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\title{A finite Smith--Toda complex $V(4)$ at the prime $1009$}
\author{}
\date{}
\begin{document}
\maketitle

\begin{abstract}
We construct a Smith--Toda complex $V(4)$ at the prime $p=1009$: an ordinary
finite $1009$-local spectrum $X$ whose Brown--Peterson homology is
\[
 \BP_*X \cong \BP_*/(p,v_1,v_2,v_3,v_4),
\]
as a graded $\BP_*\BP$-comodule, with the canonical quotient comodule and a
generator in degree zero. This answers the $V(4)$ existence question at this
prime in the ordinary (unlocalized) stable category. No product is needed on
$X$.
\end{abstract}

\tableofcontents

\section{A guided tour of the proof}

This section is an informal road map. The rigorous material begins in
Section~\ref{sec:setup}; nothing here is used later, and a reader who wants
the mathematics at full strength may skip directly to it. The remainder of
the paper is self-contained.

\subsection*{The objects}

Fix a prime $p$. The spectrum $\BP$ of Brown and Peterson has homotopy
\[
 \BP_* = \Zp[v_1,v_2,\dots],\qquad |v_i| = 2(p^i-1),
\]
with Hazewinkel generators. Killing the first few generators to their first
power, one at a time, produces the quotients
\[
 \BP_*/(p),\quad \BP_*/(p,v_1),\quad \BP_*/(p,v_1,v_2),\quad \dots
\]
A \emph{Smith--Toda complex} $V(n)$ is a \emph{finite} $p$-local spectrum
whose $\BP$-homology is the quotient $\BP_*/(p,v_1,\dots,v_n)$, with its
canonical comodule structure and a generator in degree zero.

The classical examples are easy at the bottom and hard higher up: $V(0)$
(the mod-$p$ Moore spectrum) exists at every prime; $V(1)$, $V(2)$, $V(3)$
exist exactly for $p>2$, $p>4$, $p>6$ (Adams, Smith, Toda). Existence of
$V(4)$ was open at every prime. The point of this paper is the case
$p=1009$, large enough that the arithmetic inequalities turn out generously.

\subsection*{One step of the ladder}

Suppose $V(n)$ is already built. If the class $v_{n+1}\in\BP_*V(n)$ happens
to lift to an actual homotopy class of $V(n)$ --- formally, if it is a
permanent cycle in the Adams--Novikov spectral sequence (ANSS) for $V(n)$ ---
then a map $S^{|v_{n+1}|}\to V(n)$ carrying $v_{n+1}$ exists. Smash this map
with $V(n)$ and use a multiplication on $V(n)$ to fold it into a map
\[
 \Sigma^{|v_{n+1}|}V(n)\longrightarrow V(n).
\]
Its cofiber then has $\BP$-homology $\BP_*/(p,v_1,\dots,v_{n+1})$: the class
$v_{n+1}$ is a nonzerodivisor on the quotient, so the cofiber exact sequence
pins down the next quotient. That is the whole idea. The real work is (a)
building $V(n)$ with just enough product to run this, and (b) proving that
$v_{n+1}$ really survives the spectral sequence.

\subsection*{Why survival is the hard part}

The ANSS for a finite spectrum is sparse. All of $\BP_*V(n)$ sits in internal
degrees divisible by $q=2p-2$, so the only possible positive differentials
have length $qk+1$ for $k\ge1$. In \emph{weight} $w$ (internal degree
divided by $q$), a differential of length $qk+1$ moves a class of bidegree
$(s,w)$ to $(s+qk+1,w+k)$. Since $v_{n+1}$ lives in filtration zero at weight
$u_{n+1}=1+p+\dots+p^n$, every dangerous target is forced to be
\[
 (s,w) = (qk+1,\; u_{n+1}+k),\qquad k\ge1.
\]
For the lower steps ($n<3$) these groups all vanish, which is why the
classical $V(0),V(1),V(2),V(3)$ are within reach. For $n=3$, i.e. for
$Y=V(3)$ and the class $v_4$, some of these groups can be nonzero, and one
has to show a differential between them cannot actually be nonzero.

\subsection*{The four surviving cases}

A May-page computation (the cohomology of the additive Hopf algebra, carried
out with base-$p$ filtrations) shows that the dangerous targets for $v_4$
can only occur at
\[
 k\in\{1,\;2,\;p-1,\;p\}.
\]
Everything else is already empty on the $E_2$ page. Each of the four survivors
is killed by a different tool:

\begin{itemize}
\item $k=1$. A concrete cohomology relation coming from the next generator
  $v_5$, together with injectivity of multiplication by a particular class,
  forces the would-be differential $d_{q+1}(v_4)$ to vanish
  (Section~\ref{sec:height4}).
\item $k=2$ and $k=p-1$. Here one replaces the additive Hopf algebra by an
  affine group scheme, uses a Frobenius exact sequence to split its cohomology
  into a kernel part and a quotient part, filters by weight, and computes the
  relevant quotient cohomology to be zero (Section~\ref{sec:frob}).
\item $k=p$. This is genuinely topological. Writing $\beta$ for the first
  beta-family sphere class, one proves a homotopy annihilation relation
  $\beta^{K}\gamma=0$ for the relevant filtration-five classes $\gamma$
  (Section~\ref{sec:final}). The relation holds on a page strictly earlier
  than the prospective differential, so the whole target has already died
  before $d_{qp+1}$ could fire.
\end{itemize}

With all targets eliminated, $v_4$ is a permanent cycle, the construction of
the previous paragraphs applies, and the cofiber is the desired finite
$V(4)$.

\subsection*{How to read the rest}

Section~\ref{sec:setup} fixes coordinates, weights, and the spectral-sequence
conventions (their proofs are deferred to Appendix~\ref{app:res}). Sections
~\ref{sec:may}--\ref{sec:height4} set up the May calculation and remove
$k=1$. Section~\ref{sec:frob} removes $k=2$ and $k=p-1$. Section~\ref{sec:final}
removes $k=p$, proves the page bound, and builds the cofiber. A reader who
wants to see the shape of the argument should read Section~\ref{sec:setup}
through the statement of Theorem~\ref{thm:main} at the end, then come back to
the individual lemmas.

\section{Brown--Peterson coordinates and the Adams--Novikov resolution}
\label{sec:setup}

Throughout,
\begin{equation}
 p=1009,\qquad q=2p-2=2016,\qquad
 u_i=\frac{p^i-1}{p-1}=1+p+\dots+p^{i-1}\quad(i\ge0).
 \label{eq:parameters}
\end{equation}
The number $1009$ is prime: its square root is below $32$, and it is
divisible by none of $2,3,5,7,11,13,17,19,23,29,31$. Put
\[
 R=\BP_* = \Zp[v_1,v_2,\dots],\qquad
 \Gamma=\BP_*\BP=R[t_1,t_2,\dots],\qquad
 I_m=(p,v_1,\dots,v_{m-1}).
\]
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 $w$ lies in stem $qw-s$.

We use the ordinary spectrum $\BP$, its polynomial cooperation algebra, and a
strictly unital associative model of its multiplication. The identification of
these polynomial rings with the homotopy and cooperations of $\BP$ is in
Ravenel's book (Theorems 4.1.18--4.1.19); the Hazewinkel recursion and the
integrality of its generators are in (Equation A2.2.1, Theorem A2.2.3) there,
with the formal-sum convention explained in the proof of Theorem A2.1.27.
The formal-group foundations are Quillen's (Section 5, Theorem 4); the
explicit generators and cooperations are Hazewinkel's (Sections 3.1 and 4.4).
A coherent associative model exists by Basterra--Mandell (Theorem 1.1), by
forgetting structure. We need no commutative or $E_\infty$ model of $\BP$
below.

\subsection{Coordinates and additive ranges}

Let $a_i$ be the coefficients in the formal-group $p$-series
\[
 [p]_F(x)=\mathop{\sum}\nolimits_F a_i x^{p^i},\qquad a_0=p,
\]
and write a strict coordinate change 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 argument.

\begin{lemma}[Leading coefficients and low cooperations]
\label{lem:coordinates}
For each $m\ge1$, $I_m$ is an invariant ideal and $v_m$ is primitive modulo
$I_m$. Modulo $I_m$, the coproducts of $t_i$ for $i\le m$ are the usual
additive-composition coproducts. At the specialization killing $p$ and all
$v_i$, the resulting Hopf algebra is
\begin{equation}
 P=\F_p[t_1,t_2,\dots],\qquad
 \Delta(t_i)=\sum_{\ell=0}^{i}t_\ell\otimes t_{i-\ell}^{p^\ell}.
 \label{eq:additive-hopf}
\end{equation}
\end{lemma}

\begin{proof}
Write the logarithm as $\log_F(x)=\sum_{i\ge0}l_i x^{p^i}$ with $l_0=1$. 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{eq: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 $1/p$; every other term involves lower $v_i$.
The $a_n$ are integral, so by grading and polynomial independence of the $v_i$,
\[
 a_n-c_nv_n\in(v_1,\dots,v_{n-1})R,
\]
and each $c_n$ is a $p$-local unit.

Invariance and primitivity follow by induction. 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
comparing this first surviving term on the two sides of the coordinate-change
identity fixes $c_m v_m$, hence $v_m$. This proves invariance of the next ideal
and the asserted primitivity.

After the base generators are killed the formal group is additive, and
composition of additive $p$-polynomials gives \eqref{eq:additive-hopf}.
Modulo $I_m$ a correction to this coproduct has a positive base coefficient,
of weight at least $u_m$; for $i<m$ its weight is already too large, and for
$i=m$ the only possibility is a constant multiple of $v_m$ with no
positive-weight cooperation factor, which either counit excludes.
\end{proof}

\begin{lemma}[The relation supplied by $v_5$]
\label{lem:v5-relation}
In the cobar cohomology with coefficients $R/I_4$, there is a scalar
$e\in\F_p$ with
\begin{equation}
 v_4\bigl(h_{1,4}+e\,v_4^{p-1}h_{1,0}\bigr)=0,
 \label{eq:v5-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
$v_4,v_5$, and there is no decomposable correction involving $v_4$ to the
second: $u_5=pu_4+1$ is not a multiple of $u_4$. The formal group law has no
nonlinear term of ordinary degree below $p^4$, since such a coefficient would
have positive weight below $u_4$. Thus, through exponent $p^5$, the $p$-series
is the ordinary sum of its $x^{p^4}$ and $x^{p^5}$ terms, and the nonlinear
cross terms in adding these two inputs have exponent above $p^5$.

In the identity intertwining this $p$-series with a strict coordinate change,
only $x+t_1x^p$ from the change affects these two exponents. Applying the
change 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 $v_4 t_1^{p^4}$. All nonlinear corrections to the strict change enter too
late after taking the $p^4$-th power. Hence
\[
 \eta_R(v_5)-v_5=c\,v_4t_1^{p^4}+d\,v_4^p t_1
 \quad\text{in }\Gamma/I_4\Gamma,
\]
with $c,d\in\F_p$ and $c\ne0$ (the nonzero coefficient follows from the unit
factors $c_4,c_5$ of \eqref{eq:coordinate-unit}). Dividing by $c$ and putting
$e=d/c$, and using that each $p$-power iterate of the primitive $t_1$ is
primitive modulo $I_4$, gives the stated cohomology relation.
\end{proof}

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

\begin{lemma}[Normalized positivity]
\label{lem:positivity}
If the coefficient comodule has no negative weights, then
\begin{equation}
 C^{s,w}\ne0\ \Longrightarrow\ w\ge s.
 \label{eq:positivity}
\end{equation}
\end{lemma}

\begin{proof}
Every reduced cooperation factor is a monomial in the $t_i$ and so has weight
at least one; the coefficient has nonnegative weight. Sum the weights.
\end{proof}

\begin{lemma}[A comparison in three cochain degrees]
\label{lem:range-comparison}
Suppose $s\ge1$, $w<u_{m+1}$, and $w-u_m<s-1$. Additive specialization
identifies the normalized cobar complexes for $R/I_m$ and $P$, in weight $w$,
in degrees $s-1,s,s+1$; it intertwines the adjacent differentials and so
induces an isomorphism on cohomology in degree $(s,w)$. If $w<u_m$, the same
conclusion holds in every cochain degree.
\end{lemma}

\begin{proof}
In each of the three degrees any positive coefficient monomial costs at least
$u_m$, leaving at most $w-u_m<s-1$ for the at least $s-1$ reduced bars, which
is impossible; a coordinate $t_i$ with $i\ge m+1$ exceeds the weight budget.
Thus these groups have scalar coefficients and only $t_1,\dots,t_m$ in their
bars, and their coproducts are additive by Lemma~\ref{lem:coordinates}. This
identifies the two groups adjacent to the desired one as well as that group,
so kernels and images agree. If $w<u_m$ the same exclusion works with no
length restriction.
\end{proof}

\subsection{Convergence, pairings, and the edge}

The next two propositions fix the topological spectral sequence and the
binary pairings used throughout. Their full proofs, in the particular
resolution used here (ordinary totalization and its filtration), are in
Appendix~\ref{app:res}.

