==== TABLE OF CONTENTS ====
A-finite-Smith-Toda-complex-V4-at-the-prime-1009//00-introduction.tex
A-finite-Smith-Toda-complex-V4-at-the-prime-1009//01-framework.tex
A-finite-Smith-Toda-complex-V4-at-the-prime-1009//02-lower-complexes.tex
A-finite-Smith-Toda-complex-V4-at-the-prime-1009//02a-cobar-preliminaries.tex
A-finite-Smith-Toda-complex-V4-at-the-prime-1009//03-may-and-first-obstruction.tex
A-finite-Smith-Toda-complex-V4-at-the-prime-1009//04-frobenius-elimination.tex
A-finite-Smith-Toda-complex-V4-at-the-prime-1009//05-final-obstruction.tex
A-finite-Smith-Toda-complex-V4-at-the-prime-1009//appendix-resolution.tex
A-finite-Smith-Toda-complex-V4-at-the-prime-1009//main.tex
A-finite-Smith-Toda-complex-V4-at-the-prime-1009//preamble.tex
A-finite-Smith-Toda-complex-V4-at-the-prime-1009//references.bib
===========================

==== BEGIN A-finite-Smith-Toda-complex-V4-at-the-prime-1009//00-introduction.tex ====
\section{Introduction}\label{intro:section}

The Smith--Toda realization problem asks how far one can kill the chromatic
parameters in Brown--Peterson homology, one at a time, while retaining a finite
spectrum. At a prime $p$, write
\[
 BP_* = \mathbb Z_{(p)}[v_1,v_2,\ldots],
 \qquad |v_i|=2(p^i-1),
\]
with Hazewinkel generators. A \emph{finite $p$-local spectrum} is a retract
of the $p$-localization of a finite CW spectrum. A \emph{Smith--Toda
complex $V(n)$} is such a spectrum with Brown--Peterson homology
$BP_*/(p,v_1,\ldots,v_n)$, including the canonical $BP_*BP$-comodule
structure and a generator in degree zero. In this quotient each listed
generator is killed to its first power. The comodule records the
Brown--Peterson cooperations, so its specification is part of the
realization problem, in addition to the underlying graded module.
The periodicity theorem gives generalized Moore spectra whose homology
kills suitable powers of the initial generators
\citep[Proposition~5.14]{HopkinsSmith1998}. Requiring every exponent
to be one is a more restrictive realization problem.

The mod-$p$ Moore spectrum gives $V(0)$ at every prime. The classical
constructions of $V(1)$, $V(2)$, and $V(3)$ for $p>2$, $p>4$, and $p>6$
are due to Adams, Smith, and Toda
\citep{Adams1966,Smith1970,Toda1971}; see also the historical account in
\citet[p.~493]{Nave2010}. Successive cofibers turn a suitable first-power
$v_i$ self-map into the next quotient. The difficulty is producing the
self-map on the finite spectrum: an algebraically primitive coefficient
need not survive the Adams--Novikov spectral sequence (ANSS) to an actual map.
These constructions thus connect the algebra of complex cobordism with
specific attaching maps in the stable homotopy category.

The next case remained outside those constructions. The September 2023
version of \citet[Remark~3.29]{CulverZhang2024} records the existence of
$V(4)$ as open at every prime. There are also height-dependent
nonexistence results: \citet[Theorem~1.3]{Nave2010} proves that
$V((p+1)/2)$ does not exist for $p\geq7$, and that $V((p-1)/2)$,
if it exists, cannot be a ring spectrum. Those obstructions leave
room for $V(4)$ at a sufficiently large prime.
We establish the following specified-prime realization.

\begin{theorem}\label{intro:main}
At $p=1009$ there is a finite $p$-local spectrum $X$ and an isomorphism
\[
 BP_*X\ \cong\ BP_*/(p,v_1,v_2,v_3,v_4)
\]
of graded $BP_*BP$-comodules, with the canonical quotient comodule on the
right and its generator in degree zero.
\end{theorem}

Theorem~\ref{intro:main} gives a positive answer to the Smith--Toda $V(4)$
existence problem at the stated prime. Its conclusion concerns the ordinary
$p$-local stable category; finiteness is established before any chromatic
localization. For comparison,
\citet[Theorem~1.7]{KatoShimomuraShimomura2025} construct at this prime
an $L_5$-local spectrum $W_5$ with
\[
 BP_*W_5=v_5^{-1}BP_*/(p,v_1,v_2,v_3,v_4),
\]
where $L_5$ denotes localization with respect to $v_5^{-1}BP$.
That realization has $v_5$ inverted and does not supply the ordinary
finite realization of the unlocalized quotient required here.
The choice $1009$ makes the finite inequalities in our calculation
generous. We do not require a product on the final spectrum $X$.

\subsection{Methods and proof strategy}

The algebraic setting comes from the spectrum of Brown and Peterson
\citep{BrownPeterson1966}, Quillen's formal-group description of
cobordism and its $p$-typical summand \citep{Quillen1969}, and
Hazewinkel's explicit $p$-typical generators \citep{HazewinkelIII}.
Our main tool is the sphere-source Adams--Novikov spectral sequence,
whose cobordism-theoretic construction goes back to Novikov
\citep{Novikov1967}. We use its ordinary $BP$ resolution, its cobar
$E_2$ term, and its compatibility with actual binary maps; see
\citet[Sections~2.2--2.3]{Ravenel2004}. The algebraic estimates use
filtered Hopf-algebra cohomology as developed by
\citet[Section~4, Theorem~4]{May1966},
with the odd-primary filtration of
\citet[Theorem~3.2.5]{Ravenel2004} and an additional filtration
adjustment constructed below.

A closer predecessor is the work of
\citet[author preprint, pp.~2--6]{LeeRavenel1994} on the nilpotence
order of the first beta-family sphere class $\beta_1$.
They use a modified May calculation on $V(3)$ and constrain
Adams--Novikov differentials through multiplication by $\beta_1$
and the $V(2)$-module structure of $V(3)$.
Our final obstruction calculation requires an annihilation relation
for products of a beta power with specific filtration-five classes,
together with a bound on the page at which those products vanish.
Toda's work on sphere relations \citep[p.~839, Corollary]{Toda1967}
and his subsequent cyclic extended-power constructions
\citep[Sections~1--3, especially Theorem~3]{Toda1968} provide the
topological antecedents. We give the particular coefficient-cohomology
calculations and finite-cell arguments needed for this bound below.

The construction has three stages. First we construct $Y=V(3)$ with a
binary product and a two-sided unit. These unit identities are enough
to turn a sphere map realizing $v_4$ into a self-map of $Y$. We obtain
the lower complexes and their products by finite cofiber constructions
and explicit vanishing of their attaching obstructions; we do not need
an associative product on $Y$.

Second, we prove that the filtration-zero class $v_4$ survives in the
Adams--Novikov spectral sequence for $Y$. To describe the obstruction
calculation, put $q=2p-2$ and $u_i=1+p+\cdots+p^{i-1}$ for $i\geq1$.
Divide the internal degree by $q$ and call the resulting integer the
\emph{weight}. A cobar class of cohomological degree $s$ and weight $w$
lies in stem $qw-s$. Every possible positive differential on $v_4$
therefore has target
\begin{equation}\label{intro:targets}
 (s,w)=(qk+1,u_4+k),\qquad k\geq1.
\end{equation}
A finite May calculation leaves only $k=1,2,p-1,p$. For $k=1$, a
relation supplied by the coboundary of $v_5$, together with an injective
multiplication on the target, forces the differential to vanish. For
$k=2,p-1$, we compute actual cohomology by taking the Frobenius kernel
of the group of truncated additive coordinate changes. The quotient
acts triangularly on kernel cohomology. Filtering this coefficient
module by weight identifies the first possible differential and
checks that both its source and its target are present. This step
controls the polynomial generators as well as the exterior generators
on the May page.

The remaining case $k=p$ requires a topological argument. We write
$\beta\in\pi_{pq-2}S^0_{(p)}$ for the first beta-family sphere class.
The entire $E_2$ target consists of products of a large power of its
detector with certain filtration-five classes of $Y$.
We realize those low-filtration classes by maps
$\gamma$ and construct an explicit relation
\[
 \beta^{\,p(p-2)+1}\gamma=0.
\]
The construction uses a cyclic extended power of a sphere and the
averaging retract of a smash power of a two-cell spectrum. The corresponding
Adams--Novikov product has filtration strictly smaller than the length of
the prospective
differential on $v_4$. Consequently the whole target has vanished
before that differential could occur. The bound on the page is the
point: eventual vanishing in homotopy, without such a bound, would
not settle this obstruction.

Finally the filtration-zero edge gives a map
$S^{2(p^4-1)}\to Y$ with Brown--Peterson Hurewicz image $v_4$.
Multiplication by the binary product on $Y$ extends it to
$\Sigma^{2(p^4-1)}Y\to Y$. Its cofiber is finite, and the
nonzerodivisor property of $v_4$ identifies its homology with the
required quotient. The bottom sphere map identifies the quotient
coaction and fixes the generator in degree zero.

Section~\ref{fw:section} fixes the Brown--Peterson coordinates and
the topological spectral-sequence conventions.
Section~\ref{may:preliminaries} constructs the filtered cobar calculation,
and Section~\ref{low:section} supplies the unital lower complexes.
Section~\ref{may:section} lists the height-four targets and removes the
first differential; Section~\ref{frob:section} proves the two Frobenius
vanishings. Section~\ref{top:section} establishes the homotopy relation,
its page bound, and the realizing cofiber. Appendix~\ref{app:resolution}
provides the ordinary convergent resolution and the filtered binary
pairing that connects the algebraic calculation to the actual
homotopy product.
==== END A-finite-Smith-Toda-complex-V4-at-the-prime-1009//00-introduction.tex ====

==== BEGIN A-finite-Smith-Toda-complex-V4-at-the-prime-1009//01-framework.tex ====
\section{Brown--Peterson coordinates and the Adams--Novikov resolution}
\label{fw:section}

Throughout the proof,
\begin{equation}
 p=1009,\qquad q=2p-2=2016,\qquad
 u_i=\frac{p^i-1}{p-1}\quad(i\geq 0).
 \label{fw:parameters}
\end{equation}
The integer $1009$ is prime: its square root is less than $32$, and none of
$2,3,5,7,11,13,17,19,23,29,31$ divides it.  Put
\[
 R=BP_* =\mathbb Z_{(p)}[v_1,v_2,\ldots],\qquad
 \Gamma=BP_*BP=R[t_1,t_2,\ldots],\qquad
 I_m=(p,v_1,\ldots,v_{m-1}).
\]
Both $v_i$ and $t_i$ have internal degree $qu_i$.  We call internal degree
divided by $q$ the \emph{weight}.  A cobar class of cohomological degree $s$
and weight $w$ has stem $qw-s$.

We use the ordinary Brown--Peterson spectrum, its polynomial cooperation
algebra, and a strictly unital associative model of its multiplication.
For the identification of these polynomial rings with the homotopy and
cooperations of $BP$, see
\citet[Theorems~4.1.18--4.1.19]{Ravenel2004}. The Hazewinkel recursion
and the integrality of its polynomial generators are given in
\citet[Equation~A2.2.1 and Theorem~A2.2.3]{Ravenel2004}; the formal-sum
coordinate convention is explained in the proof of Theorem~A2.1.27 there.
The formal-group foundations are due to
\citet[Section~5, Theorem~4]{Quillen1969}; the explicit generators and
cooperations are developed in \citet[Sections~3.1 and~4.4]{HazewinkelIII}.
A coherent associative model exists, in particular, by forgetting structure
from \citet[Theorem~1.1]{BasterraMandell2013}.  No commutative or
$E_\infty$ model of $BP$ is required below.  We give the particular coordinate,
convergence, and pairing arguments needed for the construction.

\subsection{Coordinates and additive ranges}

Let $a_i$ be the coefficients in the formal-group sum expression
\[
 [p]_F(x)=\mathop{\sum}\nolimits_F a_i x^{p^i},\qquad a_0=p.
\]
The strict coordinate change is written in the same convention as
$\sum_F t_i x^{p^i}$, with $t_0=1$.  Reversing the convention for composition
reverses the cobar conventions without changing any of the arguments.

\begin{lemma}[Leading coefficients and low cooperations]
\label{fw:coordinates}
For each $m\geq 1$, $I_m$ is an invariant ideal and $v_m$ is primitive
modulo $I_m$.  Modulo $I_m$, the coproducts of $t_i$, $i\leq m$, are the
additive-composition coproducts.  At the specialization that kills $p$ and
all $v_i$, the resulting Hopf algebra is
\begin{equation}
 P=\mathbb F_p[t_1,t_2,\ldots],\qquad
 \Delta(t_i)=\sum_{\ell=0}^i t_\ell\otimes t_{i-\ell}^{p^\ell}.
 \label{fw:additive-hopf}
\end{equation}
\end{lemma}

\begin{proof}
Write the logarithm as $\log_F(x)=\sum_{i\geq0}l_i x^{p^i}$, where $l_0=1$.
The Hazewinkel recursion is
\[
 pl_n=v_n+\sum_{i=1}^{n-1}l_i v_{n-i}^{p^i}.
\]
Applying the logarithm to the formal sum for $[p]_F$ and comparing the
coefficient of $x^{p^n}$ shows that the coefficient of the indecomposable
$v_n$ in $a_n$ is
\begin{equation}
 c_n=1-p^{p^n-1}.
 \label{fw:coordinate-unit}
\end{equation}
Indeed the two terms involving $l_n$ are $pl_n$ and $p^{p^n}l_n$, and the
coefficient of $v_n$ in $l_n$ is $1/p$.  All other terms involve lower
$v_i$.  The coefficients $a_n$ are integral, so the grading and polynomial
independence of the $v_i$ give
$a_n-c_nv_n\in(v_1,\ldots,v_{n-1})R$.  Each $c_n$ is a $p$-local unit.

The invariance and primitivity follow inductively.  The element $p$ is
invariant.  Once the lower coefficients vanish, the first surviving term
of the $p$-series is $c_m v_m x^{p^m}$.  A strict coordinate change has
leading term $x$, so comparison of this first surviving term on the two
sides of the coordinate-change identity fixes $c_m v_m$, and hence $v_m$.
This proves both invariance of the next ideal and the asserted primitivity.

After all base generators have been killed, the formal group is additive,
and composition of additive $p$-polynomials gives
Equation~\eqref{fw:additive-hopf}.  Modulo $I_m$, a correction to this
coproduct must have a positive base coefficient, of weight at least $u_m$.
For $i<m$ its weight is already too large.  For $i=m$ the only possibility
is a constant multiple of $v_m$ with no positive-weight cooperation factor.
Applying either counit excludes that term.  This proves the stated range.
\end{proof}

\begin{lemma}[The relation supplied by $v_5$]
\label{fw:v5-relation}
In the cobar cohomology with coefficients $R/I_4$, there is a scalar
$e\in\mathbb F_p$ such that
\begin{equation}
 v_4\bigl(h_{1,4}+e v_4^{p-1}h_{1,0}\bigr)=0,
 \label{fw:v5-cohomology-relation}
\end{equation}
where $h_{1,j}$ is represented by the primitive $t_1^{p^j}$.
\end{lemma}

\begin{proof}
Modulo $I_4$, the first two formal-sum coefficients are unit multiples of
$v_4,v_5$.  There is no decomposable correction involving $v_4$ to the
second of these coefficients: $u_5=pu_4+1$ is not a multiple of $u_4$.
The formal group law has no nonlinear term of total ordinary degree less
than $p^4$, because a coefficient of such a term would have positive weight
less than $u_4$.  Thus, through exponent $p^5$, its $p$-series is the ordinary
sum of its $x^{p^4}$ and $x^{p^5}$ terms.  The nonlinear terms in adding
these two inputs have exponent greater than $p^5$.

In the identity intertwining this $p$-series with a strict coordinate
change, only $x+t_1x^p$ from that change can affect these two exponents.
On the side where the change is applied after the $p$-series, the
coefficient of $x^{p^5}$ acquires a term proportional to $v_4^p t_1$.
On the other side it acquires a term proportional to
$v_4t_1^{p^4}$.  All nonlinear corrections to the strict formal-sum change
enter too late after taking the $p^4$-th power.  Consequently
\[
 \eta_R(v_5)-v_5=c\,v_4t_1^{p^4}+d\,v_4^p t_1
 \quad\text{in }\Gamma/I_4\Gamma,
\]
where $c,d\in\mathbb F_p$ and $c\ne0$.  The nonzero coefficient follows
also directly from the unit factors $c_4,c_5$ in
Equation~\eqref{fw:coordinate-unit}; their reductions cannot annihilate
the first displayed contribution.  Divide this coboundary by $c$ and put
$e=d/c$.  Since $t_1$ is primitive modulo $I_4$, all its $p$-power iterates
are primitive, and the resulting cohomology relation is
Equation~\eqref{fw:v5-cohomology-relation}.
\end{proof}

For a comodule concentrated in weights that are nonnegative integers, let
$C^{s,w}$ denote normalized cobar cochains of length $s$ and weight $w$.
We use the convention in which a positive-length cochain has a coefficient
followed by $s$ reduced cooperation factors.  This convention fixes the
signs of the usual cobar differential and cup product.

\begin{lemma}[Normalized positivity]
\label{fw:normalized-bound}
If the coefficient comodule has no negative weights, then
\begin{equation}
 C^{s,w}\ne0\quad\Longrightarrow\quad w\geq s.
 \label{fw:positivity}
\end{equation}
\end{lemma}

\begin{proof}
Every reduced cooperation factor is a polynomial monomial containing at
least one $t_i$, and hence has weight at least one.  The coefficient has
nonnegative weight.  Adding these weights proves the claim.
\end{proof}

\begin{lemma}[A comparison in three cochain degrees]
\label{fw:range-comparison}
Suppose $s\geq1$, $w<u_{m+1}$, and
$w-u_m<s-1$.  Additive specialization identifies the normalized cobar
complexes for $R/I_m$ and $P$, in weight $w$, in all three degrees
$s-1,s,s+1$.  It intertwines both adjacent differentials, and therefore
induces an isomorphism on cohomology in degree $(s,w)$.
The same conclusion in every cochain degree holds if $w<u_m$.
\end{lemma}

\begin{proof}
In each of the three degrees, any positive coefficient monomial consumes
at least $u_m$ weight.  It would leave at most $w-u_m<s-1$ for at least
$s-1$ reduced bar factors, which is impossible.  A coordinate $t_i$ with
$i\geq m+1$ exceeds the entire weight budget.  Thus these cochain groups
have scalar coefficients and only $t_1,\ldots,t_m$ in their bars.  Their
coproducts are additive by Lemma~\ref{fw:coordinates}.  This identifies
the two groups adjacent to the desired cohomology group as well as the
group itself, so both kernels and images agree.  If $w<u_m$, the same
exclusion of coefficients and coordinates works without a cochain-length
restriction.
\end{proof}

\subsection{Convergence, pairings, and the edge}

The following two propositions specify the topological spectral sequence and
the binary pairings used throughout the construction. The ordinary
Adams--Novikov resolution and its convergence are treated in
\citet[Theorems~2.2.3, 2.2.13--2.2.14]{Ravenel2004};
\citet[Theorem~2.3.3]{Ravenel2004} gives the binary-pairing, Leibniz,
and homotopy-compatibility statements in this form. Their complete proofs
in the particular resolution used here,
including the ordinary totalization model and its filtration, are given in
Appendix~\ref{app:resolution}.

\begin{proposition}[The bounded-below Adams--Novikov resolution]
\label{fw:anss}
For every bounded-below $p$-local spectrum $Z$, the sphere-source
Adams--Novikov spectral sequence has
\[
 E_2^{s,t}(Z)=H^{s,t}\bigl(C^\bullet_\Gamma(BP_*Z)\bigr)
 \quad\Longrightarrow\quad \pi_{t-s}Z.
\]
It converges strongly, and the induced filtration in each stem is finite.
The differential $d_r$ changes $(s,t)$ by $(r,r-1)$.  The filtration-zero
edge identifies $E_\infty^{0,*}$ with the image of the $BP$-Hurewicz map
$\pi_*Z\longrightarrow BP_*Z$.
\end{proposition}
The proof is given in Appendix~\ref{app:convergence}.


