac.uk/~v1ranick/papers/serre.pdf}
}

@misc{LurieHA,
  author = {Lurie, Jacob},
  title = {Higher Algebra},
  year = {2017},
  note = {Version dated September 18, 2017},
  url = {https://www.math.ias.edu/~lurie/papers/HA.pdf}
}

@article{BSS2020,
  author = {Barthel, Tobias and Schlank, Tomer M. and Stapleton, Nathaniel},
  title = {Chromatic homotopy theory is asymptotically algebraic},
  journal = {Inventiones Mathematicae},
  volume = {220},
  number = {3},
  year = {2020},
  pages = {737--845},
  doi = {10.1007/s00222-019-00943-9},
  note = {Locators refer to the \href{https://www.ms.uky.edu/~njst237/papers/ultra.pdf}{author version}},
  url = {https://www.ms.uky.edu/~njst237/papers/ultra.pdf}
}

@article{MNN2017,
  author = {Mathew, Akhil and Naumann, Niko and Noel, Justin},
  title = {Nilpotence and descent in equivariant stable homotopy theory},
  journal = {Advances in Mathematics},
  volume = {305},
  year = {2017},
  pages = {994--1084},
  doi = {10.1016/j.aim.2016.09.027},
  note = {Cited version: \href{https://arxiv.org/abs/1507.06869v2}{arXiv:1507.06869v2}},
  url = {https://arxiv.org/abs/1507.06869v2}
}

@article{NS2018,
  author = {Nikolaus, Thomas and Scholze, Peter},
  title = {On topological cyclic homology},
  journal = {Acta Mathematica},
  volume = {221},
  number = {2},
  year = {2018},
  pages = {203--409},
  doi = {10.4310/ACTA.2018.v221.n2.a1},
  note = {Cited version: \href{https://arxiv.org/abs/1707.01799v2}{arXiv:1707.01799v2}},
  url = {https://arxiv.org/abs/1707.01799v2}
}

@article{Novikov1962,
  author = {Novikov, S. P.},
  title = {Homotopy properties of {Thom} complexes},
  journal = {Matematicheskii Sbornik, Novaya Seriya},
  volume = {57(99)},
  number = {4},
  year = {1962},
  pages = {407--442},
  language = {Russian},
  url = {https://www.mathnet.ru/eng/sm4658},
  note = {Coefficient-theorem locator refers to the author's English translation,
    \href{https://homepage.mi-ras.ru/~snovikov/6.pdf}{Chapter 2, Section 2.4, Theorem 4}}
}

@article{Quillen1969,
  author = {Quillen, Daniel},
  title = {On the formal group laws of unoriented and complex cobordism theory},
  journal = {Bulletin of the American Mathematical Society},
  volume = {75},
  number = {6},
  year = {1969},
  pages = {1293--1298},
  doi = {10.1090/S0002-9904-1969-12401-8},
  url = {https://www.sas.rochester.edu/mth/sites/doug-ravenel/otherpapers/QuillenBP.pdf}
}

@article{HazewinkelI,
  author = {Hazewinkel, Michiel},
  title = {Constructing formal groups {I}: The local one dimensional case},
  journal = {Journal of Pure and Applied Algebra},
  volume = {9},
  number = {2--3},
  year = {1977},
  pages = {131--149},
  doi = {10.1016/0022-4049(77)90061-5},
  url = {https://ir.cwi.nl/pub/9976/9976D.pdf}
}

@article{HazewinkelIII,
  author = {Hazewinkel, Michiel},
  title = {Constructing formal groups {III}: Applications to complex cobordism and {Brown--Peterson} cohomology},
  journal = {Journal of Pure and Applied Algebra},
  volume = {10},
  number = {1},
  year = {1977},
  pages = {1--18},
  doi = {10.1016/0022-4049(77)90026-3},
  url = {https://ir.cwi.nl/pub/10012/10012D.pdf}
}

@article{JohnsonNoel2010,
  author = {Johnson, Niles and Noel, Justin},
  title = {For complex orientations preserving power operations, {$p$}-typicality is atypical},
  journal = {Topology and its Applications},
  volume = {157},
  number = {14},
  year = {2010},
  pages = {2271--2288},
  doi = {10.1016/j.topol.2010.06.007},
  note = {Cited version: \href{https://arxiv.org/abs/0910.3187v3}{arXiv:0910.3187v3}},
  url = {https://arxiv.org/abs/0910.3187v3}
}

@article{Toda1971,
  author = {Toda, Hirosi},
  title = {On spectra realizing exterior parts of the {Steenrod} algebra},
  journal = {Topology},
  volume = {10},
  number = {1},
  year = {1971},
  pages = {53--65},
  doi = {10.1016/0040-9383(71)90017-6},
  url = {https://www.sciencedirect.com/science/article/pii/0040938371900176}
}

@article{Nave2010,
  author = {Nave, Lee S.},
  title = {The {Smith--Toda} complex {$V((p+1)/2)$} does not exist},
  journal = {Annals of Mathematics},
  series = {2},
  volume = {171},
  number = {1},
  year = {2010},
  pages = {491--509},
  doi = {10.4007/annals.2010.171.491},
  url = {https://annals.math.princeton.edu/2010/171-1/p10}
}