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

\begin{proposition}[Binary pairings of the resolution]
\label{prop: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 a pairing of the Adams--Novikov spectral
sequences satisfying 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 $Z_1=Z_2=Z_3=Z$ carries a binary
two-sided unital multiplication and $\BP_*Z=R/I$ is cyclic on its unit, the
$E_2$ pairing is ordinary multiplication on cobar cohomology with
quotient-ring coefficients.
\end{proposition}

\begin{corollary}[Sparsity]
\label{cor:sparsity}
If $\BP_*Z$ is concentrated in internal degrees divisible by $q$, then every
page of its Adams--Novikov spectral sequence has the same internal sparsity.
Every possibly nonzero positive differential has length $r=qk+1$ for some
integer $k\ge1$; in weight notation it sends $(s,w)$ to $(s+qk+1,w+k)$. In
particular a primitive coefficient of weight $u_n$ in filtration zero can
only be hit by a differential targeting
\begin{equation}
 (s,w)=(qk+1,\,u_n+k),\qquad k\ge1.
 \label{eq:primitive-targets}
\end{equation}
\end{corollary}

\begin{proof}
On the cobar page the assertion is grade bookkeeping in the coefficient and
cooperation gradings. A differential preserves the sparse support only if $q$
divides $r-1$; later pages are subquotients of earlier ones, and $r\ge2$ on or
after the cohomology page, so the first allowed length is $q+1$.
\end{proof}

\begin{lemma}[The additive comparison for the $v_4$ targets]
\label{lem:additive-comparison}
Let $s=qk+1$ and $w=u_4+k$, $k\ge1$. A nonzero normalized cobar group in
degree $(s,w)$ with coefficients $R/I_4$ requires
\begin{equation}
 (q-1)k\le u_4-1,\qquad
 w\le\frac{qu_4-1}{q-1}<2u_4<p^4<u_5.
 \label{eq: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 and no $t_4$ bar occurs in any of the three
groups.
\end{lemma}

\begin{proof}
The first two inequalities are the rearrangement of $w\ge s$. 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 \eqref{eq:preliminary-v4-bound} hold at this
prime. For a cochain length $d\in\{s-1,s,s+1\}$, a coefficient containing $v_4$
would leave at most $k$ weight for $d\ge qk>k$ reduced factors, so it cannot
occur; a bar containing $t_4$ would leave at most $k$ for the other
$d-1\ge qk-1>k$ bars, and likewise cannot occur. Higher generators exceed the
weight bound. The remaining coproducts are additive by
Lemma~\ref{lem:coordinates}; equivalently apply
Lemma~\ref{lem:range-comparison}. This identifies both adjacent differentials
and proves the cohomology statement.
\end{proof}

\section{Filtered cobar preliminaries}
\label{sec:may}

We use May's filtered Hopf-algebra method, with the odd-primary May-type
filtration in Ravenel's form (Theorem 3.2.5), plus an extra digit-zero weight
needed for the injectivity argument of Section~\ref{sec:height4}. Earlier
applications to $V(3)$ and powers of $\beta_1$ are in Lee--Ravenel. This
section constructs the filtration, its initial page, and its convergence.
Throughout, cohomological degree and weight are written $(s,w)$; unless a
topological differential is named explicitly, the spectral sequences in this
section are algebraic, of cobar complexes. Put
\[
 P=\F_p[t_1,t_2,\dots],\qquad |t_i|_w=u_i,
 \qquad
 \overline\Delta(t_i)=\sum_{0<l<i}t_l\otimes t_{i-l}^{p^l}.
\]
Reversing all coproducts reverses the cobar conventions and changes nothing;
we use the displayed one.

\begin{lemma}[The filtered cobar calculation]
\label{lem:may-page}
For every integer $M\ge0$ the cobar cohomology of $P$ admits a multiplicative,
degreewise finite spectral sequence with initial cohomology page
\begin{equation}
 \Lambda(h_{i,j}:i\ge1,j\ge0)\otimes\F_p[b_{i,j}:i\ge1,j\ge0].
 \label{eq:may-initial-page}
\end{equation}
The bidegrees and auxiliary filtrations $F_M$ are
\[
\begin{array}{c|c|c}
 & (s,w)& F_M\\ \hline
 h_{i,j}&(1,p^ju_i)&2i-1+M[j=0]\\
 b_{i,j}&(2,p^{j+1}u_i)&p(2i-1+M[j=0]),
\end{array}
\]
where $[j=0]$ is $1$ or $0$ according as $j=0$ or $j\ne0$. All differentials
lower $F_M$. Writing an exterior generator as $h_{[a,b)}=h_{b-a,a}$, 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{eq:may-splitting}
\end{equation}
The same statements hold for the Hopf subalgebra
$P_r=\F_p[t_1,\dots,t_r]$, with $i\le r$ in \eqref{eq:may-initial-page}.
\end{lemma}

\begin{proof}
Write every exponent in base $p$ and give a digit factor $t_i^{p^j}$ the
filtration $2i-1+M[j=0]$, adding these over the digits with their
multiplicities and without carrying. Carrying $p$ copies of one digit to the
next lowers filtration, since its decrease is
\[
 p(2i-1+M[j=0])-(2i-1+M[j+1=0])\ge p-1>1.
\]
A nonprimitive coproduct summand of $t_i^{p^j}$ replaces it by
$t_l^{p^j}\otimes t_{i-l}^{p^{j+l}}$ and lowers filtration by exactly one.
Hence this is a multiplicative coalgebra filtration, and its associated graded
Hopf algebra is a tensor product of primitive truncated polynomial Hopf
algebras
\[
 \bigotimes_{i,j}\F_p[z_{i,j}]/(z_{i,j}^p).
\]
For one factor the dual algebra is again truncated polynomial (factorial
rescaling matches its basis with the polynomial basis up to degree $p-1$). The
free resolution alternating multiplication by its generator and its
$(p-1)$st power computes its cohomology as one exterior class in degree one
and one polynomial class in degree two, of weights the weight of $z_{i,j}$ and
$p$ times it. Tensoring these resolutions gives \eqref{eq:may-initial-page}
with the stated auxiliary filtrations; the exterior square vanishes because
the 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 of a digit $z$, a filtration-one
contribution would replace one of its $p$ copies by a split pair and leave
$p-1$ unsplit copies. In the associated-graded cohomology of the $z$ factor a
nonzero multiplicity has the form $pa+\epsilon$ with $a\ge0$ and
$\epsilon\in\{0,1\}$; the multiplicity $p-1$ has neither form, so that
component has zero cohomology. Carries cannot contribute to $d_1$ by the
strict inequality above, so $d_1b=0$.

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

For later calculations fix the notation, always keeping the bare exterior
letter $b$ apart 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{eq:low-alphabet}
\end{equation}
An exterior interval $h_{i,j}$ covers positions $j,\dots,j+i-1$ in its
weight; a polynomial letter $b_{i,j}$ covers positions $j+1,\dots,j+i$. 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 exactly
\eqref{eq: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{eq:three-position-weight}
\end{equation}

\section{The unital complexes through height three}
\label{sec:lower}

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

\begin{lemma}[Lower obstruction groups]
\label{lem:algebraic-vanishing}
The following cobar cohomology groups vanish.
\begin{enumerate}
 \item For $1\le n\le3$ and $k\ge1$, the group with coefficients $R/I_n$ in
 bidegree $(s,w)=(qk+1,\,u_n+k)$.
 \item For $0\le n\le3$, choose $e_i\in\{0,1,2\}$ for $0\le i\le n$ and put
 \[
  U=\sum_{i=0}^n e_i u_i,\qquad \rho=\sum_{i=0}^n e_i\ge2.
 \]
 For every $k\ge1$, 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 second by putting $U=u_n$ and $\rho=0$, keeping
its indicated coefficient comodule. If the normalized cochain group in degree
$(s,w)$ is nonzero, Lemma~\ref{lem:positivity} gives
\begin{equation}
 2015k\le U+\rho-1.
 \label{eq:normalized-cutoff}
\end{equation}
The values
\[
 (u_0,u_1,u_2,u_3,u_4)=(0,\,1,\,1010,\,1{,}019{,}091,\,1{,}028{,}262{,}820)
\]
give Table~\ref{tab:cutoffs}. Each upper bound substitutes the displayed maxima
in \eqref{eq:normalized-cutoff} and takes the integer part; a zero upper bound
on $k$ rules out all $k\ge1$.

\begin{table}[htbp]
\centering
\begin{tabular}{lrrrr}
\toprule
Case & Max $U$ & Max $\rho$ & Max $k$ & Max $w$\\
\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{tab:cutoffs}
\end{table}

These bounds also justify the additive comparison in the adjacent cochain
degrees. For a remaining product case $w<u_{n+1}$, so no positive coefficient
or coordinate $t_i$ with $i\ge n+1$ occurs; the retained coproducts are additive
modulo $I_{n+1}$, and the last clause of Lemma~\ref{lem: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 $d\ge s-1=qk>k$ reduced
bars when $d\in\{s-1,s,s+1\}$, so it cannot occur; a bar containing $t_n$ or a
higher coordinate leaves at most $k$ for the other $d-1\ge qk-1>k$ bars, so it
cannot occur either. Hence the same three cochain degrees are additive. In all
cases both adjacent differentials are identified, so the desired cohomology
equals that of $P$.

Every remaining weight in Table~\ref{tab:cutoffs} is at most $2{,}041{,}216$,
while
\[
 p^3=1{,}027{,}243{,}729.
\]
A May letter covering position $3$ or higher has weight at least $p^3$, so only
positions $0,1,2$ occur, with the common alphabet of \eqref{eq:low-alphabet}
and the interval coverage conventions of Section~\ref{sec:may}. Allowing this
whole alphabet only enlarges the upper bound.

Suppose a monomial of the wanted bidegree exists. Let $m\le6$ be its number of
exterior letters and $L=(s-m)/2$ its total polynomial exponent. Every
polynomial letter has weight at least $p$, so
\[
 U+k\ge pL=p(p-1)k+\frac{p(1-\rho-m)}2.
\]
For all profiles under consideration $U\le2p^2+4p+6$ and $\rho\le8$, so
\begin{equation}
 (p^2-p-1)k\le U+\frac{p(\rho+5)}2\le 2p^2+\frac{21}2p+6.
 \label{eq:polynomial-cutoff}
\end{equation}
For $k\ge3$ the left side exceeds the right by at least
\[
 p^2-\frac{27}2p-9=\frac{2{,}008{,}901}{2}>0.
\]
Thus $1\le k\le2$.

Write $U=F_0+pF_1+p^2F_2$. In a product profile the base-$p$ digits are
\[
 F_0=e_1+e_2+e_3,\qquad F_1=e_2+e_3,\qquad F_2=e_3,
\]
with missing $e_i$ taken as zero; for a $v_n$ target they are initial strings
of ones. In every case
\begin{equation}
 6\ge F_0\ge F_1\ge F_2\ge0,
 \qquad F_1\le4,\quad F_2\le2.
 \label{eq:low-digits}
\end{equation}
Let $H_r$ count the exterior letters covering position $r$ and let $n_B,n_C$
be the polynomial multiplicities 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{eq:low-weight}
\end{equation}
We now extract the digits without assuming carries.

First, $0\le H_0\le3$ and $1\le F_0+k\le8<p$, so reduction modulo $p$ of
\eqref{eq:low-weight} gives $H_0=F_0+k$. Also $B,C$ each cost at least $p^2$,
so
\[
 p^2(n_B+n_C)\le w\le2p^2+4p+8<3p^2,
\]
the last strict inequality being $p^2-4p-8=1{,}014{,}037>0$; hence
$n_B+n_C\le2$.