\begin{proposition}[Binary pairings of the resolution]
\label{fw:anss-pairing}
For bounded-below $p$-local spectra $Z_1,Z_2,Z_3$, an actual map
$f:Z_1\wedge Z_2\to Z_3$ induces pairings of the
Adams--Novikov spectral sequences, with the Leibniz rule on every page.
The $E_2$ pairing is the cobar cup pairing induced by the exterior
$BP$-homology pairing, and the $E_\infty$ pairing is the associated-graded
pairing of the actual homotopy map $f$.  No associativity of $f$ is needed.
If $Z_1=Z_2=Z_3=Z$ has a binary two-sided unital multiplication and
$BP_*Z=R/I$ is cyclic on its unit, the $E_2$ pairing is the ordinary
multiplication on cobar cohomology with quotient-ring coefficients.
\end{proposition}
The proof is given in Appendix~\ref{app:pairing}.


\begin{corollary}[Sparsity]
\label{fw:sparsity}
If $BP_*Z$ is concentrated in internal degrees divisible by $q$, then all
pages of its Adams--Novikov spectral sequence have that same internal
sparsity.  Every possibly nonzero positive differential has length
$r=qk+1$ for an integer $k\geq1$.  In weight notation it changes
$(s,w)$ to $(s+qk+1,w+k)$.  In particular, a primitive coefficient of
weight $u_n$ in filtration zero can have only targets
\begin{equation}
 (s,w)=(qk+1,u_n+k),\qquad k\geq1.
 \label{fw:primitive-targets}
\end{equation}
\end{corollary}

\begin{proof}
The coefficient and cooperation gradings give the assertion on the cobar
page.  A differential preserves the sparse support only if $q$ divides
$r-1$.  Subsequent pages are subquotients of earlier pages.  Since $r\geq2$
on or after the cohomology page, the first allowed length is $q+1$.
\end{proof}

\begin{lemma}[The additive comparison for the $v_4$ targets]
\label{fw:additive-comparison}
Let $s=qk+1$, $w=u_4+k$, $k\geq1$.  A nonzero normalized cobar group
in degree $(s,w)$ with coefficients $R/I_4$ requires
\begin{equation}
 (q-1)k\leq u_4-1,\qquad
 w\leq\frac{qu_4-1}{q-1}<2u_4<p^4<u_5.
 \label{fw:preliminary-v4-bound}
\end{equation}
For every such $k$, additive specialization is an isomorphism on
cohomology in degree $(s,w)$, by an identification of the cochain groups
of lengths $s-1,s,s+1$.  No $v_4$ coefficient or $t_4$ bar occurs in
any of those three cochain groups.
\end{lemma}

\begin{proof}
The first two inequalities are the rearrangement of $w\geq s$.
Here
\[
 u_4=1{,}028{,}262{,}820,\qquad
 2u_4=2{,}056{,}525{,}640<p^4=1{,}036{,}488{,}922{,}561,
\]
so the remaining inequalities in
Equation~\eqref{fw:preliminary-v4-bound} hold at the fixed prime.
For a cochain length $d\in\{s-1,s,s+1\}$, a coefficient containing
$v_4$ would leave at most $k$ weight for $d\geq qk>k$ reduced factors.
It therefore cannot occur.  A bar containing $t_4$ would leave at most
$k$ weight for the other $d-1\geq qk-1>k$ bars, and likewise cannot occur.
Higher generators exceed the weight bound.  The remaining coproducts
are additive by Lemma~\ref{fw:coordinates}; equivalently, apply
Lemma~\ref{fw:range-comparison}.  This identifies both adjacent
differentials and proves the cohomology assertion.
\end{proof}
==== END A-finite-Smith-Toda-complex-V4-at-the-prime-1009//01-framework.tex ====

==== BEGIN A-finite-Smith-Toda-complex-V4-at-the-prime-1009//02-lower-complexes.tex ====
\section{The unital complexes through height three}
\label{low:section}

We first construct the lower complexes, retaining a binary two-sided unit.
The algebraic input is the May upper bound of Lemma~\ref{may:page}:
a bidegree containing no initial-page monomial has zero cobar cohomology.
The two vanishing families below serve different parts of the induction:
the first makes each $v_n$ survive to a self-map, and the second permits
the two unit maps to extend to a binary product over every pair of cells.

\begin{lemma}[Lower obstruction groups]
\label{low:algebraic-vanishing}
The following cobar cohomology groups vanish.
\begin{enumerate}
 \item For $1\leq n\leq3$ and $k\geq1$, the group with coefficients
 $R/I_n$ in bidegree
 \[
  (s,w)=(qk+1,u_n+k).
 \]
 \item For $0\leq n\leq3$, choose $e_i\in\{0,1,2\}$ for $0\leq i\leq n$
 and put
 \[
  U=\sum_{i=0}^n e_i u_i,\qquad
  \rho=\sum_{i=0}^n e_i\geq2.
 \]
 For every $k\geq1$, the group with coefficients $R/I_{n+1}$ in bidegree
 \[
  (s,w)=(qk+1-\rho,U+k)
 \]
 is zero.
\end{enumerate}
\end{lemma}

\begin{proof}
Include the first case in the notation of the second by setting
$U=u_n$ and $\rho=0$, while retaining its indicated coefficient comodule.
If the normalized cochain group in degree $(s,w)$ is nonzero,
Lemma~\ref{fw:normalized-bound} gives
\begin{equation}
 2015k\leq U+\rho-1.
 \label{low:normalized-cutoff}
\end{equation}
The exact values
\[
 (u_0,u_1,u_2,u_3,u_4)=(0,1,1010,1{,}019{,}091,1{,}028{,}262{,}820)
\]
give Table~\ref{low:cutoffs}.  Each upper bound is obtained by substituting
the displayed maxima in Equation~\eqref{low:normalized-cutoff} and taking
the integer part.  A zero upper bound on $k$ excludes all $k\geq1$.

\begin{table}[htbp]
\centering
\begin{tabular}{lrrrr}
\toprule
Case & Maximum $U$ & Maximum $\rho$ & Maximum $k$ & Maximum $w$\\
\midrule
Product, $n=0$ & $0$ & $2$ & $0$ & ---\\
Product, $n=1$ & $2$ & $4$ & $0$ & ---\\
Product, $n=2$ & $2022$ & $6$ & $1$ & $2023$\\
Product, $n=3$ & $2{,}040{,}204$ & $8$ & $1012$ & $2{,}041{,}216$\\
$v_1$ target & $1$ & $0$ & $0$ & ---\\
$v_2$ target & $1010$ & $0$ & $0$ & ---\\
$v_3$ target & $1{,}019{,}091$ & $0$ & $505$ & $1{,}019{,}596$\\
\bottomrule
\end{tabular}
\caption{Preliminary bounds for all lower obstruction bidegrees.}
\label{low:cutoffs}
\end{table}

These bounds also justify the additive comparison in the adjacent cochain
degrees.  For either remaining product case, $w<u_{n+1}$, so no positive
coefficient or coordinate $t_i$ with $i\geq n+1$ can occur.  The retained
coproducts are additive modulo $I_{n+1}$, and the last assertion of
Lemma~\ref{fw:range-comparison} applies in every degree.  For a remaining
$v_n$ target, a positive coefficient costs at least $u_n$, leaving at most
$k$ weight for $d\geq s-1=qk>k$ reduced bars when
$d\in\{s-1,s,s+1\}$.  Such a coefficient cannot occur.  A bar containing
$t_n$ or a higher coordinate leaves at most $k$ for the other
$d-1\geq qk-1>k$ bars and cannot occur either.  Hence the same three
cochain degrees are additive.  In all cases, both adjacent differentials
are identified, so the desired cohomology is identified with that for $P$.

Every remaining weight in Table~\ref{low:cutoffs} is at most $2{,}041{,}216$,
whereas
\[
 p^3=1{,}027{,}243{,}729.
\]
A May letter covering position $3$ or above has weight at least $p^3$.
Thus only positions $0,1,2$ can occur. We use the common exterior and
polynomial alphabet of Equation~\eqref{may:low-alphabet}, with the interval
coverage conventions of Section~\ref{may:preliminaries}.
Allowing this entire alphabet only enlarges the upper bound if fewer
coordinates were retained in the preceding comparison.

Suppose a monomial of the desired bidegree exists.  Let $m\leq6$ be its
number of exterior letters and $L=(s-m)/2$ its total polynomial exponent.
Every polynomial letter has weight at least $p$, so
\[
 U+k\geq pL=p(p-1)k+\frac{p(1-\rho-m)}2.
\]
For all profiles under consideration,
$U\leq2p^2+4p+6$ and $\rho\leq8$.  Consequently
\begin{equation}
 (p^2-p-1)k
 \leq U+\frac{p(\rho+5)}2
 \leq 2p^2+\frac{21}2p+6.
 \label{low:polynomial-cutoff}
\end{equation}
If $k\geq3$, the left side exceeds the last expression by at least
\[
 p^2-\frac{27}2p-9=\frac{2{,}008{,}901}{2}>0.
\]
It follows that $1\leq k\leq2$.

Write $U=F_0+pF_1+p^2F_2$.  In a product profile the digits are
\[
 F_0=e_1+e_2+e_3,\qquad F_1=e_2+e_3,\qquad F_2=e_3,
\]
with missing $e_i$ set to zero.  For a $v_n$ target they are initial
strings of ones.  In every case,
\begin{equation}
 6\geq F_0\geq F_1\geq F_2\geq0,
 \qquad F_1\leq4,\quad F_2\leq2.
 \label{low:digits}
\end{equation}
Let $H_r$ count the exterior letters covering position $r$, and let
$n_B,n_C$ be the polynomial exponents of $B,C$.  Adding weights gives
the exact identity
\begin{equation}
 F_0+k+pF_1+p^2F_2
  =H_0+p(H_1+L-n_C)+p^2(H_2+n_B+n_C).
 \label{low:weight-equation}
\end{equation}
We next extract its digits without an unproved no-carry assumption.

First, $0\leq H_0\leq3$ and $1\leq F_0+k\leq8<p$.  Reduction modulo $p$
therefore gives $H_0=F_0+k$.  Also $B,C$ each cost at least $p^2$, so
\[
 p^2(n_B+n_C)\leq w\leq2p^2+4p+8<3p^2;
\]
the last strict inequality has difference
$p^2-4p-8=1{,}014{,}037>0$.  Hence $n_B+n_C\leq2$.

Put $\delta=(m+\rho-1)/2$.  It is an integer because $s-m$ is even.
For a target with $\rho=0$, that parity forces $m$ odd, hence $m\geq1$;
for a product profile $\rho\geq2$.  Thus $\delta\geq0$.  The bounds
$m\leq6$, $\rho\leq8$ and the same parity give $\delta\leq6$.
Now $L=(p-1)k-\delta$.  Subtracting the position-zero equation from
Equation~\eqref{low:weight-equation}, dividing by $p$, and rearranging gives
\[
 H_1-k-\delta-n_C-F_1
   =p(F_2-k-H_2-n_B-n_C).
\]
Since $H_1\leq4$, the left side lies in $[-14,3]$.  Its only possible
multiple of $p=1009$ is zero.  We have therefore proved the three exact
relations
\begin{equation}
 H_0=F_0+k,\qquad
 H_2+n_B+n_C=F_2-k,\qquad
 H_1=F_1+k+\delta+n_C.
 \label{low:coverage-relations}
\end{equation}

Among the exterior letters covering position zero, only $x,a$ fail to
cover position two.  Thus $H_2\geq H_0-2$.  Equations~\eqref{low:digits}
and~\eqref{low:coverage-relations} imply
\[
 F_0-F_2+2k\leq2.
\]
As $k\geq1$, equality is forced: $k=1$ and
$F_0=F_1=F_2=f$.  All the intervening inequalities are equalities, giving
\[
 n_B=n_C=0,\qquad H_0=f+1,\qquad H_2=f-1.
\]
The bounds on $H_0,H_2$ leave only $f=1,2$.

If $f=1$, the exterior letters $x,a$ must occur and $c,b,z$ must be
absent.  Only $y$ is optional, so $H_1\leq2$ and $m\geq2$.  On the other
hand, the last relation in Equation~\eqref{low:coverage-relations} gives
$H_1=2+\delta>2$, since
$\delta=(m+\rho-1)/2>0$.  If $f=2$, the letters $x,a,c$ must occur and
$b,z$ must be absent.  Again only $y$ is optional, so $H_1\leq3$ and
$m\geq3$, whereas the same relation gives $H_1=3+\delta>3$.
Both possibilities are impossible.  There is no monomial on the May
upper-bound page, and hence no cobar cohomology in any of the stated
bidegrees.
\end{proof}

\begin{lemma}[Finite local cell attachments]
\label{low:finite-localization}
A spectrum obtained from finitely many $p$-local sphere cells is the
$p$-localization of a finite CW spectrum.
\end{lemma}

\begin{proof}
Proceed one attachment at a time.  If the previously constructed spectrum
is the localization of a finite spectrum $Z$, an attaching map belongs to
$\pi_d(Z_{(p)})=\pi_d(Z)\otimes\mathbb Z_{(p)}$ and can be written $a/m$
with $p\nmid m$.  After localization, multiplication by $m$ on its sphere
source is an equivalence.  The localized cone of the integral attaching
map $a$ is therefore equivalent to the cone of $a/m$.  Induction over
the finitely many cells proves the assertion.
\end{proof}

\begin{proposition}[Construction through $V(3)$]
\label{low:V3}
For $0\leq n\leq3$ there is a finite $p$-local cell spectrum $V(n)$ with
a bottom-cell unit $\eta_n:S_{(p)}\to V(n)$ and a binary multiplication
\[
 \mu_n:V(n)\wedge V(n)\longrightarrow V(n)
\]
whose two unit identities hold.  Its cells are indexed by subsets of
$\{0,\ldots,n\}$, in dimensions
\begin{equation}
 \sum_{i\in S}(qu_i+1),
 \label{low:cell-dimensions}
\end{equation}
and its $BP$-homology is the canonical quotient comodule $R/I_{n+1}$,
generated in degree zero by the bottom unit.  In particular, we may fix
\begin{equation}
 Y=V(3),\qquad BP_*Y=R/I_4,
 \label{low:Y}
\end{equation}
with such a binary two-sided unital multiplication.
\end{proposition}

\begin{proof}
Start with $V(0)$, the cofiber of $p:S_{(p)}\to S_{(p)}$.  Since
multiplication by $p$ is injective on $R$, its $BP$-homology is $R/(p)$,
with the canonical quotient coaction and bottom generator.  Its cells
are in dimensions zero and one, as in
Equation~\eqref{low:cell-dimensions}.

We give first the product-extension argument, which applies to this
initial complex and to every later complex once its asserted homology
and cells have been obtained.  The subcomplex
\[
 U_n=(S_{(p)}\wedge V(n))
       \mathop{\cup}_{S_{(p)}\wedge S_{(p)}}
       (V(n)\wedge S_{(p)})
       \ \subseteq\ V(n)\wedge V(n)
\]
has a map to $V(n)$ given by the two identity maps, which agree on the
common bottom sphere.  Every remaining cell corresponds to a pair of
nonempty subsets of $\{0,\ldots,n\}$.  If $e_i$ counts occurrences of $i$
in that pair, its dimension is $qU+\rho$, with precisely the parameters
in the second part of Lemma~\ref{low:algebraic-vanishing}.  The obstruction
to extending across the cell lies in
$\pi_{qU+\rho-1}V(n)$.

An Adams--Novikov term in that stem has
$qw-s=qU+\rho-1$, or
\[
 w=U+k,\qquad s=qk+1-\rho.
\]
Since $2\leq\rho\leq8<q$, nonnegative filtration requires $k\geq1$.
Every corresponding $E_2$ group is zero by
Lemma~\ref{low:algebraic-vanishing}.  Strong convergence and finite
filtration, from Proposition~\ref{fw:anss}, give
$\pi_{qU+\rho-1}V(n)=0$.  Extending cell by cell therefore produces
$\mu_n$, agreeing with both identity maps on $U_n$.  This constructs the
initial product on $V(0)$ as well.

Suppose now that $V(n-1)$, including its product, has been constructed,
where $1\leq n\leq3$.  By Lemma~\ref{fw:coordinates}, $v_n$ is a primitive
element of $BP_*V(n-1)=R/I_n$.  It defines a filtration-zero $E_2$ class.
Its possible positive differential targets are exactly
Equation~\eqref{fw:primitive-targets}; the first part of
Lemma~\ref{low:algebraic-vanishing} makes them all zero.  There is no
incoming differential to filtration zero.  The edge assertion in
Proposition~\ref{fw:anss} consequently realizes $v_n$ by a homotopy map
\[
 \widetilde v_n:S^{qu_n}_{(p)}\longrightarrow V(n-1).
\]
Use the product to form
\[
 f_n=\mu_{n-1}(\widetilde v_n\wedge\operatorname{id}):
       \Sigma^{qu_n}V(n-1)\longrightarrow V(n-1),
\]
and define $V(n)$ as its cofiber.

The induced map on $BP$-homology is multiplication by $v_n$: the
coefficient pairing is $R$-bilinear, the homology is cyclic on the unit,
and $\widetilde v_n$ has the specified Hurewicz image.  This multiplication is
injective on the polynomial quotient $R/I_n$.  The cofiber exact sequence
therefore gives precisely $R/I_{n+1}$, with no additional shifted kernel
summand.  The bottom sphere map surjects onto this cyclic homology, so its
kernel $I_{n+1}$ identifies the coaction with the canonical quotient
coaction, as well as fixing the generator in degree zero.

The mapping cone adds the old cells shifted by $qu_n+1$, giving exactly
Equation~\eqref{low:cell-dimensions}.  The product-extension argument
already proved then constructs $\mu_n$.  This completes the induction.
All attachments are finite $p$-local cell attachments, and
Lemma~\ref{low:finite-localization} gives finiteness in the sense of the
stated realization problem.  No associativity of any $\mu_n$ has been
asserted or used.
\end{proof}

\begin{corollary}[The action of $p$ on $Y$]
\label{low:p-null}
For the spectrum $Y$ of Proposition~\ref{low:V3}, $\pi_0Y=\mathbb F_p$ and
$p\operatorname{id}_Y$ is nullhomotopic.
\end{corollary}

\begin{proof}
The bottom two cells form the mod-$p$ Moore spectrum.  All other cells have
dimension at least $qu_1+1=q+1>1$, so they do not change $\pi_0$.
Thus $p\eta_3=0$ in $\pi_0Y$.  Applying the left unit identity of its
binary multiplication gives
\[
 p\operatorname{id}_Y
  =\mu_3\bigl((p\eta_3)\wedge\operatorname{id}_Y\bigr)=0.
\]
\end{proof}
==== END A-finite-Smith-Toda-complex-V4-at-the-prime-1009//02-lower-complexes.tex ====

==== BEGIN A-finite-Smith-Toda-complex-V4-at-the-prime-1009//02a-cobar-preliminaries.tex ====
\section{Filtered cobar preliminaries}
\label{may:preliminaries}

We use the filtered Hopf-algebra method of \citet{May1966}, with
the odd-primary May-type filtration in the form of
\citet[Theorem~3.2.5]{Ravenel2004}, and allow an additional digit-zero
weight needed for the injectivity argument in Section~\ref{may:section}.
For earlier applications of Ravenel's filtration to $V(3)$ and powers
of $\beta_1$, see \citet{LeeRavenel1994}.
We construct this modified filtration below, including its initial page
and convergence. Throughout this section, cohomological degree and weight are denoted by
$(s,w)$.  Unless an Adams--Novikov differential is named explicitly, the
spectral sequences in this section are algebraic spectral sequences of
cobar complexes.  Put
\[
 P=\mathbb F_p[t_1,t_2,\ldots],\qquad
 |t_i|_w=u_i,
 \qquad
 \overline\Delta(t_i)=\sum_{0<l<i}t_l\otimes t_{i-l}^{p^l}.
\]
Reversing all coproducts reverses the cobar conventions and changes none of
the conclusions.  We use the displayed convention.

\begin{lemma}[The filtered cobar calculation]
\label{may:page}
For every integer $M\geq0$, the cobar cohomology of $P$ admits a
multiplicative, degreewise finite spectral sequence whose initial
cohomology page is
\begin{equation}
 \Lambda(h_{i,j}:i\geq1,j\geq0)
 \otimes\mathbb F_p[b_{i,j}:i\geq1,j\geq0].
 \label{may:initial-page}
\end{equation}
The bidegrees and auxiliary filtrations are
\[
\begin{array}{c|c|c}
 & (s,w)& F_M\\ \hline
 h_{i,j}&(1,p^ju_i)&2i-1+M[j=0]\\
 b_{i,j}&(2,p^{j+1}u_i)&p(2i-1+M[j=0]).
\end{array}
\]
Here $[j=0]$ is $1$ or $0$ according as $j=0$ or $j\neq0$.
All differentials lower $F_M$.  If an exterior generator is written
$h_{[a,b)}=h_{b-a,a}$, its first differential is
\begin{equation}
 d_1 h_{[a,b)}=\sum_{a<l<b}h_{[a,l)}h_{[l,b)},
 \qquad d_1b_{i,j}=0.
 \label{may:splitting}
\end{equation}
The same statements hold for the Hopf subalgebra
$P_r=\mathbb F_p[t_1,\ldots,t_r]$, with $i\leq r$ in
\eqref{may:initial-page}.
\end{lemma}

\begin{proof}
Write every exponent in base $p$ and assign a digit factor $t_i^{p^j}$
the filtration $2i-1+M[j=0]$.  For a monomial, add these filtrations with
the digit multiplicities, without carrying.  Carrying $p$ copies of one
digit to the next digit lowers filtration: its decrease is
\[
 p(2i-1+M[j=0])-(2i-1+M[j+1=0])\geq p-1>1.
\]
A nonprimitive coproduct summand of $t_i^{p^j}$ replaces it by
$t_l^{p^j}\otimes t_{i-l}^{p^{j+l}}$ and decreases filtration by exactly
one.  These observations show that this is a multiplicative coalgebra
filtration, and that its associated graded Hopf algebra is the tensor
product of primitive truncated polynomial Hopf algebras
\[
 \bigotimes_{i,j}\mathbb F_p[z_{i,j}]/(z_{i,j}^p).
\]
For one such factor the dual algebra is again a truncated polynomial
algebra: factorial rescaling identifies its basis in exponents less than
$p$ with the ordinary polynomial basis.  The free resolution alternating
multiplication by its generator and its $(p-1)$st power computes its
cohomology as one exterior generator in degree one and one polynomial
generator in degree two.  Their weights are respectively the weight of
$z_{i,j}$ and $p$ times that weight.  Tensoring these resolutions gives
\eqref{may:initial-page}, including the stated auxiliary filtrations.
The exterior square is zero since its cohomological degree is odd and
$p$ is odd.