@article{CulverZhang2024,
  author = {Culver, Dominic Leon and Zhang, Ningchuan},
  title = {Exotic {Picard} groups and chromatic vanishing via the {Gross--Hopkins} duality},
  journal = {Topology and its Applications},
  volume = {341},
  year = {2024},
  pages = {108742},
  doi = {10.1016/j.topol.2023.108742},
  eprint = {2203.09455v3},
  archivePrefix = {arXiv},
  primaryClass = {math.AT},
  url = {https://arxiv.org/abs/2203.09455v3},
  note = {The cited Remark 3.29 is in \href{https://arxiv.org/abs/2203.09455v3}{arXiv:2203.09455v3}, dated September 30, 2023}
}

@book{May1977,
  author = {May, J. Peter},
  title = {{$E_\infty$} Ring Spaces and {$E_\infty$} Ring Spectra},
  series = {Lecture Notes in Mathematics},
  volume = {577},
  publisher = {Springer-Verlag},
  address = {Berlin--New York},
  year = {1977},
  note = {With contributions by Frank Quinn, Nigel Ray, and J{\o}rgen Tornehave},
  url = {https://www.math.uchicago.edu/~may/BOOKS/e_infty.pdf}
}

@book{Hatcher2002,
  author = {Hatcher, Allen},
  title = {Algebraic Topology},
  publisher = {Cambridge University Press},
  year = {2002},
  url = {https://pi.math.cornell.edu/~hatcher/AT/AT.pdf}
}

@article{Milnor1960,
  author = {Milnor, John},
  title = {On the cobordism ring {$\Omega^*$} and a complex analogue, {Part I}},
  journal = {American Journal of Mathematics},
  volume = {82},
  number = {3},
  pages = {505--521},
  year = {1960},
  doi = {10.2307/2372970},
  url = {https://www.sas.rochester.edu/mth/sites/doug-ravenel/otherpapers/Milnor-MU.pdf}
}

@article{Adams1966,
  author = {Adams, J. F.},
  title = {On the groups {$J(X)$}. {IV}},
  journal = {Topology},
  volume = {5},
  year = {1966},
  pages = {21--71},
  doi = {10.1016/0040-9383(66)90004-8}
}

@article{Smith1970,
  author = {Smith, Larry},
  title = {On realizing complex bordism modules. {Applications} to the stable homotopy of spheres},
  journal = {American Journal of Mathematics},
  volume = {92},
  number = {4},
  year = {1970},
  pages = {793--856},
  doi = {10.2307/2373397}
}

@misc{RandalWilliams2026,
  author = {Randal-Williams, Oscar},
  title = {A chromatic approach to homological stability},
  year = {2026},
  eprint = {2508.20629v3},
  archivePrefix = {arXiv},
  primaryClass = {math.AT},
  note = {Version 3, July 21, 2026},
  url = {https://arxiv.org/abs/2508.20629v3}
}

@misc{KatoShimomura2025,
  author = {Kato, Ryo and Shimomura, Katsumi and Shimomura, Mao-no-suke},
  title = {A note on the {Greek} letter elements of the homotopy of the {$L_n$}-local spheres},
  year = {2025},
  eprint = {2507.02155v1},
  archivePrefix = {arXiv},
  primaryClass = {math.AT},
  note = {Version 1, July 2, 2025},
  doi = {10.48550/arXiv.2507.02155},
  url = {https://arxiv.org/abs/2507.02155v1}
}

@article{HopkinsSmith1998,
  author = {Hopkins, Michael J. and Smith, Jeffrey H.},
  title = {Nilpotence and stable homotopy theory. {II}},
  journal = {Annals of Mathematics},
  series = {2},
  volume = {148},
  number = {1},
  year = {1998},
  pages = {1--49},
  doi = {10.2307/120991},
  url = {https://www.sas.rochester.edu/mth/sites/doug-ravenel/otherpapers/HS.pdf}
}

@article{tomDieck1968,
  author = {tom Dieck, Tammo},
  title = {Steenrod-Operationen in Kobordismen-Theorien},
  journal = {Mathematische Zeitschrift},
  volume = {107},
  year = {1968},
  pages = {380--401},
  doi = {10.1007/BF01110069}
}

@article{Quillen1971,
  author = {Quillen, Daniel},
  title = {Elementary proofs of some results of cobordism theory using {Steenrod} operations},
  journal = {Advances in Mathematics},
  volume = {7},
  year = {1971},
  pages = {29--56},
  doi = {10.1016/0001-8708(71)90041-7},
  url = {https://www.sas.rochester.edu/mth/sites/doug-ravenel/otherpapers/QuillenElem.pdf}
}

@article{BrownPeterson1966,
  author = {Brown, Jr., Edgar H. and Peterson, Franklin P.},
  title = {A spectrum whose {$Z_p$} cohomology is the algebra of reduced {$p$}th powers},
  journal = {Topology},
  volume = {5},
  number = {2},
  year = {1966},
  pages = {149--154},
  doi = {10.1016/0040-9383(66)90015-2}
}
==== END Finite-Smith-Toda-Complexes-at-Varying-Primes//references.bib ====