Put $\delta=(m+\rho-1)/2$, an integer because $s-m$ is even. For a target with
$\rho=0$ that parity forces $m$ odd, hence $m\ge1$; for a product profile
$\rho\ge2$. Thus $\delta\ge0$, and $m\le6$, $\rho\le8$ give $\delta\le6$.
Now $L=(p-1)k-\delta$. Subtracting the position-zero equation from
\eqref{eq:low-weight}, dividing by $p$ and rearranging,
\[
 H_1-k-\delta-n_C-F_1=p\,(F_2-k-H_2-n_B-n_C).
\]
Since $H_1\le4$ the left side lies in $[-14,3]$, so the only possible multiple
of $p=1009$ is zero. We have thus proved the 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{eq:low-coverage}
\end{equation}

Among the exterior letters covering position zero only $x,a$ fail to cover
position two, so $H_2\ge H_0-2$. With \eqref{eq:low-digits} and
\eqref{eq:low-coverage},
\[
 F_0-F_2+2k\le2.
\]
Since $k\ge1$, equality holds: $k=1$ 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 $f=1,2$.

If $f=1$, the letters $x,a$ occur and $c,b,z$ do not; only $y$ is optional, so
$H_1\le2$ and $m\ge2$, whereas the last relation of \eqref{eq:low-coverage}
gives $H_1=2+\delta>2$ (since $\delta>0$). If $f=2$, the letters $x,a,c$
occur and $b,z$ do not; again only $y$ is optional, so $H_1\le3$ and $m\ge3$,
whereas the same relation gives $H_1=3+\delta>3$. Both are impossible. There is
no monomial on the May upper-bound page, hence no cobar cohomology in any
stated bidegree.
\end{proof}

\begin{lemma}[Finite local cell attachments]
\label{lem: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 spectrum so far is the localization
of a finite spectrum $Z$, an attaching map lies in
$\pi_d(Z_{(p)})=\pi_d(Z)\otimes\Zp$ and can be written $a/m$ with $p\nmid m$.
After localization, multiplication by $m$ on its sphere source is an
equivalence, so the localized cone of $a$ is equivalent to the cone of $a/m$.
Induction over the finitely many cells proves the claim.
\end{proof}

\begin{proposition}[Construction through $V(3)$]
\label{prop:V3}
For $0\le n\le3$ 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
$\{0,\dots,n\}$, in dimensions
\begin{equation}
 \sum_{i\in S}(qu_i+1),
 \label{eq: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{eq: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, and its cells are in dimensions zero
and one, as in \eqref{eq:cell-dimensions}.

We first give the product-extension argument, valid for this initial complex and
for every later one once its homology and cells are known. 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 $\{0,\dots,n\}$. If $e_i$ counts occurrences of $i$ in that pair, its
dimension is $qU+\rho$ with exactly the parameters of the second part of
Lemma~\ref{lem:algebraic-vanishing}, and the obstruction to extending across
the cell lies in $\pi_{qU+\rho-1}V(n)$.

An Adams--Novikov term in that stem has $qw-s=qU+\rho-1$, i.e.
\[
 w=U+k,\qquad s=qk+1-\rho.
\]
Since $2\le\rho\le8<q$, nonnegative filtration needs $k\ge1$, and every such
$E_2$ group is zero by Lemma~\ref{lem:algebraic-vanishing}. Strong convergence
and finite filtration (Proposition~\ref{prop:anss}) give
$\pi_{qU+\rho-1}V(n)=0$. Extending cell by cell therefore gives $\mu_n$
matching both identity maps on $U_n$. This also builds the initial product on
$V(0)$.

Suppose now that $V(n-1)$, including its product, is constructed, for
$1\le n\le3$. By Lemma~\ref{lem:coordinates}, $v_n$ is a primitive element of
$\BP_*V(n-1)=R/I_n$, defining a filtration-zero $E_2$ class. Its possible
positive targets are exactly \eqref{eq:primitive-targets}, all killed by the
first part of Lemma~\ref{lem:algebraic-vanishing}; nothing hits filtration
zero. The edge assertion of Proposition~\ref{prop:anss} therefore realizes
$v_n$ by a homotopy map $\widetilde v_n:S^{qu_n}_{(p)}\to V(n-1)$. Using the
product,
\[
 f_n=\mu_{n-1}(\widetilde v_n\wedge\operatorname{id}):
   \Sigma^{qu_n}V(n-1)\longrightarrow V(n-1),
\]
and $V(n)$ is 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 stated Hurewicz image. This multiplication is injective on $R/I_n$, so
the cofiber exact sequence gives exactly $R/I_{n+1}$, with no extra 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
and fixes the generator in degree zero.

The mapping cone adds the old cells shifted by $qu_n+1$, giving exactly
\eqref{eq:cell-dimensions}, and the product-extension argument then produces
$\mu_n$. This completes the induction. All attachments are finite $p$-local
cell attachments, and Lemma~\ref{lem:finite-localization} gives finiteness in
the stated sense. No associativity of any $\mu_n$ is used.
\end{proof}

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

\begin{proof}
The bottom two cells form the mod-$p$ Moore spectrum, and 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 the binary
multiplication,
\[
 p\operatorname{id}_Y=\mu_3\bigl((p\eta_3)\wedge\operatorname{id}_Y\bigr)=0.\qedhere
\]
\end{proof}

\section{The height-four obstruction calculation}
\label{sec:height4}

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

\subsection{Possible differential targets}

We use the finite spectrum $Y$ of Proposition~\ref{prop: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 out of
$v_4$ target
\begin{equation}
 s=qk+1,\qquad w=u_4+k,\qquad k\ge1.
 \label{eq:obstruction-degrees}
\end{equation}

\begin{lemma}[Explicit cutoffs]
\label{lem:target-bound}
The groups in \eqref{eq:obstruction-degrees} are identified with $H^{s,w}(P)$.
Their May page has only positions $0,1,2,3$, and is zero unless $k\le p+2$.
For $k\ge2$, only positions $0,1,2$ occur.
\end{lemma}

\begin{proof}
If $w<s$ the degree-$(s,w)$ cochain groups for both $Y$ and $P$ vanish, as
does the May page, and everything is immediate. We may suppose $w\ge s$, so the
normalized bound of Lemma~\ref{lem:positivity} gives $(q-1)k\le u_4-1$. Hence
\[
 w\le u_4+\frac{u_4-1}{q-1}<\frac q{q-1}u_4<2u_4<4p^3<p^4
 \qquad(p=1009),
\]
using $u_4<2p^3$ and $q/(q-1)<2$. So no letter reaches position four, and
$w<u_5$. A coefficient containing $v_4$ leaves at most $k$ weight for
normalized bars, while the three lengths $s-1,s,s+1$ all exceed $k$
($s-1=qk>k$); higher coefficient generators are excluded by $w<u_5$. The
adjacent-chain comparison of Lemma~\ref{lem:additive-comparison} therefore
identifies the group with its additive specialization.

There are ten possible exterior intervals on positions $0,1,2,3$. Since $s$ is
odd, $m$ is odd and $m\le9$. Each polynomial letter has weight at least $p$, so
\[
 u_4+k\ge pL\ge p\bigl((p-1)k-4\bigr),
 \qquad (p^2-p-1)k\le u_4+4p.
\]
But
\[
 (p^2-p-1)(p+3)-(u_4+4p)=p^2-9p-4>0 \qquad(p=1009),
\]
so $k\le p+2$.

For $k\ge2$ we have $L\ge2p-6$. If a letter reaches position three it has
weight at least $p^3$, while all other polynomial letters contribute at least
$(L-1)p$ (whether the distinguished letter is exterior or polynomial). Hence
\[
 w\ge p^3+2p^2-7p>p^3+p^2+2p+3=u_4+p+2\ge w,
\]
since the strict difference is $p^2-9p-3>0$ at $p=1009$. This excludes
position three in that range.
\end{proof}

\subsection{The first obstruction}

The first possible differential is $d_{q+1}$, targeting the actual cobar
cohomology in degree $(q+1,u_4+1)$. An injective multiplication on this group
will rule it out. Lemma~\ref{lem:v5-relation} gives
\[
 v_4\bigl(h_{1,4}+e\,v_4^{p-1}h_{1,0}\bigr)=0
\]
in the $E_2$ algebra for $Y$. By $q$-sparsity no earlier positive differential
occurs, so Proposition~\ref{prop:anss-pairing} makes $d_{q+1}$ a derivation on
this algebra. Let $\sigma$ denote algebraic specialization to $P$-cohomology.
Differentiating the relation and then applying $\sigma$, every term containing
$v_4$ specializes to zero and $d_{q+1}(v_4^p)=pv_4^{p-1}d_{q+1}(v_4)=0$; hence
\begin{equation}
 \sigma\bigl(d_{q+1}(v_4)\bigr)h_{1,4}=0.
 \label{eq:specialized-obstruction}
\end{equation}
Specialization is applied to an identity of cobar-cohomology classes, after the
topological differential; no spectrum realizing $\sigma$ is needed.

It remains to show that multiplication by $h_{1,4}$ is injective on
$H^{q+1,u_4+1}(P)$. Its degree shift is $(1,p^4)$, so the product lies in
$(q+2,u_5+1)$, and 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; the digit-zero
adjustment of Lemma~\ref{lem:may-page} will separate the surviving source
filtrations from the product filtration. In the alphabet \eqref{eq:low-alphabet}
the splitting differential is
\begin{equation}
 da=xy,\qquad db=yz,\qquad dc=xb+az,
 \label{eq: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{lem: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{eq:first-target-list}
\end{equation}
The first is not a splitting cycle and the second is. In bidegree
$(q+1,u_5+1)$ the complete list is Table~\ref{tab:n5}. 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 $w=u_n+1$, $n=4,5$. Since $u_n+1<p^n$,
the available exterior intervals are $[a,b)$ with $0\le a<b\le n$, and the
polynomial intervals cover positions $1,\dots,n-1$. The target weight has
base-$p$ digits $(2,1,\dots,1)$. At position zero $H_0\le n<p$, so $H_0=2$ and
there is no carry into position one. As $s=2p-1$ is odd, write
\[
 m=2D-1,\qquad L=p-D.
\]
Exactly two exterior intervals start at zero; at most $n(n-1)/2$ others occur.
Hence $m\le11$, $2\le D\le6$, and
\begin{equation}
 p-6\le L\le p-2.
 \label{eq:L-range}
\end{equation}
The bound $D\ge2$ follows from $m$ 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, so inductively $H_r+B_r+c_r\le9+(p-2)+1=p+8<2p$;
every carry is zero or one. The digit equations are
\begin{equation}
 B_r=1-H_r-c_r+p\,c_{r+1},\qquad 1\le r\le n-1.
 \label{eq:carry-equations}
\end{equation}
With no carry, $B_r\le1$ for every $r$ and $L\le\sum B_r\le4$, contrary to
$L\ge p-6=1003$.

Call a position high when $c_{r+1}=1$. A high position has $B_r\ge p-9$; a
nonhigh one has $B_r\le1$. There cannot be two blocks of high positions: for
high $a<b$ in different blocks and a nonhigh $c$ between them, an interval
covering both $a$ and $b$ also covers $c$, so
\[
 L\ge B_a+B_b-B_c\ge2(p-9)-1=2p-19>p-2,
\]
since $2p-19=1999$ and $p-2=1007$, contradicting \eqref{eq:L-range}. Thus the
high positions form a single block $[l,r]$ with $1\le l\le r\le n-2$. At the
next position \eqref{eq:carry-equations} reads $B_{r+1}=-H_{r+1}$, so
\begin{equation}
 B_{r+1}=H_{r+1}=0.
 \label{eq:gap-after-block}
\end{equation}
At the start of the block $B_l=p+1-H_l\le L$ gives $H_l\ge D+1$; if $l<r$ the
end gives $B_r=p-H_r\le L$, hence $H_r\ge D$.

A singleton block is impossible: every exterior interval covering position $l$
must end at $l+1$ by \eqref{eq:gap-after-block}, and there are at most two such
(for $l=1$ there are exactly two; for $l>1$ at most one begins before $l$ since
$H_{l-1}\le1$ and at most one begins at $l$). So $H_l\le2$, contradicting
$H_l\ge D+1\ge3$. Henceforth $l<r$.