The filtration-one part of the reduced coproduct gives the splitting
formula for $h$.  For the polynomial generator associated to a digit
$z$, a filtration-one contribution would replace one of its $p$ copies
by a split pair and leave $p-1$ unsplit copies of $z$.  In the
associated-graded cohomology of the $z$ factor, a nonzero multiplicity
has the form $pa+\epsilon$, with $a\geq0$ and
$\epsilon\in\{0,1\}$.  The multiplicity $p-1$ has neither form, so that
component has zero cohomology.  Carries cannot contribute to this first
differential by the strict inequality above.  This proves $d_1b=0$.

In a fixed positive internal weight only finitely many digit factors
occur, every monomial exponent is bounded, and a normalized bar has
positive weight.  Its length is consequently bounded as well.  Thus all
the filtrations under discussion are finite in each weight, and their
spectral sequences converge to the indicated cobar cohomology.  The
construction restricts to $P_r$ because its coproduct uses only the
first $r$ coordinates.
\end{proof}

For later calculations we fix the following notation, always
distinguishing the bare exterior letter $b$ from the indexed polynomial
letters:
\begin{equation}
\begin{gathered}
 x=h_{1,0},\quad a=h_{2,0},\quad c=h_{3,0},\qquad
 y=h_{1,1},\quad b=h_{2,1},\quad z=h_{1,2},\\
 A=b_{1,0},\qquad B=b_{2,0},\qquad C=b_{1,1}.
\end{gathered}
\label{may:low-alphabet}
\end{equation}
An exterior interval $h_{i,j}$ covers positions
$j,\ldots,j+i-1$ in its weight, and a polynomial letter $b_{i,j}$
covers positions $j+1,\ldots,j+i$.  Let $m$ be the number of exterior
letters, let $H_r$ count those covering position $r$, and let
$L=(s-m)/2$ be the number of polynomial letters, counted with
multiplicity.  If only positions $0,1,2$ occur, the alphabet is precisely
\eqref{may:low-alphabet}.  Writing $n_A,n_B,n_C$ for the polynomial
multiplicities gives
\begin{equation}
 w=H_0+p(H_1+L-n_C)+p^2(H_2+n_B+n_C),
 \qquad n_A=L-n_B-n_C.
 \label{may:three-position-weight}
\end{equation}
==== END A-finite-Smith-Toda-complex-V4-at-the-prime-1009//02a-cobar-preliminaries.tex ====

==== BEGIN A-finite-Smith-Toda-complex-V4-at-the-prime-1009//03-may-and-first-obstruction.tex ====
\section{The height-four obstruction calculation}
\label{may:section}

The algebraic spectral sequences below use the filtered cobar complexes
of Section~\ref{may:preliminaries}.

\subsection{Possible differential targets}

We now use the finite spectrum $Y$ furnished by
Proposition~\ref{low:V3}.  Its homology is $R/I_4$, its bottom cell is a
unit, and it has a binary two-sided unital multiplication.  The possible
positive Adams--Novikov differentials from $v_4$ have targets
\begin{equation}
 s=qk+1,\qquad w=u_4+k,\qquad k\geq1.
 \label{may:obstruction-degrees}
\end{equation}

\begin{lemma}[Explicit cutoffs]
\label{may:target-bound}
The groups in \eqref{may:obstruction-degrees} are identified with
$H^{s,w}(P)$.  Their May page has only positions $0,1,2,3$, and it is zero
unless $k\leq p+2$.  For $k\geq2$, only positions $0,1,2$ can occur.
\end{lemma}

\begin{proof}
If $w<s$, the degree-$(s,w)$ cochain groups for both $Y$ and $P$
are zero, as is the May page, and all the conclusions are immediate.
We may therefore suppose $w\geq s$.  The normalized bound in
Lemma~\ref{fw:normalized-bound} then gives
$(q-1)k\leq u_4-1$.  Consequently
\[
 w\leq u_4+\frac{u_4-1}{q-1}
 <\frac q{q-1}u_4<4p^3<p^4
 \quad (p=1009),
\]
where $u_4<2p^3$ and $q/(q-1)<2$.  Thus no letter can reach position
four.  Also $w<u_5$.  A coefficient containing $v_4$ leaves at most $k$
weight for normalized bars, whereas the three lengths $s-1,s,s+1$ are
all greater than $k$, because $s-1=qk>k$.  Higher coefficient generators
are excluded by $w<u_5$.  The adjacent-chain comparison of
Lemma~\ref{fw:additive-comparison} therefore identifies the required
cohomology with that of $P$.

There are ten possible exterior intervals on positions $0,1,2,3$.
Since $s$ is odd, $m$ is odd and hence $m\leq9$.  Every polynomial letter
has weight at least $p$, so
\[
 u_4+k\geq pL\geq p((p-1)k-4),
 \qquad
 (p^2-p-1)k\leq u_4+4p.
\]
But
\[
 (p^2-p-1)(p+3)-(u_4+4p)=p^2-9p-4>0
 \quad (p=1009).
\]
It follows that $k\leq p+2$.

For $k\geq2$ we have $L\geq2p-6$.  If any letter reaches position
three, that letter has weight at least $p^3$, while all other polynomial
letters have total weight at least $(L-1)p$.  This is a valid lower bound
whether the distinguished letter is exterior or polynomial.  Thus
\[
 w\geq p^3+2p^2-7p
 >p^3+p^2+2p+3=u_4+p+2\geq w,
\]
since the strict difference is $p^2-9p-3>0$ at $p=1009$.  This
contradiction excludes position three in that range.
\end{proof}

\subsection{The first obstruction}

The first possible differential is $d_{q+1}$, and its target is the
actual cobar cohomology in degree $(q+1,u_4+1)$. We first explain why
an injective multiplication on this group will rule out the differential.
Lemma~\ref{fw:v5-relation} gives
\[
 v_4h_{1,4}+e v_4^p h_{1,0}=0
\]
in the $E_2$ algebra for $Y$. By $q$-sparsity no earlier positive
differential occurs, so Proposition~\ref{fw:anss-pairing} makes
$d_{q+1}$ a derivation on this algebra. Let $\sigma$ denote the
algebraic specialization to $P$-cohomology. Differentiate the relation
and then apply $\sigma$. All terms containing $v_4$ specialize to zero,
and $d_{q+1}(v_4^p)=p v_4^{p-1}d_{q+1}(v_4)=0$. Hence
\begin{equation}
 \sigma\bigl(d_{q+1}(v_4)\bigr)h_{1,4}=0.
 \label{may:specialized-obstruction}
\end{equation}
Specialization is applied after the topological differential, to an
identity of cobar-cohomology classes; no spectrum realizing $\sigma$
is required.

It remains to prove that multiplication by $h_{1,4}$ is injective on
$H^{q+1,u_4+1}(P)$. Its degree shift is $(1,p^4)$, so its product
lies in degree $(q+2,u_5+1)$. An incoming May differential to that
product must have source degree $(q+1,u_5+1)$. The next lemma lists
the initial May monomials in the obstruction degree and in this
incoming-source degree.
We will use the digit-zero adjustment of Lemma~\ref{may:page} to
separate the surviving source filtrations from the product filtration.

In the alphabet of Equation~\eqref{may:low-alphabet}, the splitting
differential is
\begin{equation}
 da=xy,\qquad db=yz,\qquad dc=xb+az,
 \label{may:small-splitting}
\end{equation}
with the exterior order $x,a,c,y,b,z$ used for signs.

\begin{lemma}[The first target and all incoming-source intervals]
\label{may:first-list}
In bidegree $(s,w)=(q+1,u_4+1)$ the complete initial May page consists of
\begin{equation}
 acb B^{p-2},\qquad acybz B^{p-3}.
 \label{may:first-target-list}
\end{equation}
The first is not a splitting cycle and the second is a splitting cycle.
In bidegree $(q+1,u_5+1)$ the complete list is given in
Table~\ref{may:n5-table}.  The four entries of exterior length three or
five have no nonzero splitting cycle in their span.
\end{lemma}

\begin{proof}
We prove exhaustion simultaneously for $w=u_n+1$, with $n=4,5$.
Since $u_n+1<p^n$, the available exterior intervals are
$[a,b)$ with $0\leq a<b\leq n$, and the polynomial intervals cover
positions $1,\ldots,n-1$.  The target weight has base-$p$ digits
$(2,1,\ldots,1)$.  At position zero, $H_0\leq n<p$, so $H_0=2$ and
there is no carry into position one.  As $s=2p-1$ is odd, write
\[
 m=2D-1,\qquad L=p-D.
\]
There are exactly two exterior intervals starting at zero; at most
$n(n-1)/2$ others are available.  Hence $m\leq11$, $2\leq D\leq6$,
and
\begin{equation}
 p-6\leq L\leq p-2.
 \label{may:L-range}
\end{equation}
The lower bound on $D$ follows from $m$ being odd and at least two.

Let $c_r$ be the carry into position $r$, with $c_1=c_n=0$, and let
$B_r$ count polynomial intervals covering position $r$.  The largest
possible exterior coverage at a position is nine.  Inductively, the
quantity $H_r+B_r+c_r$ is at most $9+(p-2)+1=p+8<2p$;
therefore every carry is zero or one.  The digit equations are
\begin{equation}
 B_r=1-H_r-c_r+pc_{r+1},\qquad 1\leq r\leq n-1.
 \label{may:carry-equations}
\end{equation}
If there were no carry, then $B_r\leq1$ for every $r$, and
$L\leq\sum B_r\leq4$, contrary to $L\geq p-6=1003$.

Call a position high when $c_{r+1}=1$.  A high position has
$B_r\geq p-9$, and a nonhigh position has $B_r\leq1$.
There cannot be two blocks of high positions.  Indeed, choose high
positions $a<b$ in different blocks and a nonhigh position $c$ between
them.  An interval covering both $a$ and $b$ also covers $c$, whence
\[
 L\geq B_a+B_b-B_c\geq2(p-9)-1=2p-19>p-2.
\]
Here $2p-19=1999$ and $p-2=1007$.  This contradicts
\eqref{may:L-range}.  Thus the high positions form a single block
$[l,r]$ with $1\leq l\leq r\leq n-2$.
At the next position \eqref{may:carry-equations} reads
$B_{r+1}=-H_{r+1}$, so
\begin{equation}
 B_{r+1}=H_{r+1}=0.
 \label{may:gap-after-block}
\end{equation}
At the beginning of the block, $B_l=p+1-H_l\leq L$, giving
$H_l\geq D+1$.  If $l<r$, its end gives
$B_r=p-H_r\leq L$, hence $H_r\geq D$.

A singleton block is impossible.  Every exterior interval covering its
position $l$ must end at $l+1$ by \eqref{may:gap-after-block}.  When
$l=1$ there are only two such intervals.  When $l>1$, at most one begins
before $l$, since $H_{l-1}\leq1$, and at most one begins at $l$.
Thus in either case $H_l\leq2$, contradicting $H_l\geq D+1\geq3$.
We henceforth have $l<r$.

If $l>1$, at most two exterior intervals cover both $l$ and $r$: they
must end at $r+1$, and at most one starts before $l$, while at most one
starts at $l$.  At least one of the two intervals starting at zero
misses both positions, since $H_{l-1}\leq1$.  Consequently
\[
 H_l+H_r
 =m+\#\{\text{intervals meeting both}\}
       -\#\{\text{intervals meeting neither}\}
 \leq m+1.
\]
This contradicts $H_l+H_r\geq2D+1=m+2$.  Therefore $l=1$.
There are now at most two exterior intervals covering both endpoints,
namely $[0,r+1)$ and $[1,r+1)$.  Equality in the same counting identity
is forced: both are present, every exterior interval meets position $1$
or position $r$, and
\[
 H_1=D+1,\qquad H_r=D.
\]
It follows that $B_1=B_r=L$.  Every polynomial interval therefore covers
both positions, and by \eqref{may:gap-after-block} every one is exactly
$[1,r+1)$.  No exterior or polynomial interval can occur after the block.
The carry supplies the target digit at $r+1$; all further digits would
be zero.  Since the required remaining digits are all one, necessarily
$r=n-2$.  Thus
\begin{equation}
 \text{polynomial factor }b_{n-2,0}^{p-D},\qquad
 H_0=2,\quad H_1=D+1,\quad
 H_2=\cdots=H_{n-2}=D,
 \quad H_{n-1}=0.
 \label{may:forced-profile}
\end{equation}

It remains to list a small, now forced set of exterior intervals.
For $n=4$, the intervals $[0,3)$ and $[1,3)$ are mandatory.  The other
start-zero interval must be $[0,2)$; $[0,1)$ would meet neither endpoint.
The only optional intervals are $[1,2)$ and $[2,3)$.  The coverage
requirements in \eqref{may:forced-profile} require both or neither.
This is exactly \eqref{may:first-target-list}.

For $n=5$, the mandatory intervals are $[0,4)$ and $[1,4)$.
The remaining start-zero interval is $[0,t)$ with $t=2$ or $3$.
The only optional intervals are $[1,2),[1,3),[2,4),[3,4)$;
$[2,3)$ meets neither endpoint and is forbidden.  Write their inclusion
indicators as $e_{12},e_{13},e_{24},e_{34}\in\{0,1\}$, and let their
sum be $o$.  The three required coverage equations are
\begin{equation}
 e_{12}+e_{13}=o/2,\qquad
 e_{24}+e_{34}=o/2,\qquad
 e_{13}+e_{24}+[t=3]=o/2,
 \label{may:four-indicators}
\end{equation}
where $o\in\{0,2,4\}$.  For $t=2$ the solutions are no optional
intervals, the pair $\{12,24\}$, the pair $\{13,34\}$, and all four.
For $t=3$ the only solution is the pair $\{12,34\}$.  This proves the
exhaustive Table~\ref{may:n5-table}, including its length-seven entry.

\begin{table}[htbp]
\centering
\begin{tabular}{c l l}
\toprule
$m$ & Exterior intervals (exclusive right endpoints) & Polynomial factor\\
\midrule
$3$ & $02,04,14$ & $b_{3,0}^{p-2}$\\
$5$ & $03,04,12,14,34$ & $b_{3,0}^{p-3}$\\
$5$ & $02,04,13,14,34$ & $b_{3,0}^{p-3}$\\
$5$ & $02,04,12,14,24$ & $b_{3,0}^{p-3}$\\
$7$ & $02,04,12,13,14,24,34$ & $b_{3,0}^{p-4}$\\
\bottomrule
\end{tabular}
\caption{Every initial May monomial in bidegree $(q+1,u_5+1)$.
The symbol $ab$ in this table denotes the interval $[a,b)$, not a
product of the letters in \eqref{may:low-alphabet}.}
\label{may:n5-table}
\end{table}

For completeness, the splitting calculation is also explicit.
The length-three entry has, for example, a nonzero component on
$01,04,12,14$.  For the three length-five entries in their displayed
order, take the following four rows, ordering exterior intervals
lexicographically:
\[
\begin{array}{c|rrr}
01,04,12,13,14,34&1&-1&0\\
02,03,12,14,24,34&1&0&1\\
02,04,12,13,24,34&0&1&1\\
02,04,12,14,23,34&-1&-1&1
\end{array}
\]
The common polynomial factor is $b_{3,0}^{p-3}$.  The minor using the
first, second, and fourth rows is $3$, which is nonzero in
$\mathbb F_{1009}$.  Thus these three images are linearly independent.
Finally, \eqref{may:small-splitting} shows that $acb$ has a nonzero
splitting differential and that $acybz$ is closed: every term in the
latter differential repeats an exterior factor.  This finishes the
lemma.
\end{proof}

\begin{proposition}[An injective multiplication]
\label{may:injectivity}
Multiplication by the primitive class $h_{1,4}$ is injective on actual
cobar cohomology in bidegree $(q+1,u_4+1)$:
\[
 H^{q+1,u_4+1}(P)\xrightarrow{\ h_{1,4}\ }
 H^{q+2,u_5+1}(P).
\]
\end{proposition}

\begin{proof}
Use the filtration in Lemma~\ref{may:page} with the explicit choice
$M=3p$.  If the source cohomology is nonzero, any nonzero class has
leading detector a nonzero scalar multiple of
$acybz B^{p-3}$, by Lemma~\ref{may:first-list}.  Its filtration is
\[
 F_0=3p^2-9p+13+M(2+p(p-3)).
\]
The cocycle $[t_1^{p^4}]$ represents $h_{1,4}$, with degree, weight, and
filtration shifts $(1,p^4,1)$.  The proposed leading product therefore
has filtration
\begin{equation}
 F_*=3p^2-9p+14+M(2+p(p-3)).
 \label{may:product-filtration}
\end{equation}

Every possible incoming May differential to this product has source
cohomological degree $q+1$ and weight $u_5+1$.
Table~\ref{may:n5-table} is its complete initial-page source list.
The length-three and length-five source filtrations are respectively
\[
 5p^2-10p+15+M(2+p(p-2)),\qquad
 5p^2-15p+19+M(2+p(p-3));
\]
all three length-five sources have the latter filtration.
The length-three and length-five entries disappear under $d_1$, since
their displayed splitting maps are injective.  None can be a splitting
boundary for the product itself, because its polynomial factor is a
power of $B=b_{2,0}$, whereas their polynomial factor is a power of
$b_{3,0}$.  The length-seven source has filtration
\[
 F_7=5p^2-20p+23+M(2+p(p-4)).
\]
For $M=3p$ the difference is
\[
 F_7-F_*=2p^2-11p+9-pM=-p^2-11p+9<0.
\]
As all May differentials lower filtration, this last source cannot hit
the proposed product on any later page.  There are no other incoming
sources.