If $l>1$, at most two exterior intervals cover both $l$ and $r$ (they must end
at $r+1$, at most one starting before $l$ and at most one at $l$), and at least
one of the two intervals starting at zero misses both positions (since
$H_{l-1}\le1$). Then
\[
 H_l+H_r=m+\#\{\text{meet both}\}-\#\{\text{meet neither}\}\le m+1,
\]
contradicting $H_l+H_r\ge2D+1=m+2$. Therefore $l=1$. Now at most two exterior
intervals cover both endpoints, $[0,r+1)$ and $[1,r+1)$; equality is forced:
both are present, every exterior interval meets position $1$ or $r$, and
$H_1=D+1$, $H_r=D$. Hence $B_1=B_r=L$: every polynomial interval covers both
positions and, by \eqref{eq: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 $r+1$ and all later 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{eq: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 of \eqref{eq:forced-profile} requires both
or neither. This is exactly \eqref{eq: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.
Writing their inclusion indicators $e_{12},e_{13},e_{24},e_{34}\in\{0,1\}$ and
their sum $o$,
\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{eq:four-indicators}
\end{equation}
with $o\in\{0,2,4\}$. For $t=2$ the solutions are: no optional intervals, the
pair $\{12,24\}$, the pair $\{13,34\}$, and all four; for $t=3$ only the pair
$\{12,34\}$. This proves the exhaustive Table~\ref{tab:n5}, 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 $(q+1,u_5+1)$. Here $ab$ denotes
the interval $[a,b)$, not a product of the letters in \eqref{eq:low-alphabet}.}
\label{tab:n5}
\end{table}

For completeness, the splitting calculation is 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, ordering exterior intervals
lexicographically,
\[
\begin{array}{c|rrr}
01,04,12,13,14,34&1&-1&0\\
02,03,12,14,24,34&1&0&1\\
02,04,12,13,24,34&0&1&1\\
02,04,12,14,23,34&-1&-1&1
\end{array}
\]
with common polynomial factor $b_{3,0}^{p-3}$. The minor using the first,
second, and fourth rows is $3$, nonzero in $\F_{1009}$, so the three images are
linearly independent. Finally \eqref{eq:small-splitting} shows $acb$ has a
nonzero splitting differential and $acybz$ is closed (every term repeats an
exterior factor). This finishes the lemma.
\end{proof}

\begin{proposition}[An injective multiplication]
\label{prop: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 of Lemma~\ref{lem:may-page} with the choice $M=3p$. A
nonzero source class has leading detector a nonzero multiple of $acybz\,B^{p-3}$
by Lemma~\ref{lem:first-list}, of filtration
\[
 F_0=3p^2-9p+13+M\bigl(2+p(p-3)\bigr).
\]
The cocycle $[t_1^{p^4}]$ represents $h_{1,4}$, with shift $(1,p^4,1)$, so the
proposed leading product has filtration
\begin{equation}
 F_*=3p^2-9p+14+M\bigl(2+p(p-3)\bigr).
 \label{eq:product-filtration}
\end{equation}
Every possible incoming May differential to this product has source degree
$q+1$ and weight $u_5+1$; Table~\ref{tab:n5} is its complete source list. The
length-three and length-five source filtrations are respectively
\[
 5p^2-10p+15+M\bigl(2+p(p-2)\bigr),\qquad
 5p^2-15p+19+M\bigl(2+p(p-3)\bigr),
\]
all three length-five sources sharing the latter. These entries die under $d_1$
since their splitting maps are injective, and none is a splitting boundary for
the product because its polynomial factor is a power of $b_{3,0}$ whereas the
product's is a power of $B=b_{2,0}$. The length-seven source has filtration
\[
 F_7=5p^2-20p+23+M\bigl(2+p(p-4)\bigr),
\]
and for $M=3p$ the difference is
\[
 F_7-F_*=2p^2-11p+9-pM=-p^2-11p+9<0.
\]
Since May differentials lower filtration, this source cannot reach the product
on any later page, and there are no other incoming sources.

We need not know that $B$ represents a permanent class alone. Choose an actual
cocycle for the given nonzero source class and multiply that whole cocycle by
$[t_1^{p^4}]$; the result is an actual cocycle whose leading associated-graded
cup product is the nonzero monomial $acybz\,B^{p-3}h_{1,4}$ (the new exterior
letter is distinct from the old ones). An actual cocycle has no outgoing May
differential, and its leading detector can disappear only by becoming a
boundary, which the incoming-source count excludes at every page. Its class is
therefore nonzero.
\end{proof}

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

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

\subsection{The remaining targets}

\begin{proposition}[All later obstruction candidates]
\label{prop:remaining-list}
For $k\ge2$ the complete initial May page in \eqref{eq:obstruction-degrees} is
Table~\ref{tab:remaining}; all other $k$ 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]
$p-1$ & $y\,B^3A^{N_0-3}$,\quad $b\,B^2A^{N_0-2}$,\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{tab:remaining}
\end{table}
\end{proposition}

\begin{proof}
Lemma~\ref{lem:target-bound} allows only positions $0,1,2$ and $2\le k\le p+2$,
so \eqref{eq:three-position-weight} applies. There are six exterior letters and
$m$ is odd; write
\[
 m=2f+1,\quad 0\le f\le2,\qquad L=(p-1)k-f.
\]
Reducing the weight modulo $p$ gives $H_0\equiv1+k\pmod p$ with $0\le H_0\le3$,
so $k=2$ or $k=p+d$ with $-1\le d\le2$.

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{eq:three-position-weight} gives
\begin{equation}
 pT-n_C+H_1=p^2-p+3+f.
 \label{eq:k2-equation}
\end{equation}
For the four exterior subsets, $e=H_1-3-f$ is $-2$ or $-1$, so
$n_C\equiv e\pmod p$; since $0\le n_C\le L\le2p-3$, the only lift is
$n_C=p+e$ (the lift $e$ is negative and $2p+e\ge2p-2>L$). Then
\eqref{eq:k2-equation} gives $T=p$, and the counts are:
\[
\begin{array}{c|c|c|c|c|c|c}
\text{extra exterior}&f&H_1&H_2&n_C&n_B&n_A\\ \hline
\varnothing&1&2&1&p-2&1&p-2\\
y,b&2&4&2&p-1&-1&p-2\\
y,z&2&3&2&p-2&0&p-2\\
b,z&2&3&3&p-2&-1&p-1
\end{array}
\]
The negative counts exclude rows two and four; the first and third are the
$k=2$ row of Table~\ref{tab:remaining}.

Now let $k=p+d$, $-1\le d\le2$. Then $H_0=1+d$ and the weight equation becomes
\begin{equation}
 pT-n_C+H_1=(2-d)p+2+d+f.
 \label{eq:pd-equation}
\end{equation}
Since $n_C\le T$ and $H_1\ge0$,
\[
 (p-1)T\le(2-d)(p-1)+(4+f),
\]
and $4+f\le6<p-1$, so $T\le2-d\le3$ and $n_C\le3$. Rearranging,
\[
 p\bigl(T-(2-d)\bigr)=2+d+f+n_C-H_1,
\]
whose right side lies between $-3$ and $9$, strictly inside $(-p,p)$. Both
sides vanish, giving
\begin{equation}
 H_2+n_B+n_C=2-d,\qquad H_1=2+d+f+n_C.
 \label{eq:pd-exact}
\end{equation}
For $d\ge1$ these require $3+f\le H_1\le\min(2f+1,4)$, impossible for
$f=0,1,2$. For $d=0$ exactly one of $x,a,c$ occurs, so $H_1\le3$;
\eqref{eq:pd-exact} forces $f=1$, $n_C=0$, $H_1=3$, and the only triples are
$ayb$ and $cyb$, with $H_2=1$ and $2$, giving the two $k=p$ monomials. For
$d=-1$ only $y,b,z$ occur: for $m=1$ the choices $y,b,z$ have
$(H_1,H_2)=(1,0),(1,1),(0,1)$, and \eqref{eq:pd-exact} excludes $z$ and gives
$(n_C,n_B)=(0,3),(0,2)$; for $m=3$ the exterior is $ybz$, $f=1$,
$H_1=H_2=2$, giving $(n_C,n_B)=(0,1)$. Since $L=N_0-f$, these are exactly the
$k=p-1$ monomials. Every nonnegative count and exterior subset is covered.
\end{proof}

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

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

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

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

Section~\ref{sec:frob} removes the first two rows of Table~\ref{tab:remaining} in
actual cobar cohomology; the last row is treated in Section~\ref{sec:final} by a
homotopy argument with an explicit page bound.

\section{Eliminating two obstruction degrees by Frobenius}
\label{sec:frob}

Throughout this section $p=1009$. We eliminate the rows $k=2$ and $k=p-1$ of
Table~\ref{tab:remaining} in actual cobar cohomology. A Frobenius extension
reduces the task to quotient-group cohomology with coefficients in kernel
cohomology; a filtration of those coefficients then reduces it to
trivial-coefficient quotient groups, computed by May methods. We build these
three steps and identify the coefficient modules the two row
require.

\subsection{The Frobenius extension and its action}

For $r=2,3$ let $G_r$ be the affine group scheme over $\F_p$ with coordinate
algebra
\[
 P_r=\F_p[t_1,\dots,t_r],\qquad
 \Delta t_i=t_i\otimes1+1\otimes t_i+\sum_{0<l<i}t_l\otimes t_{i-l}^{p^l},
\]
with weight $\mathrm{wt}(t_i)=u_i$. On points over a commutative $\F_p$-algebra,
\begin{equation}
 (gh)_i=g_i+h_i+\sum_{0<l<i}g_l\,h_{i-l}^{p^l}.
 \label{eq:group-law}
\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 sequence for
an affine group scheme and a normal subgroup (see Jantzen, Part I,
Proposition~6.6(3) and Section~6.7, for the sequence and the quotient action on
kernel cohomology). We verify the acyclicity and compute that action.

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

\begin{proof}
The Frobenius subring makes $P_r$ free of rank $p^r$, with basis
$t_1^{e_1}\cdots t_r^{e_r}$, $0\le e_i<p$, hence faithfully flat. Its kernel
has coordinate algebra $O(K_r)=\F_p[e_1,\dots,e_r]/(e_1^p,\dots,e_r^p)$; the
mixed coproduct terms vanish, so each $e_i$ is primitive. This proves
\eqref{eq:frob-exact}.

For acyclicity, right multiplication by a kernel point is, by
\eqref{eq: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}
 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\le j_i<p.
 \label{eq:regular-decomposition}
\end{equation}
The first factor has trivial $K_r$-coaction and 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). Equation~\eqref{eq:regular-decomposition} shows every
cofree $G_r$-comodule is $K_r$-acyclic with $K_r$-invariants cofree over $Q_r$;
every injective is a summand of a cofree comodule, so this holds for injectives
too. Finally $M^{G_r}=(M^{K_r})^{Q_r}$. Resolving these two invariants functors
successively with the indicated cofree resolutions gives \eqref{eq:frob-lhss}.
This derives $K_r$-invariants and does not assert exactness on arbitrary
modules. The sequence is first quadrant, so each total degree has a finite
diagonal; weights are positive and each weight component is finite dimensional,
giving the stated convergence, also weight by weight.
\end{proof}

\begin{lemma}[Conjugation on kernel cohomology]
\label{lem:frob-action}
For $r=3$ there is an isomorphism of bigraded algebras
\begin{equation}
 H^*(K_3;\F_p)=\Lambda(x,a,c)\otimes\F_p[A,B,E].
 \label{eq:kernel-cohomology}
\end{equation}
The exterior generators have cohomological degree one and weights $1,u_2,u_3$;
the polynomial generators have degree two and weights $p,pu_2,pu_3$. 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{eq:frob-exterior-action}\\
 A&\longmapsto A,&
 B&\longmapsto B+A\xi^p,&
 E&\longmapsto E+A\eta^p+B\xi^{p^2}.
 \label{eq: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 $r=2$ statements omit $c$ and $E$.
\end{lemma}

\begin{proof}
The dual algebra of a primitive truncated coordinate $\F_p[e]/(e^p)$ is again
truncated polynomial 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, and the 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, with nonzero iterates in every
even degree). The exterior generator squares to $[e\mid e]=\tfrac12 d[e^2]=0$
at this odd prime. Tensoring these resolutions gives
\eqref{eq:kernel-cohomology} with the stated weights.

For completeness, the polynomial generators are natural but Frobenius
semilinear. A primitive coordinate $T$ has degree-two cocycle
\[
 \beta(T)=\sum_{j=1}^{p-1}\frac{1}{p}\binom pj[T^j\mid T^{p-j}],
\]
with the displayed integers reduced modulo $p$. For a linear coordinate
$T=\sum_i\lambda_i e_i$ the integral polynomial
$\bigl((\sum_i\lambda_i e_i)^p-\sum_i\lambda_i^p e_i^p\bigr)/p$ 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 the degree-two classes; this polynomial identity also
applies when the $\lambda_{ij}$ are quotient coordinate functions.