This argument does not require $B$ to represent a permanent class by
itself.  Choose an actual cocycle for the given nonzero source
cohomology class and multiply that entire cocycle by
$[t_1^{p^4}]$.  The result is an actual cocycle.  Its leading
associated-graded cup product is the nonzero monomial
$acybz B^{p-3}h_{1,4}$, since the new exterior letter is distinct from
all the old ones.  An actual cocycle has no outgoing May differential;
its leading detector could disappear only by becoming a boundary,
which the preceding incoming-source calculation excludes at every
page.  Its product cohomology class is therefore nonzero.
\end{proof}

\begin{proposition}[The first Adams--Novikov differential]
\label{may:first-differential}
In the Adams--Novikov spectral sequence for $Y$, one has
$d_{q+1}(v_4)=0$.
\end{proposition}

\begin{proof}
Equation~\eqref{may:specialized-obstruction} and
Proposition~\ref{may:injectivity} give
$\sigma(d_{q+1}(v_4))=0$. Lemma~\ref{may:target-bound} identifies
this specialization with the original target group. Hence
$d_{q+1}(v_4)=0$.
\end{proof}

\subsection{The remaining targets}

\begin{proposition}[All later obstruction candidates]
\label{may:remaining-list}
For $k\geq2$ the complete initial May page in
\eqref{may:obstruction-degrees} is Table~\ref{may:remaining-table}; all
other values of $k$ give an empty page.  Put
$N_0=(p-1)^2$ and $N=p(p-1)-2$.
\begin{table}[htbp]
\centering
\begin{tabular}{c l}
\toprule
$k$ & Initial May monomials\\
\midrule
$2$ & $xac B A^{p-2}C^{p-2}$,
       $\quad xacyz A^{p-2}C^{p-2}$\\[2pt]
$p-1$ & $y B^3 A^{N_0-3}$,
         $\quad b B^2 A^{N_0-2}$,
         $\quad ybz B A^{N_0-2}$\\[2pt]
$p$ & $cyb A^{N+1}$,
       $\quad ayb B A^N$\\
\bottomrule
\end{tabular}
\caption{Every initial May monomial for the later $v_4$ obstruction
bidegrees.}
\label{may:remaining-table}
\end{table}
\end{proposition}

\begin{proof}
Lemma~\ref{may:target-bound} allows only positions $0,1,2$ and
$2\leq k\leq p+2$.  Thus \eqref{may:three-position-weight} applies.
There are six exterior letters; $m$ is odd, so write
\[
 m=2f+1,\quad 0\leq f\leq2,\qquad L=(p-1)k-f.
\]
Reducing the weight modulo $p$ gives
$H_0\equiv1+k\pmod p$, with $0\leq H_0\leq3$.
The only possibilities are $k=2$ or $k=p+d$ with
$-1\leq d\leq2$.

When $k=2$, $H_0=3$ forces all of $x,a,c$.  The other exterior letters
are an even subset of $\{y,b,z\}$.  Put
$T=H_2+n_B+n_C$.  Substitution in
\eqref{may:three-position-weight} gives
\begin{equation}
 pT-n_C+H_1=p^2-p+3+f.
 \label{may:k2-equation}
\end{equation}
For the four possible exterior subsets the value
$e=H_1-3-f$ is $-2$ or $-1$.  Therefore
$n_C\equiv e\pmod p$.  Since
$0\leq n_C\leq L\leq2p-3$, the only possible lift is
$n_C=p+e$: the lift $e$ is negative, and $2p+e\geq2p-2>L$.
Equation~\eqref{may:k2-equation} now gives $T=p$.
All the remaining counts follow as displayed:
\[
\begin{array}{c|c|c|c|c|c|c}
\text{extra exterior letters}&f&H_1&H_2&n_C&n_B&n_A\\ \hline
\varnothing&1&2&1&p-2&1&p-2\\
y,b&2&4&2&p-1&-1&p-2\\
y,z&2&3&2&p-2&0&p-2\\
b,z&2&3&3&p-2&-1&p-1
\end{array}
\]
The negative counts exclude the second and fourth rows.  The first and
third are exactly the $k=2$ row of Table~\ref{may:remaining-table}.

Now let $k=p+d$, $-1\leq d\leq2$.  Then $H_0=1+d$, and the weight
equation becomes
\begin{equation}
 pT-n_C+H_1=(2-d)p+2+d+f.
 \label{may:pd-equation}
\end{equation}
Since $n_C\leq T$ and $H_1\geq0$, it follows that
\[
 (p-1)T\leq(2-d)(p-1)+(4+f).
\]
Here $4+f\leq6<p-1$, so $T\leq2-d\leq3$ and $n_C\leq3$.
Rearranging \eqref{may:pd-equation} now gives
\[
 p(T-(2-d))=2+d+f+n_C-H_1.
\]
The right side lies between $-3$ and $9$, strictly between $-p$ and
$p$.  Both sides therefore vanish, proving the exact equations
\begin{equation}
 H_2+n_B+n_C=2-d,\qquad H_1=2+d+f+n_C.
 \label{may:pd-exact}
\end{equation}

For $d\geq1$, these require
$3+f\leq H_1\leq\min(2f+1,4)$, impossible for each of
$f=0,1,2$.  For $d=0$, exactly one of $x,a,c$ occurs, so $H_1\leq3$.
Equation~\eqref{may:pd-exact} forces $f=1$, $n_C=0$, and $H_1=3$.
The only exterior triples are $ayb$ and $cyb$.  Their $H_2$ values are
respectively $1$ and $2$, giving the two $k=p$ monomials in the table.

For $d=-1$, only $y,b,z$ can occur.  If $m=1$, the choices $y,b,z$
have $(H_1,H_2)$ equal to $(1,0),(1,1),(0,1)$.
Equation~\eqref{may:pd-exact} excludes $z$ and gives respectively
$(n_C,n_B)=(0,3),(0,2)$ for the other two.  If $m=3$, the exterior
factor is $ybz$, with $f=1$, $H_1=H_2=2$; the same equations give
$(n_C,n_B)=(0,1)$.  Since $L=N_0-f$, these are exactly the three
$k=p-1$ monomials.  Every nonnegative count and every exterior subset
has now been considered.
\end{proof}

The first two rows of Table~\ref{may:remaining-table} use only the first
three or two coordinates, respectively. Every monomial in its last row
has a large common power of $A$. The following comparisons turn these
two observations into surjections on actual cobar cohomology, which are
the inputs to the remaining obstruction arguments.

\begin{lemma}[Two comparison surjections]
\label{may:comparison}
The following initial-page tests give surjections on actual cobar
cohomology.
\begin{enumerate}
 \item If every monomial of \eqref{may:initial-page} in bidegree
 $(s,w)$ uses only generators with $i\leq r$, then
 $H^{s,w}(P_r)\longrightarrow H^{s,w}(P)$ is surjective.
 \item Put $A=b_{1,0}$. If every initial-page monomial in bidegree
 $(s,w)$ contains $A^a$, then multiplication by the actual cocycle
 representing $A^a$ gives a surjection
 \[
 H^{s-2a,w-pa}(P)\longrightarrow H^{s,w}(P).
 \]
\end{enumerate}
These conclusions also transport through additive comparisons that
identify the source and target cohomology groups and commute with the
indicated maps.
\end{lemma}

\begin{proof}
The subalgebra map includes a subset of the exterior and polynomial
generators in \eqref{may:initial-page}; its initial-page map is therefore
injective in every bidegree. Likewise, multiplication by the polynomial
$A^a$ is injective in every bidegree on that page. It is induced by a
filtered cochain map: $t_1$ is primitive, and
\begin{equation}
 \sum_{i=1}^{p-1}\frac{\binom pi}{p}
       [t_1^i\mid t_1^{p-i}]
 \label{may:A-cocycle}
\end{equation}
is an actual cocycle, obtained by taking the integral divided reduced
coproduct and then reducing modulo $p$. It has leading class $A$.
For the second comparison shift the source cohomological degree,
weight, and auxiliary filtration by $2a$, $pa$, and $ap(1+M)$.

Take the filtered mapping cone of either cochain map. The long exact
sequence of associated-graded cohomology, together with injectivity in
\emph{all} bidegrees, identifies its initial cohomology page with the
cokernel of the specified page map; there is no extra kernel from the
next degree. That cokernel is zero in the stated target bidegree.
Finite filtration therefore gives zero cone cohomology there. The
ordinary cone long exact sequence gives the required surjection.
Transport through the stated commuting comparisons preserves that
surjectivity.
\end{proof}

Section~\ref{frob:section} removes the first two rows of
Table~\ref{may:remaining-table} in actual cobar cohomology.  The last
row is treated in Section~\ref{top:section} by a homotopy argument with an explicit
Adams--Novikov page bound.
==== END A-finite-Smith-Toda-complex-V4-at-the-prime-1009//03-may-and-first-obstruction.tex ====

==== BEGIN A-finite-Smith-Toda-complex-V4-at-the-prime-1009//04-frobenius-elimination.tex ====
\section{Eliminating two obstruction degrees by Frobenius}
\label{frob:section}

Throughout this section $p=1009$.  We eliminate the rows $k=2$ and
$k=p-1$ of Table~\ref{may:remaining-table} in actual cobar cohomology.
The Frobenius extension reduces this task to quotient-group cohomology
with coefficients in kernel cohomology.  A filtration of those coefficients
then reduces the calculation to trivial-coefficient quotient groups, which
we compute by May methods.  We first establish these three steps and identify
the coefficient modules that the two obstruction degrees require.

\subsection{The Frobenius extension and its action}

Let $G_r$, for $r=2,3$, be the affine group scheme over
$\mathbb F_p$ with coordinate algebra
\[
 P_r=\mathbb F_p[t_1,\ldots,t_r],\qquad
 \Delta t_i=t_i\otimes1+1\otimes t_i+
       \sum_{0<l<i}t_l\otimes t_{i-l}^{p^l}.
\]
The weight of $t_i$ is $u_i$.  Thus, on points over a commutative
$\mathbb F_p$-algebra,
\begin{equation}\label{frob:group-law}
 (g h)_i=g_i+h_i+\sum_{0<l<i}g_lh_{i-l}^{p^l}.
\end{equation}
We write $H^a(G_r;M)$ for rational group cohomology, equivalently the
cohomology of the normalized coordinate cobar complex with coefficients
in the $G_r$-module $M$.

The spectral sequence below is the Lyndon--Hochschild--Serre spectral
sequence for an affine group scheme and a normal subgroup; see
\citet[Part~I, Proposition~6.6(3) and Section~6.7]{Jantzen1985} for
the spectral sequence and the quotient action on kernel cohomology.
We verify the required acyclicity and compute that action for this extension.

\begin{lemma}[The Frobenius extension]\label{frob:extension}
There is a finite faithfully flat exact sequence of affine group schemes
\begin{equation}\label{frob:exact-sequence}
 1\longrightarrow K_r=(\alpha_p)^r\longrightarrow G_r
   \xrightarrow{F} Q_r\longrightarrow1,
 \qquad O(Q_r)=\mathbb F_p[t_1^p,\ldots,t_r^p].
\end{equation}
Its derived invariants give a first-quadrant spectral sequence
\begin{equation}\label{frob:lhss}
 {}^{\mathrm F}E_2^{a,v}
   =H^a\bigl(Q_r;H^v(K_r;\mathbb F_p)\bigr)
   \Longrightarrow H^{a+v}(G_r;\mathbb F_p).
\end{equation}
It preserves weight and converges in every weight and total
cohomological degree.
\end{lemma}

\begin{proof}
The inclusion of the Frobenius subring makes $P_r$ free of rank $p^r$,
with basis $t_1^{e_1}\cdots t_r^{e_r}$, $0\leq e_i<p$.
It is therefore faithfully flat.  Its kernel has coordinate algebra
\[
 O(K_r)=\mathbb F_p[e_1,\ldots,e_r]/(e_1^p,\ldots,e_r^p).
\]
All the mixed terms in its coproduct vanish, so each $e_i$ is primitive.
This proves the description of the kernel and the exactness of
\eqref{frob:exact-sequence}.

Here is the acyclicity calculation needed to derive invariants.
Right multiplication by a kernel point is, by
\eqref{frob:group-law}, simply $t_i\mapsto t_i+e_i$: every mixed
term contains a positive $p$-power of a kernel coordinate.  Hence
multiplication gives an isomorphism of $K_r$-comodules
\begin{equation}\label{frob:regular-decomposition}
 O(Q_r)\otimes O(K_r)\xrightarrow{\ \cong\ }O(G_r),
 \qquad
 f\otimes e_1^{j_1}\cdots e_r^{j_r}
 \longmapsto f t_1^{j_1}\cdots t_r^{j_r},
 \quad 0\leq j_i<p.
\end{equation}
The first tensor factor has trivial $K_r$-coaction.  The second is the
regular comodule.  In particular,
$O(G_r)^{K_r}=O(Q_r)$, with its regular quotient coaction.

A cofree comodule is injective: its cofree functor is right adjoint to
the exact forgetful functor to vector spaces.  Equation
\eqref{frob:regular-decomposition} shows that every cofree
$G_r$-comodule is $K_r$-acyclic and that its $K_r$-invariants are
cofree over $Q_r$.  Every injective is a summand of a cofree comodule,
so these assertions also hold for injectives.  Finally,
$M^{G_r}=(M^{K_r})^{Q_r}$.  Resolving these two invariants functors
successively, using the indicated cofree resolutions, gives
\eqref{frob:lhss}.  This argument derives $K_r$-invariants; it does
not assert their exactness on arbitrary modules.  The spectral
sequence is first quadrant, so each fixed total cohomological degree
has a finite diagonal.  All coordinate weights are positive and each
weight component is finite dimensional.  These observations give the
asserted convergence, also weight by weight.
\end{proof}

\begin{lemma}[Conjugation on kernel cohomology]\label{frob:action}
For $r=3$ there is an isomorphism of bigraded algebras
\begin{equation}\label{frob:kernel-cohomology}
 H^*(K_3;\mathbb F_p)
   =\Lambda(x,a,c)\otimes\mathbb F_p[A,B,E].
\end{equation}
The exterior generators have cohomological degree one and weights
$1,u_2,u_3$, and the polynomial generators have cohomological degree
two and weights $p,pu_2,pu_3$, respectively.  Write
$\xi=t_1^p$, $\eta=t_2^p$ for quotient coordinates.  With the
right-conjugation convention, the quotient coaction is
\begin{align}
 x&\longmapsto x,&
 a&\longmapsto a+x\xi,&
 c&\longmapsto c+x\eta+a\xi^p,\label{frob:exterior-action}\\
 A&\longmapsto A,&
 B&\longmapsto B+A\xi^p,&
 E&\longmapsto E+A\eta^p+B\xi^{p^2}.
 \label{frob:polynomial-action}
\end{align}
Products in these formulas mean coefficient times quotient coordinate.
In particular, $xac$ and $A$ are invariant.  Exterior degree and
polynomial degree are separately preserved.  The corresponding
statements for $r=2$ omit $c$ and $E$.
\end{lemma}

\begin{proof}
The dual algebra of a primitive truncated coordinate
$\mathbb F_p[e]/(e^p)$ is a truncated polynomial algebra after rescaling
the dual basis by the invertible factorials $j!$, $j<p$.
Its resolution alternates multiplication by its generator and by the
$(p-1)$st power of that generator.  Applying the augmentation gives
one class in each cohomological degree.  Its algebra of extensions is
an exterior degree-one generator and a polynomial degree-two
generator.  The shift by two of the periodic resolution represents
the latter, and its iterates are nonzero in every even degree.
The former has square $[e\mid e]=\tfrac12d[e^2]$, which is zero
at this odd prime.
Tensoring these resolutions gives \eqref{frob:kernel-cohomology},
including the indicated weights.

For completeness, the naturality of the polynomial generators is
Frobenius semilinear.  A primitive coordinate $T$ has degree-two cocycle
\[
 \beta(T)=\sum_{j=1}^{p-1}\frac{1}{p}\binom pj
                      [T^j\mid T^{p-j}],
\]
where the displayed coefficients are integers reduced modulo $p$.
For a linear coordinate $T=\sum_i\lambda_i e_i$, the integral
polynomial
\[
 \frac{(\sum_i\lambda_i e_i)^p-\sum_i\lambda_i^p e_i^p}{p}
\]
has reduced coproduct
$\beta(\sum_i\lambda_i e_i)-\sum_i\lambda_i^p\beta(e_i)$.
Thus a matrix $(\lambda_{ij})$ on degree-one coordinates induces
$(\lambda_{ij}^p)$ on these degree-two cohomology classes.  This is a
polynomial identity in the $\lambda_{ij}$, so it also applies when
they are quotient coordinate functions.

If $e$ is a kernel point, direct use of \eqref{frob:group-law} gives
\[
 (g^{-1}eg)_1=e_1,\qquad
 (g^{-1}eg)_2=e_2+e_1g_1^p,\qquad
 (g^{-1}eg)_3=e_3+e_1g_2^p+e_2g_1^{p^2}.
\]
These coefficients belong to $O(Q_r)$, so the action descends to
the quotient.  The degree-one and degree-two naturality just proved
gives \eqref{frob:exterior-action} and
\eqref{frob:polynomial-action}.  The exterior matrix is triangular
with diagonal entries one, so its top exterior power $xac$ is fixed.
The formulas also prove separate preservation of the two degrees.
\end{proof}

\subsection{Coefficient filtration and quotient cohomology}

\begin{lemma}[The coefficient-weight filtration]\label{frob:coefficients}
Fix an exterior degree and a polynomial degree in
\eqref{frob:kernel-cohomology}, obtaining an invariant $Q_r$-submodule $M$.
At each total weight $w$, increasing coefficient weight filters the
normalized quotient cobar complex $C^*(Q_r;M)_w$ by a finite
filtration.  Its initial cohomology page is
\begin{equation}\label{frob:coefficient-page}
 {}^{\mathrm C}E_1^{a,\lambda}
      =M_{\lambda}\otimes H^a(Q_r;\mathbb F_p)_{w-\lambda},
\end{equation}
where $M_\lambda$ is viewed with trivial quotient action.
The coefficient differential with drop $d$ has the form
$ {}^{\mathrm C}d_d:(a,\lambda)\mapsto(a+1,\lambda-d)$.
If the nonidentity coefficient action has no weight drop smaller than
$d$, all earlier coefficient differentials vanish, and the differential
with drop $d$ is induced by that part of the coaction.

For the module $\operatorname{Sym}^n(A,B)$ we may write
\begin{equation}\label{frob:finite-D}
 A^n\langle1,D,\ldots,D^n\rangle,
 \qquad A^nD^j:=A^{n-j}B^j\quad(0\leq j\leq n).
\end{equation}
This is notation for a finite-dimensional module, with action
$D\mapsto D+\xi^p$; no inverse of $A$ is introduced.
\end{lemma}

\begin{proof}
In \eqref{frob:exterior-action} and
\eqref{frob:polynomial-action}, every nonidentity term has lower
coefficient weight, its difference being the positive weight of its
quotient coordinate.  The cobar differential therefore preserves
the increasing coefficient filtration.  In the associated graded
complex only the trivial coefficient action remains, giving
\eqref{frob:coefficient-page}.  At fixed total weight the coefficient
weights lie between zero and $w$, so the filtration is finite.
The stated description of its first possible differential follows by
taking the leading nonzero weight-lowering component of the cobar
differential.  Finally, expanding
\[
 (B+A\xi^p)^jA^{n-j}
   =\sum_{l=0}^j\binom jl A^{n-l}B^l\xi^{p(j-l)}
\]
proves the stated action on \eqref{frob:finite-D}.
\end{proof}

For either obstruction bidegree $(s,w)$, our aim is to make every group
$H^a(Q_r;H^{s-a}(K_r;\mathbb F_p))_w$ zero.  This makes the entire
total-degree-$s$ diagonal of \eqref{frob:lhss} zero, and hence proves
$H^{s,w}(G_r;\mathbb F_p)=0$.  We compute these groups by the coefficient
filtration \eqref{frob:coefficient-page}, using May only to bound or identify
its trivial-coefficient quotient groups.

The resulting vector-space upper bound has exactly the exterior-polynomial
letters used for $G_r$: kernel cohomology supplies the digit-zero letters,
and the quotient's initial May page supplies all higher digits.  This description
retains the kernel degree $v$.  Thus Table~\ref{may:remaining-table} bounds
every coefficient summand on the relevant Frobenius diagonal.
Lemma~\ref{may:comparison} reduces $k=2$ to $G_3$ and $k=p-1$ to $G_2$.
Grouping the candidates by their invariant kernel modules gives the four
computations in Table~\ref{frob:coefficient-obligations}.