If $e$ is a kernel point, \eqref{eq: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 lie in $O(Q_r)$, so the action descends to the quotient. The
degree-one and degree-two naturality give
\eqref{eq:frob-exterior-action} and \eqref{eq:frob-polynomial-action}. The
exterior matrix is triangular with diagonal entries one, so its top exterior
power $xac$ is fixed; the formulas also give separate preservation of the two
degrees.
\end{proof}

\subsection{Coefficient filtration and quotient cohomology}

\begin{lemma}[The coefficient-weight filtration]
\label{lem:frob-coefficients}
Fix an exterior degree and a polynomial degree in \eqref{eq: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, with initial cohomology page
\begin{equation}
 {}^{\mathrm C}E_1^{a,\lambda}=M_\lambda\otimes H^a(Q_r;\F_p)_{w-\lambda},
 \label{eq:coefficient-page}
\end{equation}
where $M_\lambda$ is viewed with trivial quotient action. The coefficient
differential of 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 drop-$d$ differential is induced by that part of
the coaction.

For $\operatorname{Sym}^n(A,B)$ write
\begin{equation}
 A^n\langle1,D,\dots,D^n\rangle,\qquad A^nD^j:=A^{n-j}B^j\ \ (0\le j\le n).
 \label{eq:finite-D}
\end{equation}
This denotes a finite-dimensional module with action $D\mapsto D+\xi^p$; no
inverse of $A$ is introduced.
\end{lemma}

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

For either obstruction bidegree $(s,w)$ our aim is to make every group
$H^a(Q_r;H^{s-a}(K_r;\F_p))_w$ zero. This kills the whole total-degree-$s$
diagonal of \eqref{eq:frob-lhss}, hence proves $H^{s,w}(G_r;\F_p)=0$. We compute
these groups by \eqref{eq:coefficient-page}, using May only to bound or identify
the trivial-coefficient quotient groups. The resulting upper bound uses exactly
the letters of $G_r$: kernel cohomology supplies the digit-zero letters and the
quotient initial May page supplies all higher digits, retaining the kernel
degree $v$. So Table~\ref{tab:remaining} bounds every coefficient summand on the
relevant Frobenius diagonal, and Lemma~\ref{lem:comparison} reduces $k=2$ to
$G_3$ and $k=p-1$ to $G_2$. Grouping by invariant kernel module gives the four
computations of Table~\ref{tab:coefficient-obligations}.

For each module $M$ there let $\lambda_0$ be its least coefficient weight and put
\begin{equation}
 \mathcal R=\frac{w-\lambda_0}{p},
 \label{eq:residual-weight}
\end{equation}
the \emph{residual weight}; henceforth quotient weights are written after
division by $p$. Recall $N_0=(p-1)^2$.
\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{tab:coefficient-obligations}
\end{table}

For the first two rows the least weights are
$\mathrm{wt}(xacA^{p-1})=3+p+2p^2$ and $\mathrm{wt}(xacA^{p-2})=3+2p^2$; for the
last two they are $np$. These give the displayed residual weights. Replacing an
$A$ by $E$ raises the coefficient weight, after division by $p$, by $p^2+p$,
which exceeds both residual weights of the $k=2$ rows; so no coefficient
involving $E$ occurs there in any quotient degree. All four rows thus use the
finite $D$-modules of \eqref{eq: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}
 \mathrm{wt}_Q(\zeta)=\mathcal R-jp.
 \label{eq:D-weight}
\end{equation}
Its first possible coefficient differential lowers $j$ by one and raises the
quotient degree by one, with weight drop $p^2$ in original weights, or $p$ after
rescaling.

The quotient coordinates $\xi,\eta$ have rescaled weights $1,p+1$ and reduced
coproduct
\begin{equation}
 \bar\Delta\xi=0,\qquad \bar\Delta\eta=\xi\otimes\xi^p.
 \label{eq:quotient-coproduct}
\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 $1,p+1,p,p$; these are exactly the letters $y,b,z,C$ of
Table~\ref{tab:remaining}.

\begin{lemma}[The small quotient groups]
\label{lem:frob-quotient-groups}
For $Q_2$ or $Q_3$, at any positive rescaled weight $w<p^2$ the available May
letters are only $y,b,z,C$, with monomials and bidegrees
\begin{equation}
 C^jy^ez^fb^g,\quad j\ge0,\quad e,f,g\in\{0,1\},\qquad
 a=2j+e+f+g,\quad w=p(j+f+g)+e+g.
 \label{eq:small-monomials}
\end{equation}
The following lists are exhaustive, 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 $zC^{p-2}$ 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 lie outside
the asserted range, so Lemma~\ref{lem:may-page} gives exactly
\eqref{eq:small-monomials}. To verify the table, reduce its weight equation
modulo $p$: the remainder is $e+g\in\{0,1,2\}$. For weights divisible by $p$
this forces $e=g=0$, after which the degree equation fixes $j$ and $f$; for
weights congruent to two it forces $e=g=1$, and substitution into both equations
gives the entries shown. The weight-two entries follow since $j=f=g=0$ there.
All tabulated weights lie within the 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 stated nonzero
leading May monomial. The preceding cohomological degree at its weight has no
May monomial in any filtration, so it cannot become a boundary on any May page.
An actual cocycle has no outgoing May differential, so $zC^{p-2}$ is nonzero and
hence $C^{p-2}$ is nonzero.

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}
 U=[\xi\mid\eta]+\tfrac12[\xi^2\mid\xi^p]
 \label{eq:corrected-cocycle}
\end{equation}
is closed, its two differentials being $-[\xi\mid\xi\mid\xi^p]$ and
$[\xi\mid\xi\mid\xi^p]$. In the ordinary May filtration the first summand has
degree four and the correction degree three, so its leading detector is $yb$.
The table excludes both an incoming and an outgoing May differential, so
$\upsilon=[U]\neq0$. Likewise
\[
 U\mid\xi^p=[\xi\mid\eta\mid\xi^p]+\tfrac12[\xi^2\mid\xi^p\mid\xi^p]
\]
is a closed representative with leading detector $ybz$, the correction having
smaller May filtration. The absence of a degree-two May monomial at weight
$2p+2$ excludes every incoming differential and the explicit cocycle excludes
outgoing ones, so $\upsilon z\neq0$.
\end{proof}

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

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

\begin{proof}
We prove vanishing for $G_3$ by treating the first two rows of
Table~\ref{tab: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 excluding $E$ by weight, the finite module is
$xacA^{p-1}\langle1,D,\dots,D^{p-1}\rangle$. The only candidate from
Table~\ref{tab:remaining} is $D\,C^{p-2}$, suppressing the invariant factor
$xacA^{p-1}$. Its quotient weight is $p(p-2)$ by \eqref{eq:D-weight}. The
translation of $D$ gives, on the first possible coefficient page,
\begin{equation}
 D[C^{p-2}]\longmapsto\pm[zC^{p-2}].
 \label{eq:k2-differential}
\end{equation}
The target has quotient degree $2p-3$ and weight $p(p-1)$. Source and target are
both nonzero actual quotient cohomology classes by
Lemma~\ref{lem:frob-quotient-groups}, and both are present on this page because
all earlier coefficient differentials vanish. Thus \eqref{eq:k2-differential}
removes the candidate.

We record the adjacent checks. A differential on this page entering the source
would have $D$-degree two, quotient degree $2p-5$ and quotient weight $p(p-3)$,
a group that is zero by Lemma~\ref{lem:frob-quotient-groups}. The target, with
$D$-degree zero, has no outgoing coefficient differential; its first-page
incoming source has $D$-degree one, and \eqref{eq:small-monomials} in degree
$2p-4$, weight $p(p-2)$ gives only $C^{p-2}$. Thus no other source can cancel
the nonzero map of \eqref{eq:k2-differential}, and 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{tab:remaining}
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; directly, $yz=d[\eta]$ and $C$ is an
actual cocycle, so $yzC^{p-2}$ is exact. It is the only May monomial in this
quotient bidegree (put $a=2p-2$, $w=p^2-p+1$ in \eqref{eq:small-monomials}), so
it contributes no actual quotient cohomology in \eqref{eq: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{eq:frob-lhss} are zero, and 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{prop:frob-kpminus1}
Put $N_0=(p-1)^2$. At $s=2N_0+1$ and $w=u_4+p-1$, the cobar cohomology group
of Table~\ref{tab:remaining} is zero.
\end{proposition}

\begin{proof}
We treat the last two rows of Table~\ref{tab:coefficient-obligations} for
$G_2$ with the finite module \eqref{eq: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 through lower powers of $D$, and the crossed-cocycle equation
rules those out by demanding unequal mixed coefficients where every available
correction supplies equal ones. Write $\xi(g)=r$, $\eta(g)=u$,
$\xi(h)=s$, $\eta(h)=v$, so
\[
 \xi(gh)=r+s,\qquad \eta(gh)=u+v+rs^p.
\]
A normalized one-cocycle is a polynomial $f(D;g)$ with
\begin{equation}
 f(D;gh)=f(D;g)+f(D+r^p;h).
 \label{eq:crossed-equation}
\end{equation}
Since $D$ has weight $p$ and $4p>3p+1$,
\[
 f(D;g)=D^3f_3(g)+D^2f_2(g)+Df_1(g)+f_0(g),
\]
all powers lying in the finite module since $N_0\ge3$. For a polynomial
$\varphi$ write $\delta\varphi(g,h)=\varphi(gh)-\varphi(g)-\varphi(h)$. Comparing
the four coefficients of \eqref{eq:crossed-equation},
\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{eq: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*}
with all coefficients in $\F_p$. The second equation of
\eqref{eq: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 $us^p$, whereas
\begin{equation}
 \delta(\eta\xi^p)=us^p+vr^p+r^{p+1}s^p+rs^{2p}
 \label{eq:asymmetric-pair}
\end{equation}
and $\delta(\xi^{2p+1})$ contains neither $u$ nor $v$. This left side must have
equal coefficients on $r^pv$ and $us^p$; since $2b_2=-6a_3$ and $6\ne0$ in
$\F_{1009}$, we get $a_3=0$, then $b_2=c_2=0$. No term of $f_0$ affects this
coefficient-of-$D$ equation.

Now $f_1$ is primitive. Equation~\eqref{eq:asymmetric-pair} first forces
$d_1=0$; the coefficient of $r^{2p}s$ in $\delta(\xi^{2p+1})$ then forces
$e_1=0$. Finally $f_0$ is primitive: the monomial $u^2s^{p-1}$ has coefficient
$t_0$, so $t_0=0$; the monomial $us^{2p}$ then has coefficient $s_0$, so
$s_0=0$; and the coefficient of $r^{3p}s$ in $\delta(\xi^{3p+1})$ is one, so
$r_0=0$. 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 wanted 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 class $\upsilon$ of
Lemma~\ref{lem:frob-quotient-groups},
\begin{equation}
 D^2\upsilon\longmapsto\pm2\,D\upsilon z.
 \label{eq:h3-differential}
\end{equation}
The coefficient differential first prepends $z$, and the product $z\upsilon$
equals $\upsilon z$ (both are actual cocycles with the same leading monomial
$ybz$, coefficient $+1$ after two exterior interchanges, and
\eqref{eq:small-monomials} leaves no other initial monomial in this bidegree);
so the ordering in \eqref{eq:h3-differential} is direct. Both powers of $D$ lie
in the finite module since $n\ge3$, and \eqref{eq:D-weight} gives $(2,p+2)$ for
the source and $(3,2p+2)$ for the target, both of residual weight $3p+2$. All
earlier coefficient differentials vanish, so both are their actual classes on
this first page, and both are nonzero by Lemma~\ref{lem:frob-quotient-groups}.

A same-page differential into the source would need $D$-degree three and
quotient $H^1$ of weight two, which is zero. The only same-page source for the
target is the one-dimensional span $D^2H^2(Q_2)_{p+2}$ of
\eqref{eq:h3-differential}. A same-page outgoing differential from the target is
proportional to $\upsilon z^2=0$ (indeed $[\xi^p\mid\xi^p]=\tfrac12d[\xi^{2p}]$),
and the term $\xi^{2p}$ in the expansion of $D^2$ lowers two $D$-levels and
cannot cancel its first differential. Since $2\ne0$ in $\F_p$,
\eqref{eq:h3-differential} removes the target.

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

\begin{corollary}
\label{cor: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{tab:remaining}. Among its
rows with $k\ge2$, only $k=p$ remains.
\end{corollary}

\begin{proof}
Apply Propositions~\ref{prop:frob-k2} and~\ref{prop:frob-kpminus1} with the
exhaustive list of Table~\ref{tab:remaining}.
\end{proof}

\section{The final obstruction and the finite cofiber}
\label{sec:final}

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

\begin{theorem}[Main theorem]
\label{thm: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, the right side carrying the canonical quotient
comodule and its generator in degree zero.
\end{theorem}

The conclusion concerns the ordinary $p$-local stable category; finiteness is
established before any chromatic localization. For comparison, Kato, Shimomura,
and Shimomura 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 $L_5$ is localization with
respect to $v_5^{-1}\BP$; that realization inverts $v_5$ and so does not give the
ordinary finite realization of the unlocalized quotient required here. We do not
require a product on the final $X$.