For each module $M$ in that table, let $\lambda_0$ be its least coefficient
weight and put
\begin{equation}\label{frob:residual-weight}
 \mathcal R=\frac{w-\lambda_0}{p}.
\end{equation}
We call this the residual weight, and henceforth express quotient weights
after division by $p$.  Recall that $N_0=(p-1)^2$.
\begin{table}[htbp]
\centering
\begin{tabular}{ccccc}
\toprule
$k$ & $r$ & Kernel module $M$ & Quotient degree $a$ & $\mathcal R$\\
\midrule
$2$ & $3$ & $xac\operatorname{Sym}^{p-1}(A,B,E)$ & $2p-4$ & $p(p-1)$\\
$2$ & $3$ & $xac\operatorname{Sym}^{p-2}(A,B,E)$ & $2p-2$ & $p^2-p+1$\\
$p-1$ & $2$ & $\operatorname{Sym}^{N_0}(A,B)$ & $1$ & $3p+1$\\
$p-1$ & $2$ & $\operatorname{Sym}^{N_0-1}(A,B)$ & $3$ & $3p+2$\\
\bottomrule
\end{tabular}
\caption{The coefficient-cohomology groups that must vanish. Each row
means $H^a(Q_r;M)_w$, with $w=u_4+k$ in original weights.}
\label{frob:coefficient-obligations}
\end{table}

For the first two rows, the least weights are respectively
$\operatorname{wt}(xacA^{p-1})=3+p+2p^2$ and
$\operatorname{wt}(xacA^{p-2})=3+2p^2$; for the last two they are $np$,
where $n$ is the indicated polynomial degree.  These give the displayed
residual weights.  Replacing an $A$ by $E$ increases the coefficient weight,
after division by $p$, by $p^2+p$.  This exceeds both residual weights in
the $k=2$ rows, so no coefficient involving $E$ occurs in either fixed-weight
complex, in any quotient degree.  All four rows therefore use the finite
$D$-modules of \eqref{frob:finite-D}, multiplied by $xac$ in the first two
rows.  Suppressing the common invariant factor $A^n$ or $xacA^n$, a term
$D^j\zeta$ requires
\begin{equation}\label{frob:D-weight}
 \operatorname{wt}_Q(\zeta)=\mathcal R-jp.
\end{equation}
Their first possible coefficient differential lowers $j$ by one and raises
the quotient degree by one: its weight drop is $p^2$ in original weights,
or $p$ in these rescaled weights.

We now calculate the trivial-coefficient quotient groups needed for these
four computations and their adjacent degrees.
The coordinates $\xi,\eta$ have rescaled weights $1,p+1$ and
\begin{equation}\label{frob:quotient-coproduct}
 \overline\Delta\xi=0,\qquad
 \overline\Delta\eta=\xi\otimes\xi^p.
\end{equation}
In these rescaled weights, the letters
\[
 y=h_{1,0}^{Q},\quad b=h_{2,0}^{Q},\quad
 z=h_{1,1}^{Q},\quad C=b_{1,0}^{Q}
\]
have weights $1,p+1,p,p$, respectively; these are exactly the
letters $y,b,z,C$ used in Table~\ref{may:remaining-table}.

\begin{lemma}[The small quotient groups]\label{frob:quotient-groups}
For $Q_2$ or $Q_3$, at any positive rescaled weight
$w<p^2$ all available May letters are $y,b,z,C$.
Their monomials and bidegrees are
\begin{equation}\label{frob:small-monomials}
 C^j y^e z^f b^g,\quad
 j\geq0,\quad e,f,g\in\{0,1\},\qquad
 a=2j+e+f+g,\quad w=p(j+f+g)+e+g.
\end{equation}
The following lists are exhaustive, with a dash denoting the empty
list:
\begin{center}
\begin{tabular}{lll}
\toprule
Rescaled weight & Cohomological degree & May monomials\\
\midrule
$p(p-1)$ & $2p-4$ & ---\\
$p(p-1)$ & $2p-3$ & $zC^{p-2}$\\
$p(p-2)$ & $2p-4$ & $C^{p-2}$\\
$p(p-3)$ & $2p-5$ & ---\\
$p+2$ & $1,3$ & ---\\
$p+2$ & $2$ & $yb$\\
$2p+2$ & $2$ & ---\\
$2p+2$ & $3$ & $ybz$\\
$3p+2$ & $3$ & ---\\
$2$ & $1,3$ & ---\\
\bottomrule
\end{tabular}
\end{center}
The classes $C^{p-2}$ and $zC^{p-2}$ in the table are nonzero in actual
quotient cohomology.  There is also a nonzero class $\upsilon$ of degree two
and weight $p+2$, detected by $yb$, and $\upsilon z$ is nonzero
in degree three and weight $2p+2$.
\end{lemma}

\begin{proof}
The next exterior weights are $p^2$ and $p^2+p$; the third-coordinate
weight, if present, is $p^2+p+1$.  The next polynomial weight is
$p^2$.  All are outside the asserted range.  Lemma~\ref{may:page} therefore
gives exactly \eqref{frob:small-monomials}.  To verify the table,
first reduce its weight equation modulo $p$: its remainder is
$e+g\in\{0,1,2\}$.  For weights divisible by $p$ it forces
$e=g=0$, after which the degree equation fixes $j$ and $f$.
For the listed weights congruent to two it forces $e=g=1$;
substitution into both equations gives the entries shown.
The weight-two entries follow in the same way, since $j=f=g=0$
at that weight.  All weights in the table are within the asserted
range, because $3p+2<p(p-1)$ at $p=1009$.

The cocycles $z=[\xi^p]$ and $C=\beta(\xi)$ come from the
primitive coordinate $\xi$.  Their product $zC^{p-2}$ is an
actual cocycle with the indicated nonzero leading May monomial.
The preceding cohomological degree at its weight has no May
monomial in any filtration.  It consequently cannot become a
boundary on any May page.  An actual cocycle has no outgoing May
differential, proving its nonvanishing.  The factor $C^{p-2}$ is therefore
nonzero as well.

For the other two classes, use the cobar convention
$d[u\mid v]=d[u]\mid v-u\mid d[v]$ with
$d[\eta]=[\xi\mid\xi^p]$.  The corrected cocycle
\begin{equation}\label{frob:corrected-cocycle}
 U=[\xi\mid\eta]+\frac12[\xi^2\mid\xi^p]
\end{equation}
is closed: its two differentials are
$-[\xi\mid\xi\mid\xi^p]$ and
$[\xi\mid\xi\mid\xi^p]$, respectively.  In the ordinary
May filtration its first summand has degree four and the correction
has degree three.  Its leading detector is thus $yb$.  The table
excludes both an incoming and an outgoing May differential, so
$\upsilon=[U]\neq0$.  Similarly
\[
 U\mid\xi^p
   =[\xi\mid\eta\mid\xi^p]
       +\frac12[\xi^2\mid\xi^p\mid\xi^p]
\]
is a closed representative with leading detector $ybz$.
Its correction has smaller May filtration.  The absence of a
degree-two May monomial at weight $2p+2$ excludes every possible
incoming differential, and the explicit cocycle excludes outgoing
differentials.  Hence $\upsilon z\neq0$.
\end{proof}

\subsection{The row \texorpdfstring{$k=2$}{k=2}}

\begin{proposition}[The row $k=2$]\label{frob:k2}
At $s=4p-3$ and $w=u_4+2$, the cobar cohomology group in
Table~\ref{may:remaining-table} is zero.
\end{proposition}

\begin{proof}
We prove vanishing for $G_3$ by treating the first two rows of
Table~\ref{frob:coefficient-obligations}.  Their kernel degrees are
$2p+1$ and $2p-1$, so their coefficient differentials cannot mix.

For $M=xac\operatorname{Sym}^{p-1}(A,B,E)$, we have
$\mathcal R=p(p-1)$ and quotient degree $a=2p-4$.
After the weight exclusion of $E$, the finite module is
$xacA^{p-1}\langle1,D,\ldots,D^{p-1}\rangle$.
The sole candidate from Table~\ref{may:remaining-table} is
$D C^{p-2}$, suppressing the invariant factor $xacA^{p-1}$.
Its quotient weight is $p(p-2)$ by \eqref{frob:D-weight}.
The translation of $D$ gives, on the first possible coefficient page,
\begin{equation}\label{frob:k2-differential}
 D[C^{p-2}]\longmapsto \pm[zC^{p-2}].
\end{equation}
The target has quotient degree $2p-3$ and weight $p(p-1)$.
Both source and target are nonzero actual quotient-cohomology classes
by Lemma~\ref{frob:quotient-groups}; they are present on this page because
all earlier coefficient differentials vanish.  Thus
\eqref{frob:k2-differential} removes the candidate.

We record the adjacent-degree checks.  A differential on this page
entering its source would have $D$-degree two, quotient degree
$2p-5$, and quotient weight $p(p-3)$; this group is zero
by Lemma~\ref{frob:quotient-groups}.  The target, having $D$-degree
zero, has no outgoing coefficient differential.  Its first-page
incoming source can only have $D$-degree one, and
\eqref{frob:small-monomials} in degree $2p-4$, weight $p(p-2)$,
gives only $C^{p-2}$.  Thus no other source can cancel the nonzero
map in \eqref{frob:k2-differential}.  In the original total degree,
the same equation, or the exhaustive candidate table, leaves no
other coefficient-leading term in this vertical summand.  Hence
a change of representative cannot replace the removed class.

Now take the full second summand
$M=xac\operatorname{Sym}^{p-2}(A,B,E)$, with $\mathcal R=p^2-p+1$
and quotient degree $2p-2$.
Table~\ref{may:remaining-table} leaves only its $D$-degree-zero candidate,
whose coefficient $xacA^{p-2}$ is invariant.
Its quotient detector $yzC^{p-2}$ is already a
splitting boundary, since $db=yz$ and $dC=0$ on the May page.
This also has a direct cobar realization: $yz=d[\eta]$ and $C$
is an actual cocycle, so $yzC^{p-2}$ is exact.  It is the only
May monomial in this quotient bidegree, as follows by putting
$a=2p-2$, $w=p^2-p+1$ in \eqref{frob:small-monomials}.
It therefore contributes no actual quotient cohomology in
\eqref{frob:coefficient-page}.  These are all initial candidates
of total degree $4p-3$ and weight $u_4+2$.  The coefficient spectral
sequences consequently vanish there, so all corresponding terms
of \eqref{frob:lhss} are zero.  Its convergence proves the result.
\end{proof}

\subsection{The row \texorpdfstring{$k=p-1$}{k=p-1}}

\begin{proposition}[The row $k=p-1$]\label{frob:kpminus1}
Put $N_0=(p-1)^2$.  At $s=2N_0+1$ and
$w=u_4+p-1$, the cobar cohomology group in
Table~\ref{may:remaining-table} is zero.
\end{proposition}

\begin{proof}
We treat the last two rows of Table~\ref{frob:coefficient-obligations}
for $G_2$, using the finite module \eqref{frob:finite-D} in each case.
For $n=N_0$, the residual weight is $\mathcal R=3p+1$ and the quotient
degree is one.  We show directly that every one-cocycle vanishes.
The leading candidates could survive only if lower powers of $D$
supplied corrections.  The crossed-cocycle equation will rule these out:
it requires unequal mixed coefficients where every available correction
has equal ones.

Write $\xi(g)=r$, $\eta(g)=u$,
$\xi(h)=s$, $\eta(h)=v$, so that
\[
 \xi(gh)=r+s,\qquad\eta(gh)=u+v+rs^p.
\]
A normalized one-cocycle is a polynomial $f(D;g)$ satisfying
\begin{equation}\label{frob:crossed-equation}
 f(D;gh)=f(D;g)+f(D+r^p;h).
\end{equation}
Since $D$ has weight $p$ and $4p>3p+1$, it must have the form
\[
 f(D;g)=D^3f_3(g)+D^2f_2(g)+Df_1(g)+f_0(g).
\]
All these powers are in the finite module, since $N_0\geq3$.
For a polynomial function $\varphi$ write
$\delta\varphi(g,h)=\varphi(gh)-\varphi(g)-\varphi(h)$.
Comparing the four coefficients in
\eqref{frob:crossed-equation} gives
\begin{align}
 \delta f_3&=0,\nonumber\\
 \delta f_2&=3r^pf_3(h),\nonumber\\
 \delta f_1&=2r^pf_2(h)+3r^{2p}f_3(h),
       \label{frob:crossed-coefficients}\\
 \delta f_0&=r^pf_1(h)+r^{2p}f_2(h)+r^{3p}f_3(h).
       \nonumber
\end{align}
Homogeneity forces
\begin{align*}
 f_3&=a_3\xi,\\
 f_2&=b_2\eta+c_2\xi^{p+1},\\
 f_1&=d_1\eta\xi^p+e_1\xi^{2p+1},\\
 f_0&=r_0\xi^{3p+1}+s_0\eta\xi^{2p}
                       +t_0\eta^2\xi^{p-1}.
\end{align*}
Here the coefficients $a_3,b_2,c_2,d_1,e_1,r_0,s_0,t_0$ lie in $\mathbb F_p$.
The second equation of \eqref{frob:crossed-coefficients}, together
with
\[
 \delta\eta=rs^p,\qquad
 \delta(\xi^{p+1})=r^ps+rs^p,
\]
forces $c_2=3a_3$ and $b_2=-3a_3$.
The next equation has coefficient $2b_2$ on $r^pv$ and zero
on $us^p$.  On the other hand,
\begin{equation}\label{frob:asymmetric-pair}
 \delta(\eta\xi^p)
       =us^p+vr^p+r^{p+1}s^p+rs^{2p},
\end{equation}
and $\delta(\xi^{2p+1})$ contains neither $u$ nor $v$.
Its left side must therefore have equal coefficients on $r^pv$
and $us^p$.  Since $2b_2=-6a_3$ and $6\neq0$ in
$\mathbb F_{1009}$, we obtain $a_3=0$, and then
$b_2=c_2=0$.  No term in $f_0$ can alter this
coefficient-of-$D$ equation.

Now $f_1$ is primitive.  Equation~\eqref{frob:asymmetric-pair}
first forces $d_1=0$, and the coefficient of $r^{2p}s$
in $\delta(\xi^{2p+1})$ then forces $e_1=0$.
Finally $f_0$ is primitive.  In its reduced coproduct the monomial
$u^2s^{p-1}$ has coefficient $t_0$, so $t_0=0$.
The monomial $us^{2p}$ then has coefficient $s_0$, so
$s_0=0$.  The coefficient of $r^{3p}s$ in
$\delta(\xi^{3p+1})$ is one, giving $r_0=0$.
Every normalized cocycle is thus zero, proving the required
$H^1$ vanishing, including all lower-$D$ corrections.

For $n=N_0-1$, the residual weight is $\mathcal R=3p+2$ and the desired
quotient degree is three.  The first coefficient differential
comes from the term $2D\xi^p$ in
\[
 (D+\xi^p)^2-D^2=2D\xi^p+\xi^{2p}.
\]
Using the actual class $\upsilon$ of
Lemma~\ref{frob:quotient-groups}, it is
\begin{equation}\label{frob:h3-differential}
 D^2\upsilon\longmapsto\pm2D\upsilon z.
\end{equation}
The coefficient differential first prepends $z$. Its product
$z\upsilon$ agrees with $\upsilon z$: both are actual cocycles
with the same leading May monomial $ybz$, with coefficient $+1$
after two exterior interchanges, and the list in Equation~\eqref{frob:small-monomials}
leaves no other initial May monomial or lower-filtration cohomology
in this bidegree. Thus the product order in
Equation~\eqref{frob:h3-differential} is justified directly.
Both powers of $D$ lie in the finite module, since $n\geq3$.
Equation~\eqref{frob:D-weight} gives quotient degree and weight
$(2,p+2)$ for the source and $(3,2p+2)$ for the target.
Both therefore have the required residual weight $3p+2$.
All earlier coefficient differentials vanish, so source and target are
still their actual quotient-cohomology classes on this first possible
page.  They are both nonzero by
Lemma~\ref{frob:quotient-groups}.

A same-page incoming differential to the source would require
$D$-degree three and quotient $H^1$ of weight two,
which is zero.  The only same-page incoming source for the target
is $D^2H^2(Q_2)_{p+2}$, the one-dimensional span displayed in
\eqref{frob:h3-differential}.  A same-page outgoing differential
from the target is proportional to $\upsilon z^2=0$.
Indeed $[\xi^p\mid\xi^p]=\tfrac12d[\xi^{2p}]$, so this
last vanishing also follows directly in the cobar complex.
The term $\xi^{2p}$ in the expansion of $D^2$ lowers two
$D$-levels and cannot cancel its first differential.  Since
$2\neq0$ in $\mathbb F_p$, \eqref{frob:h3-differential}
removes the target.

There is no other degree-three coefficient-leading term.  In fact,
the possibilities for $D$-degrees $0,1,2,3$ require quotient
degree three at respective weights
$3p+2,2p+2,p+2,2$; by
Lemma~\ref{frob:quotient-groups}, only the second has a
candidate, namely $ybz$.  Larger $D$-degrees exceed the residual
weight.  Thus there is no replacement in a lower coefficient
filtration.  The initial candidate table excludes every other
kernel degree or exterior summand at this total bidegree.
Both possible terms of \eqref{frob:lhss} consequently vanish,
and the proposition follows.
\end{proof}

\begin{corollary}\label{frob:vanishing}
The obstruction groups in bidegrees
\[
 (s,w)=(2q+1,u_4+2),\qquad
 (s,w)=(q(p-1)+1,u_4+p-1)
\]
vanish in the additive cobar calculation of
Table~\ref{may:remaining-table}.  Among its rows with $k\geq2$,
only the row $k=p$ remains to be addressed.
\end{corollary}

\begin{proof}
Apply Propositions~\ref{frob:k2} and~\ref{frob:kpminus1},
and the exhaustive list in Table~\ref{may:remaining-table}.
\end{proof}
==== END A-finite-Smith-Toda-complex-V4-at-the-prime-1009//04-frobenius-elimination.tex ====

==== BEGIN A-finite-Smith-Toda-complex-V4-at-the-prime-1009//05-final-obstruction.tex ====
\section{The final obstruction and the finite cofiber}
\label{top:section}

We retain the prime $p=1009$ and put
\begin{equation}
\label{top:constants}
 N=p(p-1)-2,\qquad K=p(p-2)+1=(p-1)^2,
 \qquad W=2p^2+4p+1.
\end{equation}
The spectrum $Y$ of Proposition~\ref{low:V3} has
$BP_*Y=R/I_4$ and a binary two-sided unital multiplication
$\mu:Y\wedge Y\longrightarrow Y$ with unit $\eta:S^0_{(p)}\to Y$.
All spheres and smash products in this section are $p$-local.
We shall use the product and convergence assertions of
Proposition~\ref{fw:anss-pairing} and Proposition~\ref{fw:anss}.
In this section the second superscript of $E_r^{s,w}$ is weight:
it denotes the group of internal degree $qw$ in the earlier notation.

The May comparison and the remaining-target calculation,
Lemma~\ref{may:comparison} and Table~\ref{may:remaining-table},
give a surjection
\begin{equation}
\label{top:An-surjection}
 A^N\colon E_2^{5,W}(Y)\longrightarrow
 E_2^{qp+1,u_4+p}(Y).
\end{equation}
Indeed, every initial May monomial in the target is divisible by
$A^N$, and the two degree identities are
\[
 5+2N=qp+1,\qquad W+pN=u_4+p.
\]
For the source, $W<u_4$, so specialization to the additive Hopf
algebra identifies the normalized cobar groups in all three degrees
$4,5,6$. For the target, $u_4+p<u_5$; a positive $v_4$ coefficient
would leave weight $p$, less than the shortest adjacent tensor length
$qp=(qp+1)-1$. Thus the target groups in degrees $qp,qp+1,qp+2$
also agree with their additive specializations, by
Lemma~\ref{fw:normalized-bound}. The two triples differ by $2N$.
Cup multiplication by the actual carry cocycle representing $A^N$
commutes with specialization, so the filtered surjection of
Lemma~\ref{may:comparison} transports through this commuting square
to the displayed surjection for $Y$.