The May comparison and the remaining-target calculation,
Lemmas~\ref{lem:comparison} and \ref{lem:target-bound} with
Table~\ref{tab:remaining}, give a surjection
\begin{equation}
 A^N:E_2^{5,W}(Y)\longrightarrow E_2^{qp+1,u_4+p}(Y).
 \label{eq:An-surjection}
\end{equation}
Indeed every initial May monomial in the target is divisible by $A^N$, and the
degrees match, $5+2N=qp+1$ and $W+pN=u_4+p$. For the source $W<u_4$, so additive
specialization identifies the normalized cobar groups in all three degrees
$4,5,6$. For the target $u_4+p<u_5$, and a positive $v_4$ coefficient would
leave weight $p$, less than the shortest adjacent tensor length $qp=(qp+1)-1$,
so the target groups in degrees $qp,qp+1,qp+2$ also agree with their additive
specializations by Lemma~\ref{lem:positivity}. The two triples differ by $2N$,
and cup multiplication by the actual carry cocycle representing $A^N$ commutes
with specialization, so the filtered surjection of Lemma~\ref{lem:comparison}
transports to \eqref{eq:An-surjection}.

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

Put $n=p-1$. Since $n!$ is a $p$-local unit, the averaging idempotent
\[
 e=\frac{1}{n!}\sum_{\sigma\in\Sigma_n}\sigma
 \quad\text{on }J^{\wedge n}
\]
has a finite retract $T=\operatorname{Sym}^{n}J$. Choose splitting maps
$i:T\to J^{\wedge n}$ and $\mathrm{pr}:J^{\wedge n}\to T$ with $i\,\mathrm{pr}=e$,
and put $\varepsilon_T=\varepsilon^{\wedge n}i:T\to S^0$. The key map is
\[
 f=(\eta\beta)\circ\Sigma^d\varepsilon_T:\Sigma^dT\longrightarrow Y.
\]
The choice of $p-1$ factors is concrete: 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 $1,r,\dots,r^{p-1}$ of weights $0,\dots,p-1$, the exact coefficient powers
in $((r+x)^p-r^p-x^p)/p$. Lemma~\ref{lem:symmetric-comodule} determines the
translated coaction, and Proposition~\ref{prop: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 $f$ null.

We first show every source class is represented by $\gamma\in\pi_{qW-5}Y$ with
$\alpha\gamma=0$ 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
inequality $5+2K<qp+1$, with $K<N$, turns this 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{lem:small-stems}
Every class of $E_2^{5,W}(Y)$ survives to homotopy, and
\begin{equation}
 \pi_{q(W+1)-6}Y=0.
 \label{eq:alpha-gamma-vanishing}
\end{equation}
\end{lemma}

\begin{proof}
By $q$-sparsity, a positive differential out of $(5,W)$ targets
$(s,w)=(6+qj,W+j)$ with $j\ge1$. All Adams--Novikov terms in the stem of
\eqref{eq:alpha-gamma-vanishing} have $(s,w)=(6+qj,W+1+j)$ with $j\ge0$. Treat
both by
\begin{equation}
 (s,w)=(6+qj,\,W+\delta+j),\qquad
 \begin{cases}\delta=0,&j\ge1,\\ \delta=1,&j\ge0.\end{cases}
 \label{eq:small-stem-degrees}
\end{equation}

First the bounds permitting the additive calculation. The bound
\eqref{eq:positivity} gives $(q-1)j\le W+\delta-6$. At this prime
\[
 q-1=2015,\quad W=2{,}040{,}199,\quad
 2015\cdot1013=2{,}041{,}195>W+1-6.
\]
Hence $j\le1012$ and
\begin{equation}
 w\le2{,}041{,}212<p^3=1{,}027{,}243{,}729<u_4=1{,}028{,}262{,}820.
 \label{eq:small-stem-cutoff}
\end{equation}
So no nonconstant coefficient of $R/I_4$ occurs at these weights; only
positions $0,1,2$ arise, with additive cooperation formulas, and
Lemma~\ref{lem:range-comparison} identifies the cobar calculation with that of
$P$. Using the six exterior letters $x,a,c,y,b,z$ and the three polynomial
letters $A,B,C$, with $m$ exterior letters and $L=(s-m)/2$ polynomial letters,
we have $m\le6$ and $L\ge(p-1)j$, $w\ge pL$, so
\[
 \bigl(p(p-1)-1\bigr)j\le W+\delta.
\]
Since $p(p-1)-1=1{,}017{,}071$ and $3\cdot1{,}017{,}071>W+1$, this improves the
bound to $j\le2$.

We enumerate this remaining finite case. Let $H_i$ count exterior letters
covering position $i$ and $n_A,n_B,n_C$ the polynomial exponents. The weight
equation is
\begin{equation}
 w=H_0+p(H_1+L-n_C)+p^2(H_2+n_B+n_C).
 \label{eq:top-weight-equation}
\end{equation}
In \eqref{eq: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 and \eqref{eq:small-stem-cutoff} gives
$w<3p^2$, so $\kappa_2\le2$ and $n_C\le2$. Substituting $L=3+(p-1)j-m/2$ into
\eqref{eq:top-weight-equation},
\[
 p(\kappa_2+j-2)=1+j+m/2+n_C-H_1.
\]
For $0\le j\le2$ the right side lies between $-3$ and $8$, and the only multiple
of $p=1009$ there is zero. Hence
\begin{equation}
 H_0=1+\delta+j,\qquad \kappa_2=2-j,\qquad H_1=1+j+m/2+n_C.
 \label{eq:small-stem-coverage}
\end{equation}

For $j=2$, $H_0=4$ is impossible and $H_0=3$ forces $c$ to occur whereas
$\kappa_2=0$. For $j=1,\delta=1$, all of $x,a,c$ occur and $\kappa_2=1$ excludes
$b,z$, so $H_1\le3$, but the last equation of \eqref{eq:small-stem-coverage},
with even $m\ge4$, requires $H_1\ge4$. For $j=1,\delta=0$, $H_0=2$: if $m=2$ the
required $H_1\ge3$ is impossible; if $m=4$ one needs $H_1=4$ and $n_C=0$, but the
four letters covering position one are exactly $a,c,y,b$, with $H_2=2$ contrary
to $\kappa_2=1$; and $m=6$ would give $H_0=3$. All positive-$j$ groups in
\eqref{eq: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{eq:small-stem-coverage} permits only
$ac$, with $n_B=1,n_C=0,n_A=1$. For $m=4$ choose two of $x,a,c$ and two of
$y,b,z$; then $n_C=H_1-3$, $n_B=2-H_2-n_C$, $n_A=1-n_B-n_C$. Enumerating the nine
choices, a dash marking a negative exponent,
\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}
So the complete initial-page list at $(6,W+1)$ is
\begin{equation}
 acAB,\qquad xaybB,\qquad xcybA,\qquad acyzA.
 \label{eq:four-terms}
\end{equation}
With exterior order $x,a,c,y,b,z$ and $da=xy$, $db=yz$, $dc=xb+az$, the
splitting differentials are
\[
 d(acAB)=(xab-xcy)AB,\quad d(xaybB)=0,\quad
 d(xcybA)=-xaybzA,\quad d(acyzA)=-xaybzA.
\]
The first image has polynomial factor $AB$ and cannot cancel the others; the two
remaining cycle directions are boundaries,
\[
 d(acyB)=-xaybB,\qquad d(acbA)=-xcybA+acyzA.
\]
The splitting cohomology of \eqref{eq:four-terms} is zero, and the May spectral
sequence, finite in these weights, proves all the claimed $E_2$ vanishings.

For orientation, the same equations enumerate the source $(5,W)$. There
$H_0=1$, $m$ is odd, $L=(5-m)/2$, and they force $H_2+n_B+n_C=2$ and
$H_1+L-n_C=4$. If $m=1$ then $H_1+L\le1+2<4$; if $m=5$ then at least two of
$x,a,c$ occur, contrary to $H_0=1$. Thus $m=3$, forcing $n_C=0$ and $H_1=3$; the
only monomials are $aybB$ and $cybA$, and we do not need to know whether either
survives the May spectral sequence.

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

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

\begin{lemma}[The sphere classes]
\label{lem: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 cobar class $A$ of \eqref{eq:An-surjection}.
\end{lemma}

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

The integral polynomial
\begin{equation}
 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}
 \label{eq:beta-cocycle}
\end{equation}
gives 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 from
expanding $(x+y+z)^p$), and its additive image is nonzero in cohomology: at
weight $p$ the only coordinate is $t_1$, the only one-cochain is $t_1^p$ with
zero differential in characteristic $p$, while \eqref{eq:beta-cocycle} is a
nonzero two-cochain. So it is not a boundary.

The first positive topological differential length permitted by sparsity is
$q+1$. From $(2,p)$ an outgoing differential targets $(qj+3,p+j)$ with $j\ge1$,
violating $w\ge s$, and none can reach filtration two; the same holds at
$(1,1)$. These cocycles therefore survive to the stated sphere classes.

For a positive stem $i$ with a nonzero normalized-cobar term, either $s=0$ and
$i=qw\ge q$, or $s\ge1$ and $i=qw-s\ge(q-1)s\ge q-1$. This gives the lower
sphere vanishing; in stem $q-1$ the same inequality permits only $(s,w)=(1,1)$,
so no extension ambiguity, and $\pi_{q-1}S^0_{(p)}=\Z/p\{\alpha\}$. Finally the
sphere action and its cobar pairing in Proposition~\ref{prop: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 $2k$, from Steenrod--Epstein (Chapter VI, Section 1,
and Chapter VIII, Theorem 2.2). 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
(Proceedings of the Japan Academy 43 (1967), 839) and later treated such
relations with cyclic extended powers (Proceedings 44 (1968), 198). We give the
image relation needed in $Y$ through a finite-cell construction.

\begin{lemma}[A finite three-cell model]
\label{lem:three-cell-model}
Let $U_0$ be a connective $p$-local spectrum with
\[
 H_0(U_0;\Zp)=\Zp,\qquad H_{q-1}(U_0;\Zp)=\Z/p,
\]
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 $b_0$, with
$u\in\Zp^\times$ and $a\in\F_p^\times$.
\end{lemma}

\begin{proof}
Let $C$ be the cofiber of $b_0:S^0\to U_0$. The assumed integral homology and
successive connective Hurewicz show $C$ is $(q-2)$-connected and
\[
 \pi_{q-1}C=H_{q-1}(C;\Zp)=\Z/p.
\]
Choose a generator $S^{q-1}\to C$. Its obstruction to lifting to $U_0$ lies in
$\pi_{q-2}S^0=0$ by Lemma~\ref{lem:sphere-classes}; a lift, with $b_0$, gives
$S^0\vee S^{q-1}\to U_0$. Its cofiber $D$ is $(q-1)$-connected, and the integral
homology sequence gives
\[
 H_q(D;\Zp)=\ker\bigl(\Zp\to\Z/p\bigr)=p\Zp,
\]
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 a unit times $p$, since its homology image is exactly $p\Zp$, and its
component on the bottom lies in $\pi_{q-1}S^0=\Z/p\{\alpha\}$, hence is
$a\alpha$.