Let $\alpha\in\pi_{q-1}S^0$ and
$\beta\in\pi_{pq-2}S^0$ be the sphere classes constructed in
Lemma~\ref{top:sphere-classes}, detected by $[t_1]$ and $A$,
respectively, and write $d=|\beta|=pq-2$.
The two-cell spectrum that we use for lifts is
\begin{equation}
\label{top:J-definition}
 J=\Sigma^{-q}C(\alpha),\qquad
 \varepsilon:J\longrightarrow S^0,
\end{equation}
where $C(\alpha)$ is the cofiber of $\alpha:S^{q-1}\to S^0$
and $\varepsilon$ is its desuspended top quotient. Thus $J$ has
cells in degrees $-q$ and zero. Its connecting map is the corresponding
suspension of $\alpha$, so the cofiber exact sequence gives
\begin{equation}
\label{top:lift-criterion}
 x\in\pi_gY\text{ lifts to }\pi_g(J\wedge Y)
 \quad\Longleftrightarrow\quad
 \alpha x=0\in\pi_{g+q-1}Y.
\end{equation}

Put $n=p-1$. Since $n!$ is a $p$-local unit, the averaging idempotent
\[
 e=\frac1{n!}\sum_{\sigma\in\Sigma_n}\sigma
 \quad\text{on }J^{\wedge n}
\]
has a finite retract $T=\operatorname{Sym}^{n}J$.
Choose splitting maps $i:T\to J^{\wedge n}$ and
$\operatorname{pr}:J^{\wedge n}\to T$, with
$i\operatorname{pr}=e$, and put
$\varepsilon_T=\varepsilon^{\wedge n}i:T\to S^0$.
The key map will be
\[
 f=(\eta\beta)\circ\Sigma^d\varepsilon_T:\Sigma^dT\longrightarrow Y.
\]
The choice of $p-1$ factors has a concrete algebraic purpose:
writing $DT$ for the Spanier--Whitehead dual of $T$, we will identify
$BP_*(DT\wedge Y)$ as a free $R/I_4$-module with basis
$1,r,\ldots,r^{p-1}$ of weights $0,\ldots,p-1$.
These supply exactly the coefficient powers in the integral polynomial
$((r+x)^p-r^p-x^p)/p$.
Lemma~\ref{top:symmetric-comodule} determines the translated coaction,
and Proposition~\ref{top:symmetric-null-map} shows that substituting
$x=t_1$ gives a cochain whose cobar differential is the constant
cocycle $A$. A filtration bound then proves that $f$ is an actual
null map.

We first show that every source class is represented by
$\gamma\in\pi_{qW-5}Y$ and that
$\alpha\gamma=0$ and $\alpha\cdot(\eta\beta^p)=0$ in $\pi_*Y$.
One lift of $\gamma$ and $p-2$ lifts of $\eta\beta^p$ give, after
multiplying the $Y$ factors and averaging the $J$ factors, a map to
$T\wedge Y$ with top value $\beta^{p(p-2)}\gamma$.
The null map $f$ then gives $\beta^K\gamma=0$.
The strict inequality $5+2K<qp+1$, together with $K<N$, converts
this homotopy relation into disappearance of the whole target before
$d_{qp+1}(v_4)$.

\subsection{The small stems used for the lifts}

\begin{lemma}[Small-stem lifts]
\label{top:small-stems}
Every class of $E_2^{5,W}(Y)$ survives to homotopy. Moreover,
\begin{equation}
\label{top:alpha-gamma-vanishing}
 \pi_{q(W+1)-6}Y=0.
\end{equation}
\end{lemma}

\begin{proof}
By $q$-sparsity, a positive differential out of $(5,W)$ has target
\[
 (s,w)=(6+qj,W+j),\qquad j\geq 1.
\]
All possible Adams--Novikov terms in the stem in
\eqref{top:alpha-gamma-vanishing} have
\[
 (s,w)=(6+qj,W+1+j),\qquad j\geq 0.
\]
We treat both by writing
\begin{equation}
\label{top:small-stem-degrees}
 (s,w)=(6+qj,W+\delta+j),\qquad
 \begin{cases}
 \delta=0,&j\geq1,\\
 \delta=1,&j\geq0.
 \end{cases}
\end{equation}

First we give explicit bounds permitting the additive calculation.
The normalized-cobar inequality of Lemma~\ref{fw:normalized-bound}
implies
\[
 (q-1)j\leq W+\delta-6.
\]
At the specified prime,
\[
 q-1=2015,\quad W=2{,}040{,}199,\quad
 2015\cdot1013=2{,}041{,}195>W+1-6.
\]
Consequently $j\leq1012$ and
\begin{equation}
\label{top:small-stem-cutoff}
 w\leq2{,}041{,}212
 <p^3=1{,}027{,}243{,}729
 <u_4=1{,}028{,}262{,}820.
\end{equation}
Thus no nonconstant coefficient of $R/I_4$ occurs, in any cobar
degree at these weights. The only possible digit positions are
$0,1,2$, and the cooperation formulas are additive.
Lemma~\ref{fw:range-comparison} therefore identifies the relevant
cobar calculation with the indicated range of $P$.

Use the six exterior letters $x,a,c,y,b,z$ and the three polynomial
letters $A,B,C$ of Equation~\eqref{may:three-position-weight}.
If $m$ is the number of exterior letters and $L=(s-m)/2$ the number
of polynomial letters, then $m\leq6$ and
\[
 L\geq (p-1)j,\qquad w\geq pL.
\]
It follows that
\[
 \bigl(p(p-1)-1\bigr)j\leq W+\delta.
\]
Since $p(p-1)-1=1{,}017{,}071$ and
$3\cdot1{,}017{,}071>W+1$, this improves the bound to $j\leq2$.

For completeness, we enumerate this remaining finite calculation.
Let $H_i$ count the exterior letters covering position $i$, and let
$n_A,n_B,n_C$ be the polynomial exponents. The weight equation is
\begin{equation}
\label{top:weight-equation}
 w=H_0+p(H_1+L-n_C)+p^2(H_2+n_B+n_C).
\end{equation}
In \eqref{top:small-stem-degrees}, $m$ is even and reduction modulo
$p$ gives $H_0=1+\delta+j$, whenever a solution exists. Put
$\kappa_2=H_2+n_B+n_C$. All position contributions are nonnegative;
\eqref{top:small-stem-cutoff} also gives $w<3p^2$, so
$\kappa_2\leq2$ and $n_C\leq2$. Substituting
$L=3+(p-1)j-m/2$ into \eqref{top:weight-equation} yields
\[
 p(\kappa_2+j-2)=1+j+m/2+n_C-H_1.
\]
For $0\leq j\leq2$, its right side lies between $-3$ and $8$.
Its only multiple of $p=1009$ is zero. Hence
\begin{equation}
\label{top:small-stem-coverage}
 H_0=1+\delta+j,\qquad \kappa_2=2-j,\qquad
 H_1=1+j+m/2+n_C.
\end{equation}

For $j=2$, either $H_0=4$, which is impossible, or $H_0=3$ forces
$c$ to occur whereas $\kappa_2=0$. For $j=1,\delta=1$, all of $x,a,c$
occur and $\kappa_2=1$ excludes $b,z$. Thus $H_1\leq3$, but the last
equation of \eqref{top:small-stem-coverage}, with even $m\geq4$,
requires $H_1\geq4$. For $j=1,\delta=0$, one has $H_0=2$.
If $m=2$, the required inequality $H_1\geq3$ is impossible.
If $m=4$, one must have $H_1=4$ and $n_C=0$; the four letters
covering position one are exactly $a,c,y,b$, which have $H_2=2$,
contrary to $\kappa_2=1$. Finally $m=6$ would give $H_0=3$.
All positive-$j$ groups in \eqref{top:small-stem-degrees} therefore
have empty initial May pages.

It remains to examine $j=0,\delta=1$, where $H_0=2$ and $m$ is
$2$ or $4$. For $m=2$ both letters start at zero, and
\eqref{top:small-stem-coverage} permits only $ac$, followed by
$n_B=1,n_C=0,n_A=1$. For $m=4$ choose two of $x,a,c$ and two of
$y,b,z$. The equations become
\[
 n_C=H_1-3,\qquad n_B=2-H_2-n_C,\qquad
 n_A=1-n_B-n_C.
\]
The following table lists all nine choices; a dash means that at
least one exponent is negative.
\begin{center}
\begin{tabular}{c c c c}
\toprule
Start-zero pair & $yb$ & $yz$ & $bz$\\
\midrule
$xa$ & $xaybB$ & --- & ---\\
$xc$ & $xcybA$ & --- & ---\\
$ac$ & --- & $acyzA$ & ---\\
\bottomrule
\end{tabular}
\end{center}
Thus the complete initial-page list at $(6,W+1)$ is
\begin{equation}
\label{top:four-terms}
 acAB,\qquad xaybB,\qquad xcybA,\qquad acyzA.
\end{equation}
With exterior order $x,a,c,y,b,z$ and
$da=xy$, $db=yz$, $dc=xb+az$, their splitting differentials are
\[
\begin{aligned}
 d(acAB)&=(xab-xcy)AB, &d(xaybB)&=0,\\
 d(xcybA)&=-xaybzA, &d(acyzA)&=-xaybzA.
\end{aligned}
\]
The first image has polynomial factor $AB$ and cannot cancel either
of the other images. The two remaining cycle directions are
boundaries, since
\[
 d(acyB)=-xaybB,\qquad
 d(acbA)=-xcybA+acyzA.
\]
The splitting cohomology of \eqref{top:four-terms} is zero.
The May spectral sequence, finite in each of these weights, proves
all the claimed $E_2$ vanishings.

For orientation, the same equations also enumerate the source
$(5,W)$. Here $H_0=1$, $m$ is odd, and $L=(5-m)/2$. They force
$H_2+n_B+n_C=2$ and $H_1+L-n_C=4$. If $m=1$, then
$H_1+L\leq1+2<4$; if $m=5$, then at least two of $x,a,c$
occur, contrary to $H_0=1$. Thus $m=3$, which forces
$n_C=0$ and $H_1=3$. The only monomials are $aybB$ and $cybA$. We need no assertion that either
one survives the May spectral sequence.

No topological Adams--Novikov differential can hit filtration
five: its length would have to be at least $q+1>5$.
The outgoing target groups have just been proved zero. Thus every
actual class of $E_2^{5,W}(Y)$ survives and is represented by a
class $\gamma\in\pi_{qW-5}Y$. The vanishings for $\delta=1$ exhaust
all filtrations in the stem $q(W+1)-6$; strong convergence proves
\eqref{top:alpha-gamma-vanishing}.
\end{proof}

\subsection{Two sphere classes and an extended-power relation}

\begin{lemma}[The sphere classes]
\label{top:sphere-classes}
There are sphere classes
\[
 \alpha\in\pi_{q-1}S^0_{(p)},\qquad
 \beta\in\pi_{pq-2}S^0_{(p)}
\]
detected respectively by $[t_1]$ and the divided coproduct
$b_{1,0}$. The former generates a cyclic group of order $p$, and
\[
 \pi_iS^0_{(p)}=0\qquad(0<i<q-1).
\]
Multiplication by $\beta$ on the Adams--Novikov spectral sequence
of $Y$ induces multiplication by the actual cobar class $A$ used
in \eqref{top:An-surjection}.
\end{lemma}

\begin{proof}
The cooperation $t_1$ is primitive. In weight one the normalized
sphere cobar is
\[
 \mathbb Z_{(p)}\{v_1\}\xrightarrow{\ p\ }
 \mathbb Z_{(p)}\{t_1\},
\]
because $\eta_R(v_1)-v_1=pt_1$. Hence its degree-one cohomology is
$\mathbb Z/p$ on $[t_1]$.

The integral polynomial
\begin{equation}
\label{top:beta-cocycle}
 b(x,y)=\frac{(x+y)^p-x^p-y^p}{p}
 =\sum_{i=1}^{p-1}\frac{\binom pi}{p}x^iy^{p-i}
\end{equation}
defines the two-cobar cocycle $b_{1,0}$ by substituting the two
primitive copies of $t_1$. Its differential is zero: the
two-cocycle identity follows by expanding $(x+y+z)^p$, or by
dividing the identity $d^2(t_1^p)=0$ in the torsion-free integral
cobar. Its image in the additive specialization is nonzero in
cohomology. At weight $p$ the only coordinate available is $t_1$;
the only one-cochain of that weight is $t_1^p$, whose differential
is zero in characteristic $p$, whereas \eqref{top:beta-cocycle}
is a nonzero two-cochain. Thus it is not a boundary.

The first positive topological differential length permitted by
sparsity is $q+1$. From $(2,p)$ an outgoing differential has target
$(qj+3,p+j)$ for $j\geq1$, violating $w\geq s$; no incoming
differential can reach filtration two. The same argument applies
to $(1,1)$. Therefore these cocycles survive to the stated sphere
classes.

For any positive stem $i$ with a nonzero normalized-cobar term,
either $s=0$ and $i=qw\geq q$, or $s\geq1$ and
\[
 i=qw-s\geq(q-1)s\geq q-1.
\]
This proves the lower sphere vanishing. In stem $q-1$ the same
inequality permits only $(s,w)=(1,1)$, so there is no extension
ambiguity and $\pi_{q-1}S^0_{(p)}=\mathbb Z/p\{\alpha\}$.
Finally, the sphere action and its cobar pairing in
Proposition~\ref{fw:anss-pairing} identify the detector of
multiplication by $\beta$ with $b_{1,0}=A$.
\end{proof}

We next prove that $\alpha\beta^p$ has zero image in $\pi_*Y$. We use
the odd-primary reduced powers, including the Cartan formula and
$\mathcal P^k(x)=x^p$ for a space class of degree $2k$, in
\citet[Chapter~VI, Section~1, and Chapter~VIII, Theorem~2.2]{SteenrodEpstein1962}.
Their compatibility with suspension gives operations on spectra; the
instability normalization below is applied to a Thom space before
desuspension. Toda proved the corresponding sphere relation
\citep[p.~839, Corollary]{Toda1967} and subsequently treated such
relations using cyclic extended powers
\citep[Sections~1--3, especially Theorem~3]{Toda1968}.
We give the image relation needed in $Y$ through the following
finite-cell construction.

\begin{lemma}[A finite three-cell model]
\label{top:three-cell-model}
Let $U_0$ be a connective $p$-local spectrum with
\[
 H_0(U_0;\mathbb Z_{(p)})=\mathbb Z_{(p)},\qquad
 H_{q-1}(U_0;\mathbb Z_{(p)})=\mathbb Z/p,
\]
and with all other positive integral homology through degree $q$
zero. Let $b_0:S^0\to U_0$ induce an isomorphism on $H_0$, and
suppose the reduced power $\mathcal P^1$ of the bottom mod-$p$
cohomology class is nonzero. Then there is a three-cell spectrum
\[
 Q=\bigl(S^0\vee S^{q-1}\bigr)
       \cup_{(a\alpha,\,up)}e^q
\]
and a map $Q\to U_0$ whose restriction to $S^0$ is $b_0$.
Here $u\in\mathbb Z_{(p)}^\times$ and
$a\in\mathbb F_p^\times$.
\end{lemma}

\begin{proof}
Let $C$ be the cofiber of $b_0:S^0\to U_0$.
The assumed integral homology groups and successive applications
of connective Hurewicz show that $C$ is $(q-2)$-connected and
\[
 \pi_{q-1}C=H_{q-1}(C;\mathbb Z_{(p)})=\mathbb Z/p.
\]
Choose a generator $S^{q-1}\to C$. Its obstruction to lifting to
$U_0$ lies in $\pi_{q-2}S^0$, which is zero by
Lemma~\ref{top:sphere-classes}. A lift, together with $b_0$, gives
$S^0\vee S^{q-1}\to U_0$.
Its cofiber $D$ is $(q-1)$-connected. The integral homology exact
sequence gives
\[
 H_q(D;\mathbb Z_{(p)})
 =\ker\bigl(\mathbb Z_{(p)}\longrightarrow\mathbb Z/p\bigr)
 =p\mathbb Z_{(p)}.
\]
In particular this is a free rank-one module.
Use Hurewicz to choose a generator of $\pi_qD$ and take its
boundary in $\pi_{q-1}(S^0\vee S^{q-1})$. Attaching the
corresponding cell gives a map $Q\to U_0$ extending the two lower
cells. Its component on $S^{q-1}$ is multiplication by a unit
times $p$, since its homology image is exactly
$p\mathbb Z_{(p)}$. Its component on the bottom belongs to
$\pi_{q-1}S^0=\mathbb Z/p\{\alpha\}$, so it is $a\alpha$ for
some $a\in\mathbb F_p$.

The map $Q\to U_0$ induces isomorphisms on integral homology in
degrees zero and $q-1$. The universal coefficient sequence then
gives isomorphisms on mod-$p$ cohomology in degrees zero and $q$;
the latter is the class arising from the torsion in degree $q-1$.
The assumed nonzero reduced power therefore pulls back to a
nonzero reduced power of the bottom class of $Q$.
If $a=0$, the actual attaching map would give a wedge
\[
 Q\simeq S^0\vee\operatorname{cofib}
       \bigl(up:S^{q-1}\to S^{q-1}\bigr).
\]
The degree-zero class in this wedge is pulled back from $S^0$,
so its positive reduced powers vanish by naturality. This is a
contradiction. Hence $a\ne0$.
The construction used an integral homology gap and relative
Hurewicz in a finite range; no assertion that $U_0$ itself is
finite is involved.
\end{proof}

\begin{proposition}[The extended-power relation]
\label{top:extended-power-relation}
The image of $\alpha\beta^p$ in $\pi_*Y$ is zero.
\end{proposition}

\begin{proof}
Write $d=pq-2=2m$, where
\[
 m=p(p-1)-1\equiv-1\pmod p.
\]
Let $\rho_{\mathbb C}$ be the complex regular representation of
$C_p$. The cyclic extended power of the even sphere $S^{2m}$ is
the Thom spectrum of $m\rho_{\mathbb C}$ over $BC_p$:
\[
 (EC_p)_+\wedge_{C_p}(S^{2m})^{\wedge p}
 \simeq\operatorname{Th}(m\rho_{\mathbb C}).
\]
Applying this functor to a sphere map representing $\beta$, then
collapsing $BC_p$ in the extended power of $S^0$, gives a map
\begin{equation}
\label{top:extended-beta-map}
 \Phi:\operatorname{Th}(m\rho_{\mathbb C})\longrightarrow S^0
\end{equation}
whose restriction to a fiber sphere $S^{pd}$ is $\beta^p$.
Only the symmetric multiplication of the sphere spectrum is used
in constructing this map.

The group $\mathbb F_p^\times$ acts by permutations of the
cyclically labelled smash coordinates and by the corresponding
automorphisms of $C_p$. This normalizer action is compatible with
\eqref{top:extended-beta-map}, on whose target it is trivial.
Normalize the bottom degree by setting
\[
 U=\Sigma^{-pd}\operatorname{Th}(m\rho_{\mathbb C}),\qquad
 e_0=\frac1{p-1}\sum_{a\in\mathbb F_p^\times}a:U\longrightarrow U.
\]
The idempotent $e_0$ splits in spectra. Denote its image by $U_0$,
with maps $i_0:U_0\to U$, $r_0:U\to U_0$ satisfying
$i_0r_0=e_0$ and $r_0i_0=1$.
The projected Thom fiber $b_0:S^0\to U_0$ induces an isomorphism
on $H_0(-;\mathbb Z_{(p)})$: the complex orientation makes every
normalizer permutation act as one on the bottom class.
Moreover, normalizer invariance of $\Phi$ gives a map
\[
 \Phi_0:\Sigma^{pd}U_0\longrightarrow S^0,
 \qquad \Phi_0\Sigma^{pd}b_0=\beta^p.
\]

We describe the homology and the reduced power of this summand.
The complex bundle is integrally oriented, so the integral Thom
isomorphism identifies $H_*(U;\mathbb Z_{(p)})$ with
$H_*(BC_p;\mathbb Z_{(p)})$. The periodic cyclic-group resolution,
with maps $g-1$ and $1+g+\cdots+g^{p-1}$, gives
\[
 H_0(BC_p;\mathbb Z_{(p)})=\mathbb Z_{(p)},\quad
 H_{2j-1}(BC_p;\mathbb Z_{(p)})=\mathbb Z/p\ (j\geq1),\quad
 H_{2j}(BC_p;\mathbb Z_{(p)})=0\ (j\geq1).
\]
These groups are finitely generated degreewise, as are their
direct summands for $U_0$. The same resolution, or the circle
bundle of Chern class $p$ over infinite complex projective space,
gives
\[
 H^*(BC_p;\mathbb F_p)=\Lambda(s)\otimes\mathbb F_p[t],
 \qquad |s|=1,\quad |t|=2,\quad t=\operatorname{Bockstein}(s).
\]
An automorphism $a\in\mathbb F_p^\times$ multiplies both $s$ and
$t$ by $a$. Equivalently it acts by $a^j$ on integral homology
in degree $2j-1$. Averaging consequently retains, through degree
$q$, exactly the mod-$p$ classes in degrees $0,q-1,q$ and the
integral groups
\begin{equation}
\label{top:U0-homology}
 H_0(U_0;\mathbb Z_{(p)})=\mathbb Z_{(p)},\quad
 H_{q-1}(U_0;\mathbb Z_{(p)})=\mathbb Z/p,
 \quad H_j(U_0;\mathbb Z_{(p)})=0\ (0<j\leq q,\ j\ne q-1).
\end{equation}

Let $u$ be the normalized Thom class. We claim
\begin{equation}
\label{top:Thom-P1}
 \mathcal P^1u=-m\,t^{p-1}u\ne0.
\end{equation}
For a universal complex line bundle with first Chern class $l$
and Thom class $u_l$, write
$\mathcal P^1u_l=c\,l^{p-1}u_l$ by the Thom isomorphism.
Pullback along the zero section and the identity
$\mathcal P^1(l)=l^p$ imply $c=1$ in the polynomial cohomology of
infinite complex projective space. Naturality gives the formula
for every line bundle. The Cartan formula for a direct sum of
lines therefore gives the sum of their $(p-1)$st Chern powers as
the multiplier of $\mathcal P^1$ on its Thom class.
The regular representation splits into the characters with first
Chern classes $it$, $i\in\mathbb F_p$. Thus the multiplier here is
\[
 m\sum_{i\in\mathbb F_p}(it)^{p-1}
 =-m\,t^{p-1}.
\]
Both this class and $u$ are normalizer-invariant. The reduced
power commutes with the idempotent, so the same nonzero formula
holds in $U_0$.

Apply Lemma~\ref{top:three-cell-model} to
\eqref{top:U0-homology} and \eqref{top:Thom-P1}. It supplies a
three-cell spectrum
\[
 Q=(S^0\vee S^{q-1})\cup_{(a\alpha,\,up)}e^q\longrightarrow U_0
\]
with $a$ nonzero modulo $p$ and $u$ a $p$-local unit.

Restrict $\Phi_0$ to $\Sigma^{pd}Q$. Its bottom value is
$\beta^p$; let its value on the other lower cell be $\zeta$.
The top cell gives the relation
\[
 a\alpha\beta^p+up\zeta=0
 \quad\text{in }\pi_{pd+q-1}S^0.
\]
The action of $p$ on $Y$ is null by Corollary~\ref{low:p-null}.
Apply the unit to this relation and use $a\ne0$ modulo $p$ to
obtain $\alpha\beta^p=0$ in $\pi_*Y$.
\end{proof}

\subsection{The symmetric comodule and the null map}

We now prove that the map $f:\Sigma^dT\to Y$ defined above is null.
Finite duality turns it into the bottom image of $\beta$ in
$\pi_d(DT\wedge Y)$. The next lemma identifies the coefficient
comodule in which we will make this image a cobar boundary.

\begin{lemma}[The symmetric coefficient comodule]
\label{top:symmetric-comodule}
For $T=\operatorname{Sym}^{p-1}(\Sigma^{-q}C(\alpha))$, the module
$BP_*(DT\wedge Y)$ is free over $R/I_4$ with basis
$1,r,\ldots,r^{p-1}$, where $r^k$ has weight $k$. Its coaction is
\begin{equation}
\label{top:translated-comodule}
 r^k\longmapsto\sum_{j=0}^k\binom kjr^jt_1^{k-j}.
\end{equation}
The induced homology map of
$D\varepsilon_T\wedge1_Y:Y\to DT\wedge Y$ is the inclusion
of constants. The powers of $r$ denote basis elements; no
multiplication on $DT$ is asserted.
\end{lemma}

\begin{proof}
\smallskip\noindent\emph{The two-cell coaction.}
We first compute the coefficient comodule needed for duality.
The dual $DJ$ is the cofiber of
$\alpha':S^{q-1}\to S^0$, where $\alpha'=\pm\alpha$ after fixing
cell orientations, and its bottom inclusion is $D\varepsilon$.
Put $\mathcal F=\operatorname{fib}(S^0\xrightarrow{\eta_{BP}}BP)$.
A chosen $BP$-nullhomotopy of $\alpha'$ gives a lift
$\widetilde\alpha:S^{q-1}\to\mathcal F$ and a map $H:DJ\to BP$
extending the unit, in the following map of cofiber sequences:
\[
\begin{array}{ccccccc}
 S^{q-1}&\xrightarrow{\alpha'}&S^0&\longrightarrow&DJ&
       \longrightarrow&S^q\\[2pt]
 \big\downarrow\mathrlap{\,\scriptstyle\widetilde\alpha}&&
 \big\Vert&&\big\downarrow\mathrlap{\,\scriptstyle H}&&
 \big\downarrow\mathrlap{\,\scriptstyle\Sigma\widetilde\alpha}\\[2pt]
 \mathcal F&\longrightarrow&S^0&\xrightarrow{\eta_{BP}}&BP&
       \longrightarrow&\Sigma\mathcal F .
\end{array}
\]
The $BP$-module $BP\wedge DJ$ splits because $BP_{q-1}=0$.
Let $e_0$ be its bottom generator and choose an upper generator
$z\in BP_qDJ$ whose top quotient is $1$ and for which
$\rho_*z=0$, where $\rho=\mu_{BP}(BP\wedge H):BP\wedge DJ\to BP$
retracts the bottom inclusion. With $\epsilon:\Gamma\to R$
the counit, the element $g=(BP\wedge H)_*z$ satisfies
\[
 \epsilon(g)=0,\qquad
 \Gamma_q=\eta_L(R_q)\oplus\mathbb Z_{(p)}\{t_1\},
 \qquad g=ct_1.
\]
Here $\eta_L$ is induced by the bottom map
$BP\wedge\eta_{BP}:BP\to BP\wedge BP$.
After smashing the cofiber diagram with $BP$, its connecting
map $\partial:\Gamma_q\to\pi_{q-1}(BP\wedge\mathcal F)$ has kernel
$\eta_L(R_q)$ and gives
\[
 \partial g=(BP\wedge\widetilde\alpha)_*(1).
\]
Thus $g$ represents the first Adams--Novikov detector of
$\alpha'$. It is also the actual off-diagonal coaction:
writing $\psi(z)=1\otimes z+\omega\otimes e_0$, naturality
under $H$ gives
$\Delta g=1\otimes g+\omega\otimes1$.
Apply $\operatorname{id}\otimes\epsilon$ and use
$\epsilon(g)=0$ to obtain $\omega=g$.
By Lemma~\ref{top:sphere-classes} this detector is
$\pm[t_1]$, so $c\not\equiv0\pmod p$.

For completeness, fix the left-comodule convention
$\psi(am)=\eta_L(a)\psi(m)$ with tensor product using $\eta_R$,
so $\eta_R(v_1)-\eta_L(v_1)=pt_1$.
Replacing $z$ by $z+av_1e_0$ changes its off-diagonal term to
$g+a(\eta_L-\eta_R)(v_1)=g-pa\,t_1$.
Changing $H$ by the composite
$DJ\to S^q\xrightarrow{av_1}BP$ instead changes $g$ by
$\eta_R(av_1)$; renormalizing the upper generator to
$z-av_1e_0$ subtracts $\eta_L(av_1)$, giving $g+pa\,t_1$.
Opposite conventions reverse these signs, and dual cell
orientations change the detector by a sign; none changes its
nonzero residue modulo $p$. The underlying $BP$-module splitting
therefore need not split the comodule.
Replacing $z$ by $c^{-1}z$ now makes the off-diagonal term
exactly $t_1$. Its top quotient is then $c^{-1}$ rather than $1$;
we no longer need the earlier top normalization. This change fixes
$e_0$ and the bottom inclusion $D\varepsilon$, which are the data
used in the following symmetric-power calculation.

\smallskip\noindent\emph{The symmetric coefficient module.}
Duality changes each permutation into its inverse, and hence
preserves the average. Consequently
\[
 DT\simeq\operatorname{Sym}^{n}(DJ).
\]
Under this identification $D\varepsilon_T$ is the pure bottom
tensor $e_0^{\otimes n}$. All the cell degrees of $DJ$ are even.
If $z_S$ is the tensor with upper generator in positions $S$ and
bottom generator elsewhere, define its normalized symmetric
basis by
\begin{equation}
\label{top:symmetric-basis}
 r_k=\binom nk^{-1}\sum_{|S|=k}z_S,
 \qquad 0\leq k\leq n.
\end{equation}
Every denominator is a $p$-local unit. Expanding the translation
of each tensor gives the coefficient
\[
 \frac{\binom{n-j}{k-j}\binom nj}{\binom nk}=\binom kj
\]
on $r_jt_1^{k-j}$ in the coaction of $r_k$.
Thus we can denote this basis by $1,r,\ldots,r^{p-1}$, with
coaction~\eqref{top:translated-comodule}. Each $r^k$ has weight $k$
because it is represented by tensors with $k$ upper generators of
degree $q$.

The module $BP_*DT$ is free. Smashing its free $BP$-module
splitting with $Y$ gives
\[
 BP_*(DT\wedge Y)=BP_*DT\otimes_R(R/I_4),
\]
with the same translated basis and with $D\varepsilon_T$ identified
with constants. This argument uses a free factor, so it requires
no flatness of $R/I_4$.
\end{proof}

\begin{proposition}[The symmetric null map]
\label{top:symmetric-null-map}
For $d=|\beta|=pq-2$, the actual stable map
\begin{equation}
\label{top:null-composite}
 f:\Sigma^dT\xrightarrow{\ \Sigma^d\varepsilon_T\ }S^d
       \xrightarrow{\ \beta\ }S^0
       \xrightarrow{\ \eta\ }Y
\end{equation}
is nullhomotopic.
\end{proposition}

\begin{proof}
Finite duality identifies
\[
 [\Sigma^dT,Y]\cong[S^d,DT\wedge Y]
\]
and carries $f$ to
$(D\varepsilon_T\wedge1_Y)(\eta\beta)$.
By Lemma~\ref{top:symmetric-comodule}, its induced $E_2$ class is
represented by the constant-valued beta cocycle. We first show that this cocycle
is a boundary in the translated coefficient module.
Consider the integral polynomial
\begin{equation}
\label{top:carry-polynomial}
 F(r,x)=\frac{(r+x)^p-r^p-x^p}{p}
       =\sum_{i=1}^{p-1}\frac{\binom pi}{p}r^{p-i}x^i.
\end{equation}
The displayed sum defines it before reduction modulo $p$; it uses
only the allowed coefficient powers $r,\ldots,r^{p-1}$.
The exact integral identity
\begin{equation}
\label{top:carry-boundary}
 F(r,x)+F(r+x,y)-F(r,x+y)
       =\frac{(x+y)^p-x^p-y^p}{p}
\end{equation}
follows by cancellation of the four $p$th powers.
After reduction, the coefficient coaction
\eqref{top:translated-comodule} and primitivity of $t_1$ make the
left side the cobar differential of $F(r,t_1)$, with the common
overall sign determined by the cobar convention. The right side
is the constant cocycle \eqref{top:beta-cocycle}. Thus the bottom
image of $\beta$ has zero image on $E_2$.

There is no possible higher-filtration homotopy value hidden by
this boundary. The spectrum $DT\wedge Y$ is connective and its
coefficient weights are nonnegative. In stem $pq-2$, its
$q$-sparse terms have
\[
 (s,w)=(2+qj,p+j).
\]
For $j\geq1$ the inequality $w\geq s$ would give
$p-2\geq(q-1)j$, which is impossible. Naturality places the
image of $\beta$ in filtration at least two, and the boundary
just found raises it to filtration at least three. All later
associated-graded groups are zero, so convergence with finite
filtration makes the actual image zero.

The bottom beta image is therefore zero, and finite duality makes
$f$ itself nullhomotopic.
\end{proof}

\subsection{The averaged lift and the page on which it vanishes}

\begin{proposition}[Averaging and annihilation]
\label{top:annihilation}
If $\gamma\in\pi_{qW-5}Y$ represents a class of
$E_2^{5,W}(Y)$, then
\begin{equation}
\label{top:beta-annihilation}
 \beta^K\gamma=0,\qquad K=p(p-2)+1.
\end{equation}
Only the binary two-sided unit identities for $\mu$ are needed.
\end{proposition}

\begin{proof}
Lemma~\ref{top:small-stems} gives $\alpha\gamma=0$ in precisely
the stem $q(W+1)-6$. Proposition~\ref{top:extended-power-relation}
gives $\alpha\beta^p=0$ after applying the unit to $Y$.
By \eqref{top:lift-criterion}, choose lifts
\[
 \widetilde\gamma:S^{qW-5}\longrightarrow J\wedge Y,
 \qquad
 \widetilde b:S^{pd}\longrightarrow J\wedge Y
\]
of $\gamma$ and $\eta\beta^p$, respectively.
Fix any parenthesization of the $n$-fold iteration of $\mu$.
Smash one copy of $\widetilde\gamma$ with $p-2$ copies of
$\widetilde b$, collect the $J$ factors, and apply that chosen
iteration to the $Y$ factors. This gives
\[
 L:S^{qW-5+(p-2)pd}\longrightarrow J^{\wedge n}\wedge Y.
\]
Its top projection is $\beta^{p(p-2)}\gamma$.
To verify this without an associativity assumption, for any
sphere scalar $a$ use the two unit homotopies to identify
\[
 \mu(\eta a,x)=a x,\qquad
 \mu(x,\eta a)=(-1)^{|a||x|}a x.
\]
The scalar $\beta^p$ is even. In the chosen binary tree, every
subtree without $\gamma$ is a scalar times the unit, and the
unique subtree containing $\gamma$ is the product of its scalars
times $\gamma$. This proves the asserted top value by induction
on the tree.

Now project to the symmetric retract after those $Y$ factors
have already been multiplied:
\[
 L_T=(\operatorname{pr}\wedge1_Y)L:
 S^{qW-5+(p-2)pd}\longrightarrow T\wedge Y.
\]
For every permutation $\sigma$ of the $J$ factors, naturality of
the symmetry and the zero-dimensional top sphere give an equality
of stable maps
\[
 \varepsilon^{\wedge n}\sigma=\varepsilon^{\wedge n}.
\]
Consequently
\begin{equation}
\label{top:averaged-top-map}
 (\varepsilon_T\wedge1_Y)L_T
 = (\varepsilon^{\wedge n}e\wedge1_Y)L
 = (\varepsilon^{\wedge n}\wedge1_Y)L
 =\beta^{p(p-2)}\gamma.
\end{equation}
No permutation of the multiplied $Y$ factors occurs in this
identity.

The actual null map $f$ of Proposition~\ref{top:symmetric-null-map}
now gives the following homotopy-commutative diagram:
\begin{equation}
\label{top:null-map-diagram}
\begin{array}{ccc}
\Sigma^d(T\wedge Y)&\xrightarrow{\ f\wedge1_Y\ }&Y\wedge Y\\[2pt]
\mathllap{\scriptstyle\Sigma^d(\varepsilon_T\wedge1_Y)\,}\big\downarrow
&&\big\downarrow\mathrlap{\,\scriptstyle\mu}\\[2pt]
\Sigma^dY&\xrightarrow{\ \beta\wedge1_Y\ }&Y.
\end{array}
\end{equation}
Commutativity is the left unit identity for $\mu$ and the
definition of $f$. Its top arrow is null, so composing the
diagram with $\Sigma^dL_T$ and using
\eqref{top:averaged-top-map} proves
$\beta^{p(p-2)+1}\gamma=0$.
\end{proof}

\begin{proposition}[Disappearance before the last differential]
\label{top:early-disappearance}
Every class in $E_2^{qp+1,u_4+p}(Y)$ is zero on the
Adams--Novikov page $E_{qp+1}$.
\end{proposition}

\begin{proof}
Let $\theta\in E_2^{5,W}(Y)$. By Lemma~\ref{top:small-stems},
it is a permanent cycle represented by a homotopy class $\gamma$.
The sphere class $\beta$ is also permanent. The pagewise product
$A^K\theta$ therefore supports no nonzero outgoing differential;
if either factor has zero image on a page, the product is already
zero. By Proposition~\ref{top:annihilation} its filtered homotopy
product is zero. Product compatibility and finite convergence
then imply that $A^K\theta$ is zero at $E_\infty$.

If it is nonzero at $E_2$, it must consequently be hit by an
incoming differential. Its cohomological filtration is
\[
 S=5+2K=2p^2-4p+7.
\]
An incoming differential of length $r$ has source filtration
$S-r\geq0$, so $r\leq S$. This bounds the page of disappearance,
rather than merely its eventual abutment. The exact comparisons
are
\begin{equation}
\label{top:strict-page-bound}
 qp+1-S=2p-6>0,\qquad N-K=p-3>0.
\end{equation}
Numerically,
\[
 K=1{,}016{,}064,\quad N=1{,}017{,}070,\quad
 S=2{,}032{,}133<qp+1=2{,}034{,}145.
\]
Thus $A^K\theta$ is zero before the page of the prospective
$d_{qp+1}(v_4)$. Multiplication by the further power $A^{N-K}$ is
defined on every page by the sphere permanent cycle $\beta$;
it shows that $A^N\theta$ is zero there as well.
The surjection \eqref{top:An-surjection} accounts for every
$E_2$ class of the target, proving the assertion.
\end{proof}

\subsection{The realizing cofiber}\label{top:main-construction}

All possible targets of a differential from $v_4$ have now been
eliminated on the required pages. We finish by producing the ordinary
finite spectrum and identifying its homology as a comodule.

\begin{proof}[Proof of Theorem~\ref{intro:main}]
The only possible positive differentials from $v_4$ have target
$(s,w)=(qk+1,u_4+k)$. Proposition~\ref{may:first-differential}
rules out $k=1$. Lemma~\ref{may:target-bound} and
Table~\ref{may:remaining-table} leave, for $k\geq2$, only
$k=2,p-1,p$. Corollary~\ref{frob:vanishing} eliminates the
first two target groups already at $E_2$, and
Proposition~\ref{top:early-disappearance} makes the last target
zero before its possible differential. No differential can hit
filtration zero. Hence $v_4\in E_2^{0,u_4}(Y)$ is permanent.

The edge homomorphism of the convergent Adams--Novikov spectral
sequence supplies a map
$a:S^{qu_4}\to Y$ whose $BP$-Hurewicz image is exactly $v_4$.
Extend it by the existing binary multiplication to
\[
 \widetilde v_4:\Sigma^{qu_4}Y
 \xrightarrow{\ a\wedge1_Y\ }Y\wedge Y
 \xrightarrow{\ \mu\ }Y,
\]
and take its cofiber $X$.
The right unit identity shows that the bottom generator is sent
to $v_4$. Since $BP_*Y=R/I_4$ is cyclic as an $R$-module, this
determines the entire induced map as multiplication by $v_4$.
That element is a nonzerodivisor in
$R/I_4=\mathbb F_p[v_4,v_5,\ldots]$. The homology exact sequence
of the cofiber therefore gives the short exact sequence
\[
 0\longrightarrow\Sigma^{qu_4}R/I_4
 \xrightarrow{\ v_4\ }R/I_4
 \longrightarrow BP_*X\longrightarrow0.
\]
The composite bottom map $S^0\to Y\to X$ induces the quotient
map $R\to R/(I_4,v_4)$. It is a comodule map and is surjective,
so it identifies the induced coaction with the canonical quotient
coaction, and identifies its generator in degree zero.