The map $Q\to U_0$ is an integral homology isomorphism in degrees zero and
$q-1$, so by the universal coefficient sequence it is an isomorphism on mod-$p$
cohomology in degrees zero and $q$ (the latter from the torsion in degree
$q-1$). The assumed nonzero reduced power pulls back to a nonzero reduced power
of the bottom class of $Q$. If $a=0$ the attaching map would give
\[
 Q\simeq S^0\vee\operatorname{cofib}\bigl(up:S^{q-1}\to S^{q-1}\bigr),
\]
and the degree-zero class in this wedge is pulled back from $S^0$, so its
positive reduced powers vanish by naturality, a contradiction. Hence $a\ne0$.
The construction used an integral homology gap and relative Hurewicz in a finite
range; it does not assert that $U_0$ itself is finite.
\end{proof}

\begin{proposition}[The extended-power relation]
\label{prop: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 to a sphere map representing $\beta$, then collapsing $BC_p$ in the
extended power of $S^0$, gives a map
\begin{equation}
 \Phi:\operatorname{Th}(m\rho_{\mathbb C})\longrightarrow S^0
 \label{eq:extended-beta-map}
\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 $\F_p^\times$ acts by permuting the cyclically labelled smash
coordinates and by the corresponding automorphisms of $C_p$. This normalizer
action is compatible with \eqref{eq:extended-beta-map}, on whose target it is
trivial. Normalize the bottom degree by
\[
 U=\Sigma^{-pd}\operatorname{Th}(m\rho_{\mathbb C}),\qquad
 e_0=\frac{1}{p-1}\sum_{a\in\F_p^\times}a:U\longrightarrow U.
\]
The idempotent $e_0$ splits in spectra. Let $U_0$ be its image, 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 $b_0:S^0\to U_0$ is an $H_0(-;\Zp)$-isomorphism (the complex
orientation makes every normalizer permutation act as one on the bottom class),
and normalizer invariance of $\Phi$ gives
\[
 \Phi_0:\Sigma^{pd}U_0\longrightarrow S^0,
 \qquad \Phi_0\Sigma^{pd}b_0=\beta^p.
\]

We compute the homology and the reduced power of this summand. The bundle is
integrally oriented, so the integral Thom isomorphism identifies
$H_*(U;\Zp)$ with $H_*(BC_p;\Zp)$. The periodic cyclic-group resolution, with
maps $g-1$ and $1+g+\cdots+g^{p-1}$, gives
\[
 H_0(BC_p;\Zp)=\Zp,\quad
 H_{2j-1}(BC_p;\Zp)=\Z/p\ (j\ge1),\quad
 H_{2j}(BC_p;\Zp)=0\ (j\ge1).
\]
These 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
\[
 H^*(BC_p;\F_p)=\Lambda(s)\otimes\F_p[t],
 \qquad |s|=1,\ |t|=2,\ t=\operatorname{Bockstein}(s).
\]
An automorphism $a\in\F_p^\times$ multiplies both $s$ and $t$ by $a$, equivalently
acts by $a^j$ on integral homology in degree $2j-1$. Averaging thus retains,
through degree $q$, exactly the mod-$p$ classes in degrees $0,q-1,q$ and integral
groups
\begin{equation}
 H_0(U_0;\Zp)=\Zp,\quad H_{q-1}(U_0;\Zp)=\Z/p,\quad
 H_j(U_0;\Zp)=0\ (0<j\le q,\ j\ne q-1).
 \label{eq:U0-homology}
\end{equation}

Let $u$ be the normalized Thom class. We claim
\begin{equation}
 \mathcal P^1u=-m\,t^{p-1}u\ne0.
 \label{eq:Thom-P1}
\end{equation}
For a 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; pulling back along the
zero section and using $\mathcal P^1(l)=l^p$ gives $c=1$ in the polynomial
cohomology of infinite complex projective space, and naturality gives it for
every line bundle. The Cartan formula then makes the multiplier of $\mathcal P^1$
on a Thom class the sum of the $(p-1)$st Chern powers of the summands. The
regular representation splits into the characters with first Chern classes
$it$, $i\in\F_p$, so the multiplier is
\[
 m\sum_{i\in\F_p}(it)^{p-1}=-m\,t^{p-1}.
\]
Both this class and $u$ are normalizer-invariant, and the reduced power commutes
with the idempotent, so \eqref{eq:Thom-P1} holds in $U_0$.

Apply Lemma~\ref{lem:three-cell-model} to \eqref{eq:U0-homology} and
\eqref{eq:Thom-P1}. It gives 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 $\beta^p$, its value on the other lower cell
is some $\zeta$, and the top cell gives
\[
 a\,\alpha\beta^p+up\,\zeta=0\quad\text{in }\pi_{pd+q-1}S^0.
\]
The action of $p$ on $Y$ is null by Corollary~\ref{cor:p-null}. Apply the unit to
this relation and use $a\ne0$ modulo $p$ to get $\alpha\beta^p=0$ 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 earlier 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 make this image a cobar
boundary.

\begin{lemma}[The symmetric coefficient comodule]
\label{lem: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 $1,r,\dots,r^{p-1}$, where
$r^k$ has weight $k$. Its coaction is
\begin{equation}
 r^k\longmapsto\sum_{j=0}^k\binom kj r^jt_1^{k-j}.
 \label{eq:translated-comodule}
\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}
\textbf{The two-cell coaction.} We first compute the coefficient comodule used
for duality. The dual $DJ$ is the cofiber of $\alpha':S^{q-1}\to S^0$, where
$\alpha'=\pm\alpha$ after fixing cell orientations, with bottom inclusion
$D\varepsilon$. 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 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 with $\rho_*z=0$, 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
\[
 \epsilon(g)=0,\qquad
 \Gamma_q=\eta_L(R_q)\oplus\Zp\{t_1\},\qquad g=c\,t_1,
\]
where $\eta_L$ is induced by $\BP\wedge\eta_{\BP}:\BP\to\BP\wedge\BP$. 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 off-diagonal coaction:
writing $\psi(z)=1\otimes z+\omega\otimes e_0$, naturality under $H$ gives
$\Delta g=1\otimes g+\omega\otimes1$; applying $\operatorname{id}\otimes\epsilon$
and using $\epsilon(g)=0$ gives $\omega=g$. By Lemma~\ref{lem: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 the off-diagonal term to
$g+a(\eta_L-\eta_R)(v_1)=g-pa\,t_1$; changing $H$ by
$DJ\to S^q\xrightarrow{av_1}\BP$ changes $g$ by $\eta_R(av_1)$, and renormalizing
$z-av_1e_0$ subtracts $\eta_L(av_1)$, giving $g+pa\,t_1$. Opposite conventions
reverse signs and dual orientations change the detector by a sign; none changes
its nonzero residue modulo $p$, so the module splitting need not split the
comodule. Replacing $z$ by $c^{-1}z$ makes the off-diagonal term exactly $t_1$
(its top quotient becomes $c^{-1}$, which we no longer need). This fixes $e_0$
and $D\varepsilon$, the data used below.

\textbf{The symmetric coefficient module.} Duality inverts each permutation and
preserves the average, so $DT\simeq\operatorname{Sym}^{n}(DJ)$, and under this
identification $D\varepsilon_T$ is the pure bottom tensor $e_0^{\otimes n}$. All
cell degrees of $DJ$ are even. If $z_S$ is the tensor with upper generator in
positions $S$ and bottom generator elsewhere, define the normalized symmetric
basis
\begin{equation}
 r_k=\binom nk^{-1}\sum_{|S|=k}z_S,\qquad 0\le k\le n.
 \label{eq:symmetric-basis}
\end{equation}
Every denominator is a $p$-local unit. Expanding the translation of each tensor
gives the coefficient
$\binom{n-j}{k-j}\binom nk/\binom nk=\binom kj$ on $r_jt_1^{k-j}$ in the
coaction of $r_k$. So we may write this basis as $1,r,\dots,r^{p-1}$ with
coaction \eqref{eq:translated-comodule}. Each $r^k$ has weight $k$ because it is a
tensor with $k$ upper generators of degree $q$.

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

\begin{proposition}[The symmetric null map]
\label{prop:symmetric-null-map}
For $d=|\beta|=pq-2$, the stable map
\begin{equation}
 f:\Sigma^dT\xrightarrow{\ \Sigma^d\varepsilon_T\ }S^d
   \xrightarrow{\ \beta\ }S^0\xrightarrow{\ \eta\ }Y
 \label{eq:null-composite}
\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{lem:symmetric-comodule}
its induced $E_2$ class is represented by the constant-valued beta cocycle. We
show this cocycle is a boundary in the translated coefficient module. Consider
the integral polynomial
\begin{equation}
 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.
 \label{eq:carry-polynomial}
\end{equation}
The sum defines it before reduction modulo $p$ and uses only the allowed powers
$r,\dots,r^{p-1}$. The exact integral identity
\begin{equation}
 F(r,x)+F(r+x,y)-F(r,x+y)=\frac{(x+y)^p-x^p-y^p}{p}
 \label{eq:carry-boundary}
\end{equation}
follows by cancellation of the four $p$th powers. After reduction, the coaction
\eqref{eq:translated-comodule} and primitivity of $t_1$ make the left side the
cobar differential of $F(r,t_1)$ (up to the overall sign fixed by the cobar
convention), and the right side is the constant cocycle \eqref{eq:beta-cocycle}.
So the bottom image of $\beta$ is zero on $E_2$.

No higher-filtration homotopy value is hidden by this boundary. The spectrum
$DT\wedge Y$ is connective with nonnegative coefficient weights, and in stem
$pq-2$ its $q$-sparse terms are $(s,w)=(2+qj,p+j)$; for $j\ge1$ the inequality
$w\ge s$ would give $p-2\ge(q-1)j$, impossible. Naturality places the image of
$\beta$ in filtration at least two, and the boundary raises it to at least three,
while all later associated-graded groups are zero. Convergence with finite
filtration then makes the actual image zero. 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{prop:annihilation}
If $\gamma\in\pi_{qW-5}Y$ represents a class of $E_2^{5,W}(Y)$, then
\begin{equation}
 \beta^K\gamma=0,\qquad K=p(p-2)+1.
 \label{eq:beta-annihilation}
\end{equation}
Only the binary two-sided unit identities for $\mu$ are used.
\end{proposition}

\begin{proof}
Lemma~\ref{lem:small-stems} gives $\alpha\gamma=0$ in the stem $q(W+1)-6$, and
Proposition~\ref{prop:extended-power-relation} gives $\alpha\beta^p=0$ after
applying the unit to $Y$. By \eqref{eq: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 $\eta\beta^p$. Fix any parenthesization of the $n$-fold iterate of
$\mu$. Smash one $\widetilde\gamma$ with $p-2$ copies of $\widetilde b$, collect
the $J$ factors, and apply that iterate to the $Y$ factors:
\[
 L:S^{qW-5+(p-2)pd}\longrightarrow J^{\wedge n}\wedge Y.
\]
Its top projection is $\beta^{p(p-2)}\gamma$. To see this without associativity,
for any sphere scalar $a$ the two unit homotopies give
\[
 \mu(\eta a,x)=ax,\qquad \mu(x,\eta a)=(-1)^{|a||x|}ax.
\]
The scalar $\beta^p$ is even, so 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 $\gamma$; induction on the tree gives the top
value.

Project to the symmetric retract after the $Y$ factors are multiplied:
\[
 L_T=(\mathrm{pr}\wedge1_Y)L:
 S^{qW-5+(p-2)pd}\longrightarrow T\wedge Y.
\]
For every permutation $\sigma$ of the $J$ factors, naturality of the symmetry and
the zero-dimensional top sphere give $\varepsilon^{\wedge n}\sigma=\varepsilon^{\wedge n}$;
consequently
\begin{equation}
 (\varepsilon_T\wedge1_Y)L_T=(\varepsilon^{\wedge n}e\wedge1_Y)L
 =(\varepsilon^{\wedge n}\wedge1_Y)L=\beta^{p(p-2)}\gamma.
 \label{eq:averaged-top-map}
\end{equation}
No permutation of the multiplied $Y$ factors occurs here.

The null map $f$ of Proposition~\ref{prop:symmetric-null-map} now gives a
homotopy-commutative diagram
\begin{equation}
\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}
\label{eq:null-map-diagram}
\end{equation}
Its commutativity is the left unit identity for $\mu$ and the definition of $f$.
Its top arrow is null, so composing with $\Sigma^dL_T$ and using
\eqref{eq:averaged-top-map} gives $\beta^{p(p-2)+1}\gamma=0$.
\end{proof}

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

\begin{proof}
Let $\theta\in E_2^{5,W}(Y)$. By Lemma~\ref{lem:small-stems} it is a permanent
cycle represented by $\gamma$, and the sphere class $\beta$ is permanent. The
pagewise product $A^K\theta$ therefore supports no nonzero outgoing differential;
if either factor vanishes on a page the product already vanishes. By
Proposition~\ref{prop:annihilation} its filtered homotopy product is zero, and
product compatibility with finite convergence makes $A^K\theta$ zero at
$E_\infty$. If it is nonzero at $E_2$ it must be hit by an incoming differential:
its cohomological filtration is
\[
 S=5+2K=2p^2-4p+7,
\]
and an incoming differential of length $r$ has source filtration $S-r\ge0$, so
$r\le S$. This bounds the page of disappearance, not merely its eventual abutment.
The exact comparisons are
\begin{equation}
 qp+1-S=2p-6>0,\qquad N-K=p-3>0,
 \label{eq:strict-page-bound}
\end{equation}
and 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 $\beta$, and shows $A^N\theta$ is zero there as well. The
surjection \eqref{eq:An-surjection} accounts for every $E_2$ class of the target.
\end{proof}

\subsection{The realizing cofiber}

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{thm:main}]
The only possible positive differentials from $v_4$ target
$(s,w)=(qk+1,u_4+k)$. Proposition~\ref{prop:first-differential} rules out $k=1$.
Lemmas~\ref{lem:target-bound} and \ref{lem:first-list} with
Table~\ref{tab:remaining} leave, for $k\ge2$, only $k=2,p-1,p$.
Corollary~\ref{cor:frob-vanishing} eliminates the first two target groups already
at $E_2$, and Proposition~\ref{prop:early-disappearance} makes the last target
zero before its possible differential. No differential can hit filtration zero,
so $v_4\in E_2^{0,u_4}(Y)$ is permanent.

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

Finally, $X$ is the cofiber of a map between finite $p$-local spectra, hence
finite $p$-local in the sense of the introduction; equivalently the finite cell
construction is realized by a finite spectrum before localization by clearing
prime-to-$p$ denominators in the attaching maps
(Lemma~\ref{lem:finite-localization}). Explicitly, the mapping cone adds to the
cells of $Y$ their translates by $qu_4+1$, giving a finite $p$-local CW model
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}

\appendix

\section{The ordinary resolution and its filtered pairing}
\label{app:res}

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 (Hovey--Shipley--Smith, Section 6.2, Theorems 6.3.8
and 6.4.1, Corollary 6.4.2, in the cited version); comparison with ordinary
spectra follows from Theorems 6.3.8 and 4.3.2 there. All smash products are
derived; point-set maps use cofibrant inputs. Rectify the coherent associative
structure on $\BP$ as an $S$-algebra (Elmendorf--Kriz--Mandell--May, Part II,
Corollary 3.6 and the convention comparison of Section 4). The equivalence for
associative algebras (Schwede, Main Theorem and Theorem 5.1(ii)) places it in
this category; choose a cofibrant associative symmetric ring representative
(Mandell--May--Schwede--Shipley, Theorem 12.1(iv)--(v)), whose underlying
spectrum is stably cofibrant. Smashing with it preserves stable equivalences
(Proposition 12.3 there), so the point-set smash powers below compute derived
ones. 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, $K_{-1}(Z)=*$, and $K(Z)=\operatorname{Tot}T^\bullet(Z)$; ordinary
totalization uses all ordinal maps, including codegeneracies.

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

\begin{proof}[Proof of Proposition~\ref{prop:anss}]
We identify the ordinary totalization tower. Let $\mathcal P_0([n])$ be the
poset of nonempty subsets of $\{0,\dots,n\}$. The functor
\[
 u_n:\mathcal P_0([n])\to\Delta_{\le n},\qquad S\mapsto[|S|-1],
\]
sending an inclusion to an order-preserving injection, is homotopy-initial. To
see this, for $m\le n$ identify $(u_n\downarrow[m])$ with the nonempty face poset
of the simplicial complex $L_{n,m}$ with vertices $(i,j)$, $0\le i\le n$,
$0\le j\le m$, and simplices of strictly increasing first and weakly increasing
second coordinate; its nerve is the barycentric subdivision of $L_{n,m}$.

The complex $L_{n,m}$ is contractible for $n\ge m$, by induction on $n$: the
case $n=m=0$ is a point; starting with the cone with vertex $(n,m)$ on the whole
subcomplex $L_{n-1,m}$, adjoin the cones with vertices $(n,j)$, $0\le j<m$, on
their bases $L_{n-1,j}$. Two distinct vertices with first coordinate $n$ are in
no simplex, so each new cone meets the preceding union in exactly its base,
which is contractible by induction; attachment along a simplicial subcomplex is
a homotopy pushout and preserves contractibility.

Homotopy-initiality applied to the cosimplicial unit resolution gives, naturally
in $n$ and $Z$,
\begin{equation}
 K_n(Z)\simeq\operatorname*{holim}_{\varnothing\ne S\subseteq\{0,\dots,n\}}
       \BP^{\wedge|S|}\wedge Z,
 \label{eq:finite-cube-comparison}
\end{equation}
the standard cubical description of partial Amitsur totalizations (compare
Mathew--Naumann--Noel, Propositions 2.11 and 2.14). Adjoining the empty vertex $Z$
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\to K_n(Z))\simeq\mathcal F^{\wedge(n+1)}\wedge Z.
 \label{eq:tot-fiber-index}
\end{equation}
Restriction from $n+1$ to $n$ is induced by the last factor
$\mathcal F\to S_{(p)}$, with no suspension in \eqref{eq:tot-fiber-index}.

The unit induces an isomorphism on $\pi_0$ and $\pi_1\BP=0$, 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, so
\eqref{eq:tot-fiber-index} makes $Z\to K(Z)$ an equivalence and identifies the
relative totalization tower as
\begin{equation}
 G^s(Z):=\operatorname{fib}(K(Z)\to K_{s-1}(Z))
        \simeq\mathcal F^{\wedge s}\wedge Z\quad(s\ge0).
 \label{eq:relative-tower}
\end{equation}
Its homotopy groups vanish in each fixed stem for large $s$, giving 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)\to G^s(Z))
 \simeq\BP\wedge\mathcal F^{\wedge s}\wedge Z
 \simeq\Sigma^{-s}\BP\wedge\overline{\BP}^{\wedge s}\wedge Z,
 \label{eq:tot-layer-index}
\end{equation}
with $\overline{\BP}=\operatorname{cofib}(S_{(p)}\to\BP)$; this also equals
$\operatorname{fib}(K_s(Z)\to K_{s-1}(Z))$, with the displayed $s$ desuspensions.
The iterated extended terms compute the cobar complex: $\BP\wedge\BP$, as a
$\BP$-module, is a wedge of suspended free $\BP$-modules on the monomials in the
$t_i$, and iteration identifies their homotopy with iterated tensors of $\Gamma$
over $R$, with no derived tensor corrections whether or not $\BP_*Z$ is flat over
$R$. Cofaces give the cobar differential and codegeneracy splittings give its
normalized form; the exact-couple grading gives the differential shift. The first
map is $Z\to\BP\wedge Z$, whose 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{prop:anss-pairing}]
We build maps $G^s(Z_1)\wedge G^t(Z_2)\to G^{s+t}(Z_3)$ inducing the stated cobar
pairing and recovering the actual map $f$.

\textbf{The box pairing.} We use Batanin's cosimplicial box product
(McClure--Smith, Definition 2.1, Proposition 2.3, Remark 2.4, which attributes the
Kan-extension description to Cordier and Porter). 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
\[
 \Delta\times\Delta\to\Delta,\qquad([a],[b])\mapsto[a+b+1].
\]
In degree $n$ it is the zigzag pushout of the terms $A^a\wedge B^b$ with $a+b=n$,
neighboring terms identified by $d^{a+1}\wedge1$ and $1\wedge d^0$. For the unit
resolutions, concatenating the two $\BP$ blocks and applying $f$ once gives maps
$T^a(Z_1)\wedge T^b(Z_2)\to T^{a+b+1}(Z_3)$, 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 a cosimplicial map
\begin{equation}
 m:T(Z_1)\mathbin\square T(Z_2)\longrightarrow T(Z_3).
 \label{eq:original-box-pairing}
\end{equation}
On the degree-$n$ summand $a+b=n$ it multiplies the last factor of the first
$\BP$ block with the first factor of the second; this is the single overlapping
slot, and the two descriptions agree by the codegeneracy identifying the two
junction vertices of $[a+b+1]$.

\textbf{Deriving the pairing.} 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, the representables and boundaries defining the
generating Reedy cells (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 boundary differs from the latching subobject in target degree $n$, which
factors through a proper injection into $[n]$). Kan extension gives
$h_a\square h_b=h_{a+b+1}$. In the relative box boundary
$h_{a+b+1}/(\operatorname{im}\partial h_a\square h_b\cup\operatorname{im}h_a\square\partial h_b)$
the remaining maps are the monotone maps injective on each block. Such a map is
either globally injective or identifies only the two junction vertices; a map not
surjective onto its target lies in that target degree's latching subobject, and
the only remaining surjections are the junction identification onto $[a+b]$ and
the identity of $[a+b+1]$. So the junction-identifying and globally injective
non-latching cells occur in degrees $a+b$ and $a+b+1$. Attach the junction cell
first (every further collapse of it is already in the boundary, since it collapses
two vertices within one block), then the globally injective cell (its junction
codegeneracy lands in the first cell, its other codegeneracies in the boundary).
This gives a two-cell Reedy filtration of the relative box boundary; each
attachment is tensored with the ordinary spectral pushout-product of the
underlying generating cofibrations, so the spectral pushout-product axiom makes
both cofibrations, trivial if either input is trivial. Closure under pushouts,
composition, and retracts proves it for all cofibrations.

Choose Reedy-cofibrant replacements $A_i\to T(Z_i)$ 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{eq:original-box-pairing} gives $A_1\square A_2\to R_3$, and by the Quillen
check $A_1\square A_2\to R_1\square R_2$ is a trivial Reedy cofibration; the lifting
axiom extends to
\begin{equation}
 m_R:R_1\square R_2\longrightarrow R_3.
 \label{eq:derived-box-pairing}
\end{equation}
The space of such extensions is contractible up to relative homotopy. This
transports the pairing and 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. When some $Z_i$
coincide, their source and target replacements need not be identical.

\textbf{The filtered diagonal.} We use the simplex subdivision of McClure--Smith
(Lemma 3.6, credited to Grayson). For $t=(t_0,\dots,t_n)\in\Delta^n$ put
$I_i=[\sum_{j<i}t_j,\sum_{j\le i}t_j]$,
$x_i=2\operatorname{length}(I_i\cap[0,\tfrac12])$, $y_i=2t_i-x_i$. For a cut index
$k$, $x$ is supported in $0,\dots,k$ and $y$ in $k,\dots,n$; the cut region is a
copy of $\Delta^k\times\Delta^{n-k}$ with inverse $t=(x+y)/2$, whose boundary seams
identify the last face of the first factor with the first face of the second, as
in the box pushout. Thus the cut defines
$\delta:\Delta^\bullet_+\to\Delta^\bullet_+\square\Delta^\bullet_+$, natural under
all monotone maps (they aggregate consecutive intervals, and aggregation commutes
with the cut), hence a map of ordinary cosimplicial objects including
codegeneracies.

Define the relative simplex object
\[
 P_s^\bullet=\Delta^\bullet_+/\operatorname{sk}_{s-1}\Delta^\bullet_+,
 \qquad(s\ge0),
\]
with skeleton the basepoint for $s=0$. The simplex, its skeleta, and these
quotients are Reedy cofibrant (latching maps are boundary inclusions or
identities), so
\begin{equation}
 G^s(Z_i)=\operatorname{Map}_\Delta(P_s,R_i)
 \label{eq: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 skeleton
cutoff, and naturality applies to every face of a larger simplex, so the cut
descends to maps
\begin{equation}
 \delta_{s,t}:P_{s+t}\longrightarrow P_s\square P_t.
 \label{eq:relative-cut}
\end{equation}
Enriched evaluation followed by $m_R$ and precomposition with $\delta_{s,t}$ gives
the 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*}
commuting with the tower inclusions in both inputs; by \eqref{eq:relative-tower}
these are the exact relative fibers of the unit-resolution filtration with indices
$s,t,s+t$.

\textbf{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 by one). The boundary formula for a product of relative cells,
$\partial(a\cdot b)=\partial a\cdot b+(-1)^{|a|}a\cdot\partial b$, gives a pairing
of the exact couples and the Leibniz rule on every page; on the dimension-$(s+t)$
layer the only contributing cut cell has dimensions $s,t$, and the inverse
$t=(x+y)/2$ gives its oriented front-face/back-face cup map with coefficient one.
So the normalized cobar identification gives exactly the asserted $E_2$ cup
pairing.

We verify the homotopy target without assuming 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 all-unit map to $T(Z_i)$
lifts to $\alpha_i:\Delta^\bullet_+\wedge Z_i\to A_i$ along the trivial fibration
$A_i\to T(Z_i)$. The composite with $A_i\to R_i$ represents the coaugmentation,
and the box product of the $\alpha_i$, precomposed with $\delta$, lands in
$A_1\square A_2$; since $m_R$ extends the original map there, the composite is 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 equivalences as proved, so 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, and every element of $R/I$ is a scalar multiple of that unit,
so the pairing is $(r\bmod I,r'\bmod I)\mapsto rr'\bmod I$. This uses neither
flatness of the quotient nor associativity of the target map.
\end{proof}

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\end{document}