Finally, $X$ is the cofiber of a map between finite $p$-local
spectra. It is finite $p$-local in the sense defined in the introduction;
equivalently, the finite cell construction can be realized by a
finite spectrum before localization by clearing prime-to-$p$
denominators in its attaching maps, as in
Lemma~\ref{low:finite-localization}. More explicitly, the mapping cone
adds to the cells of $Y$ their translates by $qu_4+1$.
Together with Equation~\eqref{low:cell-dimensions}, this gives a finite
$p$-local CW model for $X$ with one cell in each dimension
\[
 \sum_{i\in S}(qu_i+1)=\sum_{i\in S}(2p^i-1),
 \qquad S\subseteq\{0,1,2,3,4\}.
\]
In particular, this model has $32$ cells.
\end{proof}
==== END A-finite-Smith-Toda-complex-V4-at-the-prime-1009//05-final-obstruction.tex ====

==== BEGIN A-finite-Smith-Toda-complex-V4-at-the-prime-1009//appendix-resolution.tex ====
\section{The ordinary resolution and its filtered pairing}
\label{app:resolution}

We use a symmetric monoidal stable model of spectra with the usual
pushout-product axiom and tensors over based spaces. Such a model is
provided by topological symmetric spectra:
\citet[Section~6.2, Theorems~6.3.8 and~6.4.1, and
Corollary~6.4.2]{HoveyShipleySmith1998v2}; its comparison with ordinary
spectra follows from Theorems~6.3.8 and~4.3.2 there, in the cited
preprint version.
All smash products
are derived; when point-set maps are used, their inputs are cofibrant.
Rectify the coherent associative structure on $BP$ as an $S$-algebra by
\citet[Part~II, Corollary~3.6 and the associative convention comparison
in Section~4]{ElmendorfKrizMandellMay1997}. The equivalence for associative
algebras in \citet[Main Theorem and Theorem~5.1(ii)]{Schwede2001} then
places it in this same category of topological symmetric spectra.
Choose a cofibrant associative symmetric ring representative using
\citet[Theorem~12.1(iv)--(v)]{MandellMaySchwedeShipley2001}; its
underlying spectrum is then stably cofibrant. Smash with such a spectrum
preserves stable equivalences by Proposition~12.3 there, so the following
point-set smash powers compute derived smash powers.
The comparison uses the positive stable model structure, whose weak
equivalences agree with the ordinary stable ones; we use the ordinary
stable structure here. With this strictly unital associative model, write
\[
 T^n(Z)=BP^{\wedge(n+1)}\wedge Z.
\]
The cofaces insert units and the codegeneracies multiply adjacent $BP$
factors.  Associativity and the unit identities give a cosimplicial
spectrum.  Write $K_n(Z)=\operatorname{Tot}_n T^\bullet(Z)$ for its
derived ordinary partial totalization, set $K_{-1}(Z)=*$, and put
$K(Z)=\operatorname{Tot}T^\bullet(Z)$.  Ordinary totalization here uses
all ordinal maps, including codegeneracies.

\subsection{Convergence and the edge}\label{app:convergence}

\begin{proof}[Proof of Proposition~\ref{fw:anss}]
We first identify the ordinary totalization tower exactly.  Let
$\mathcal P_0([n])$ be the poset of nonempty subsets of
$\{0,\ldots,n\}$.  The functor
\[
 u_n:\mathcal P_0([n])\longrightarrow\Delta_{\leq n},
 \qquad S\longmapsto[|S|-1],
\]
sends an inclusion to the corresponding order-preserving injection.
It is homotopy-initial.  To prove this, for $m\leq n$ identify
$(u_n\downarrow[m])$ with the nonempty face poset of the simplicial
complex $L_{n,m}$ whose vertices are $(i,j)$, $0\leq i\leq n$,
$0\leq j\leq m$, and whose simplices have strictly increasing first
coordinates and weakly increasing second coordinates.  Its nerve is
the barycentric subdivision of $L_{n,m}$.

The complex $L_{n,m}$ is contractible for $n\geq m$, by induction on
$n$.  The case $n=m=0$ is a point.  For the inductive step start with
the cone with vertex $(n,m)$ on the entire subcomplex $L_{n-1,m}$.
Adjoin the cones with vertices $(n,j)$, $0\leq j<m$, on their bases
$L_{n-1,j}$.  Two distinct vertices with first coordinate $n$ cannot
belong to a simplex.  Each new cone therefore intersects the preceding
union in exactly its base.  That base is contractible by induction,
since $n-1\geq j$.  Attachment along this simplicial subcomplex is a
homotopy pushout and preserves contractibility.  This proves the claim.

Homotopy-initiality, applied to the cosimplicial unit resolution, now
gives, naturally in $n$ and $Z$,
\begin{equation}
 K_n(Z)\simeq
 \operatorname*{holim}_{\varnothing\ne S\subseteq\{0,\ldots,n\}}
       BP^{\wedge|S|}\wedge Z.
 \label{fw:finite-cube-comparison}
\end{equation}
This is the standard cubical description of the partial Amitsur
totalizations; compare \citet[Propositions~2.11 and~2.14]{MathewNaumannNoel2017}.
Indeed derived partial totalization is the homotopy limit over
$\Delta_{\leq n}$: the restricted standard simplex is a Reedy-cofibrant
resolution of the constant point diagram.  Adjoining the empty vertex
$Z$ on the right of Equation~\eqref{fw:finite-cube-comparison} gives
the $(n+1)$-fold cube formed by smashing copies of the unit
$S_{(p)}\to BP$.  If $\mathcal F$ is its fiber, iterated fibers and
exactness of smash identify its total fiber with
$\mathcal F^{\wedge(n+1)}\wedge Z$.  Thus
\begin{equation}
 \operatorname{fib}(Z\longrightarrow K_n(Z))
       \simeq\mathcal F^{\wedge(n+1)}\wedge Z.
 \label{fw:tot-fiber-index}
\end{equation}
The comparisons commute with restriction from $n+1$ to $n$: on the
fibers this is the map induced by the last factor
$\mathcal F\to S_{(p)}$.  There is no suspension in
Equation~\eqref{fw:tot-fiber-index}.

The unit induces an isomorphism on $\pi_0$, and $\pi_1BP=0$, so
$\mathcal F$ has no homotopy in degrees at most zero.  If $Z$ is
$c$-connective, $\mathcal F^{\wedge s}\wedge Z$ is at least
$(c+s)$-connective.  Equation~\eqref{fw:tot-fiber-index} therefore
identifies the coaugmentation $Z\to K(Z)$ as an equivalence, and
identifies the relative totalization tower as
\begin{equation}
 G^s(Z):=\operatorname{fib}(K(Z)\longrightarrow K_{s-1}(Z))
        \simeq\mathcal F^{\wedge s}\wedge Z
        \quad(s\geq0).
 \label{fw:relative-tower}
\end{equation}
Its homotopy groups vanish in each fixed stem for sufficiently large
$s$.  The exact couple consequently has the asserted abutment, no
inverse-limit ambiguity, and finite filtration in each stem.

For completeness, the successive layer is
\begin{equation}
 \operatorname{cofib}(G^{s+1}(Z)\longrightarrow G^s(Z))
 \simeq BP\wedge\mathcal F^{\wedge s}\wedge Z
 \simeq\Sigma^{-s}BP\wedge\overline{BP}^{\wedge s}\wedge Z,
 \label{fw:tot-layer-index}
\end{equation}
where $\overline{BP}=\operatorname{cofib}(S_{(p)}\to BP)$.
This is also $\operatorname{fib}(K_s(Z)\to K_{s-1}(Z))$, with the
usual normalization and the displayed $s$ desuspensions.

The iterated extended terms compute the cobar complex.  Indeed
$BP\wedge BP$, as a $BP$-module, is a wedge of suspended free
$BP$-modules with basis the polynomial monomials in the $t_i$.
Iteration identifies their homotopy with the iterated tensors of
$\Gamma$ over $R$.  There are no derived tensor corrections, whether
or not $BP_*Z$ is flat over $R$.  Cofaces give the cobar differential;
the codegeneracy splittings give its normalized form.  The exact-couple
grading gives the displayed differential shift.  Finally the first
map is $Z\to BP\wedge Z$.  Its image is exactly the permanent
filtration-zero part by the finite-filtration convergence just proved.
\end{proof}

\subsection{The filtered binary pairing}\label{app:pairing}

\begin{proof}[Proof of Proposition~\ref{fw:anss-pairing}]
We construct maps
\[
 G^s(Z_1)\wedge G^t(Z_2)\longrightarrow G^{s+t}(Z_3)
\]
which induce the stated cobar pairing and recover the actual map $f$.

\smallskip\noindent\emph{The box pairing.}
We use Batanin's cosimplicial box product in the form of
\citet[Definition~2.1, Proposition~2.3, and Remark~2.4]{McClureSmith2004};
they attribute its Kan-extension description to Cordier and Porter.
We construct the pairing before any fibrant replacement and then transport
the entire pairing. For cosimplicial spectra define $A\mathbin\square B$
as the left Kan extension of $(a,b)\mapsto A^a\wedge B^b$ along ordinal
concatenation
\[
 \Delta\times\Delta\longrightarrow\Delta,
 \qquad([a],[b])\longmapsto[a+b+1].
\]
Equivalently, in degree $n$ it is the zigzag pushout of the terms
$A^a\wedge B^b$ with $a+b=n$, with neighboring terms identified by
the maps
\[
 A^a\wedge B^b\xrightarrow{d^{a+1}\wedge1}
 A^{a+1}\wedge B^b,
 \qquad
 A^a\wedge B^b\xrightarrow{1\wedge d^0}
 A^a\wedge B^{b+1},
 \qquad a+b=n-1.
\]
The same definition applies to cosimplicial based spaces.
For the original unit resolutions, concatenating the two $BP$ blocks
and applying $f$ once gives maps
\[
 T^a(Z_1)\wedge T^b(Z_2)\longrightarrow T^{a+b+1}(Z_3).
\]
They are natural in both ordinals: cofaces and codegeneracies within a
block insert a unit or multiply adjacent factors within that block.
Kan-extension adjunction therefore gives an actual cosimplicial map
\begin{equation}
 m:T(Z_1)\mathbin\square T(Z_2)\longrightarrow T(Z_3).
 \label{fw:original-box-pairing}
\end{equation}
On the degree-$n$ summand $a+b=n$ in the pushout description, this map
multiplies the last factor of the first $BP$ block with the first factor
of the second block.  This is the single overlapping slot.  The two
descriptions agree by applying the codegeneracy that identifies the
two junction vertices of $[a+b+1]$.

\smallskip\noindent\emph{Deriving the pairing.}
We record the model-category check that permits us to derive this
construction.  The box product is a Quillen bifunctor for the Reedy
model structures.  Let $h_a([n])=\Delta([a],[n])_+$, and let
$\partial h_a$ be the subdiagram of noninjective maps.  These are the
representables and boundaries defining the generating Reedy cells.
Indeed, the boundary of the covariant representable consists of maps
factoring through a nonidentity inverse map out of $[a]$, hence of
noninjective maps by the unique surjection--injection factorization in
$\Delta$. This representable boundary is distinct from the latching
subobject in target degree $n$, which consists of maps factoring through
a proper injection into $[n]$.
Kan extension gives $h_a\square h_b=h_{a+b+1}$.  In the relative box
boundary
\[
 \frac{h_{a+b+1}}
 {\operatorname{im}(\partial h_a\square h_b)
       \cup\operatorname{im}(h_a\square\partial h_b)},
\]
the remaining maps are exactly the monotone maps injective on each of
the two blocks.  Such a map is either globally injective or identifies
only the two junction vertices. A map that is not surjective onto its
target lies in that target degree's latching subobject. The only
remaining surjections are the junction identification onto $[a+b]$
and the identity of $[a+b+1]$. Thus the junction-identifying and globally
injective non-latching cells occur in degrees $a+b$ and $a+b+1$,
respectively.  Attach the junction cell first;
every further collapse of it is already in the boundary, since it
collapses two vertices within one block.  Attach the globally injective
cell second: its junction codegeneracy lands in the first cell, and
its other codegeneracies land in the boundary. This gives a two-cell
Reedy filtration of the relative box boundary. For two generating
Reedy cofibrations, each of these attachments is tensored with the
ordinary spectral pushout-product of their underlying generating
cofibrations. The spectral pushout-product axiom therefore makes both
attachments cofibrations, and makes them trivial cofibrations if either
input is trivial. This proves the Reedy pushout-product axiom, including
the acyclic case, for generating cofibrations.  Closure under pushouts,
composition, and retracts proves the assertion for all cofibrations.

Choose Reedy-cofibrant replacements $A_i\to T(Z_i)$, $i=1,2$, that
are trivial fibrations, and trivial cofibrations $A_i\to R_i$ with
$R_i$ Reedy fibrant.  For the target choose a Reedy fibrant replacement
$T(Z_3)\to R_3$ directly.  Equation~\eqref{fw:original-box-pairing}
gives $A_1\square A_2\to R_3$.  By the preceding Quillen check,
$A_1\square A_2\to R_1\square R_2$ is a trivial Reedy cofibration.
The lifting axiom extends the map to
\begin{equation}
 m_R:R_1\square R_2\longrightarrow R_3.
 \label{fw:derived-box-pairing}
\end{equation}
The space of such extensions is contractible up to the usual relative
homotopy.  This transports the original pairing; it places no condition
on the literal $BP$ slots of the fibrant replacements.
All resulting tower and spectral-sequence pairings are transported along
the equivalences with the original diagrams.  In particular, when some
$Z_i$ coincide, their source and target replacements need not be identical.

\smallskip\noindent\emph{The filtered diagonal.}
We use the simplex subdivision map given in
\citet[Lemma~3.6]{McClureSmith2004}, credited to Grayson in their
Remark~3.7. We record its formula to verify the relative filtration. For
$t=(t_0,\ldots,t_n)\in\Delta^n$, put
\[
 I_i=\left[\sum_{j<i}t_j,\sum_{j\leq i}t_j\right],\qquad
 x_i=2\,\operatorname{length}(I_i\cap[0,1/2]),\qquad
 y_i=2t_i-x_i.
\]
For a cut index $k$, $x$ is supported in $0,\ldots,k$ and $y$ in
$k,\ldots,n$.  The cut region is a copy of
$\Delta^k\times\Delta^{n-k}$, with inverse $t=(x+y)/2$.
Its boundary seams identify the last face of the first factor with
the first face of the second factor, precisely as in the box pushout.
Thus the cut defines
\[
 \delta:\Delta^\bullet_+\longrightarrow
              \Delta^\bullet_+\square\Delta^\bullet_+.
\]
It is natural under all monotone maps: they aggregate consecutive
intervals, and aggregation commutes with the cut.  In particular this
is a map of ordinary cosimplicial objects, including codegeneracies.

Define the relative simplex object
\[
 P_s^\bullet=\Delta^\bullet_+/\operatorname{sk}_{s-1}
                      \Delta^\bullet_+\qquad(s\geq0),
\]
where the skeleton for $s=0$ is the basepoint.  The simplex, its
skeleta, and these quotients are Reedy cofibrant: their latching maps
are the corresponding boundary inclusions or identities.  Consequently
\begin{equation}
 G^s(Z_i)=\operatorname{Map}_\Delta(P_s,R_i)
 \label{fw:relative-end-model}
\end{equation}
is a derived relative end and the actual fiber of the fibration
$\operatorname{Tot}R_i\to\operatorname{Tot}_{s-1}R_i$.
The degree-$n$ box summands have dimensions $k$ and $n-k$.
If $n<s+t$, at least one is below its respective skeleton cutoff.
Naturality applies the same observation to every face of a larger
simplex, including faces with missing ambient vertices.  Hence the
cut descends to maps
\begin{equation}
 \delta_{s,t}:P_{s+t}\longrightarrow P_s\square P_t.
 \label{fw:relative-cut}
\end{equation}
Enriched evaluation, followed by $m_R$ and precomposition with
$\delta_{s,t}$, gives the actual relative pairing
\begin{align*}
 \operatorname{Map}_\Delta(P_s,R_1)\wedge
 \operatorname{Map}_\Delta(P_t,R_2)
 &\longrightarrow
 \operatorname{Map}_\Delta(P_s\square P_t,R_1\square R_2)\\
 &\longrightarrow
 \operatorname{Map}_\Delta(P_{s+t},R_3).
\end{align*}
It commutes with the tower inclusions in both inputs.  By
Equation~\eqref{fw:relative-tower}, these are the exact relative fibers
of the unit-resolution filtration, with filtration indices $s,t,s+t$.

\smallskip\noindent\emph{The cobar and homotopy pairings.}
Put $L_s=\operatorname{cofib}(G^{s+1}\to G^s)$.  The relative pairing
induces $L_s(Z_1)\wedge L_t(Z_2)\to L_{s+t}(Z_3)$: raising either
input filtration by one raises the output filtration by one.
The boundary formula for the product of relative cells is
$\partial(a\cdot b)=\partial a\cdot b+(-1)^{|a|}a\cdot\partial b$.
It gives a pairing of the exact couples and the Leibniz rule on every
derived page.  On the layer of dimension $s+t$, the only contributing
cut cell has front and back dimensions $s,t$.  The inverse
$t=(x+y)/2$ gives its usual oriented front-face/back-face cup map,
with coefficient one.  The normalized cobar identification therefore
gives exactly the asserted $E_2$ cup pairing.

We finally verify the homotopy target without assuming that a replacement
preserves units.  For cofibrant $Z_i$, the diagram
$\Delta^\bullet_+\wedge Z_i$ is Reedy cofibrant and maps levelwise
equivalently to the constant diagram $Z_i$.  Its original all-unit map
to $T(Z_i)$ lifts to a map
$\alpha_i:\Delta^\bullet_+\wedge Z_i\to A_i$ along the trivial
fibration $A_i\to T(Z_i)$.  The composite of $\alpha_i$ with
$A_i\to R_i$ represents the coaugmentation.  The box product of the
$\alpha_i$, precomposed with $\delta$, lands in $A_1\square A_2$.
Since $m_R$ extends the
original map on that object, the composite is exactly the target
coaugmentation composed with $f$, transported through $T(Z_3)\to R_3$.
On totalizations the pairing therefore satisfies
\[
 m_{\mathrm{Tot}}\circ(e_1\wedge e_2)=e_3\circ f.
\]
The coaugmentations $e_i:Z_i\to K(Z_i)$ are the equivalences proved
above.  Thus the limit pairing is the actual homotopy map $f$, and
the $E_\infty$ pairing is its associated-graded pairing for the
filtration by the images of $\pi_*G^s\to\pi_*K$.

For the final assertion, the coefficient pairing is $R$-bilinear and
sends the two units to the unit.  Every element of $R/I$ is a scalar
multiple of that unit.  The pairing is consequently
$(r\bmod I,r'\bmod I)\mapsto rr'\bmod I$.  This uses neither flatness
of the quotient nor associativity of the target map.
\end{proof}
==== END A-finite-Smith-Toda-complex-V4-at-the-prime-1009//appendix-resolution.tex ====

==== BEGIN A-finite-Smith-Toda-complex-V4-at-the-prime-1009//main.tex ====
\documentclass[11pt]{article}
\input{preamble}
\title{A finite Smith--Toda complex $V(4)$ at the prime $1009$}
\author{OpenAI}
\date{September 23, 2026}
\begin{document}
\maketitle
\begin{abstract}
We construct a Smith--Toda complex $V(4)$ at the prime $1009$. It is an
ordinary finite $1009$-local spectrum whose Brown--Peterson homology is
$BP_*/(1009,v_1,v_2,v_3,v_4)$, with its canonical comodule structure and
generator in degree zero.
\end{abstract}
\tableofcontents
\clearpage
\input{sections/00-introduction}
\input{sections/01-framework}
\input{sections/02a-cobar-preliminaries}
\input{sections/02-lower-complexes}
\input{sections/03-may-and-first-obstruction}
\input{sections/04-frobenius-elimination}
\input{sections/05-final-obstruction}
\appendix
\input{sections/appendix-resolution}
\bibliographystyle{plainnat}
\bibliography{references}
\end{document}
==== END A-finite-Smith-Toda-complex-V4-at-the-prime-1009//main.tex ====

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==== BEGIN A-finite-Smith-Toda-complex-V4-at-the-prime-1009//references.bib ====

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==== END A-finite-Smith-Toda-complex-V4-at-the-prime-1009//references.bib ====