\documentclass[11pt]{article}
\usepackage[margin=1in]{geometry}
\usepackage{amsmath,amssymb,amsthm,booktabs,array}
\usepackage[hidelinks]{hyperref}
\setlength{\emergencystretch}{3em}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\theoremstyle{definition}\newtheorem{definition}[theorem]{Definition}
\theoremstyle{remark}\newtheorem{remark}[theorem]{Remark}
\newcommand{\R}{\mathbf R}\newcommand{\C}{\mathbf C}
\newcommand{\Z}{\mathbf Z}\newcommand{\tr}{\operatorname{tr}}
\newcommand{\Nd}{N_{\mathrm d}}\newcommand{\one}{\mathbf 1}
\title{Cubic augmentation of a two-moment model:\\
an exact finite absorption theorem and its limits}
\author{Kunal Tyagi\thanks{\texttt{hello.jay.tyagi@gmail.com}.
\emph{Use of AI.} The original work used Claude (Anthropic). This revision,
proof review, and companion certificates used OpenAI Codex. The author directed
the work and is responsible for the content. Code and publication artifacts:
\url{https://github.com/x67ai/riemann-rh-program}.}}
\date{August 26, 2026; revised October 18, 2026 (version 2)}
\hypersetup{pdftitle={Cubic augmentation of a two-moment model: an exact finite absorption theorem and its limits},pdfauthor={Kunal Tyagi}}
\begin{document}\maketitle
\begin{abstract}
We study adjoining a half-bandwidth signed cubic trace and a spectral count
ladder to a bandwidth-one trace/Frobenius model motivated by zeta-zero
certificates. In a finite circle model with mass $N=64$, flat windows, and
explicit budget centers, we prove exact absorption over the full atom-only
class: the augmented and baseline optima agree for both the distinct-count
and simple-fraction objectives. Three explicitly specified integer-marked
grid columns attain the lower bounds. Their mixing weights are defined
algebraically, and an Arb certificate proves all strict feasibility and
inertia inequalities. The witness satisfies the ladder at every positive
threshold, not just on a finite threshold grid. We also give the exact
conjugate-pair spectrum, the isolated-block cubic identity, grid Parseval,
and the sharp conditional spectral-capacity constant $2\sqrt2$.
The result is a finite model theorem, not an unconditional statement about
zeta zeros. Version 2 corrects an infeasible witness, a cubic-tolerance
mismatch, and an incorrect rationality claim in version 1. Global absorption
in the pair-containing or pinned classes remains unproved; equality on a
searched column dictionary and heuristic pricing do not establish it.
\end{abstract}

\section{Problem, contribution and revision}
The two-moment certificate in the work of Alp\"oge and Furman \cite{AF26}
motivates adding a signed cubic trace at a second bandwidth. Classical
higher-level correlations \cite{RS96} suggest suitable model budgets, but
passing from those correlation statements to every row of a particular
finite model is an additional task. We make the model and its proved scope
explicit.

The main result is Theorem~\ref{thm:main}: one precisely specified $N=64$
flat-window atom-only model has an exactly base-optimal law satisfying its
half-bandwidth Frobenius row, cubic row and all-threshold count ladder.
The proof combines an analytic atom-only lower bound with a finite
algebraic construction and certified interval inequalities. It does not
require optimizing over a numerically generated dictionary.

\paragraph{Correction of version 1.}
Version 1, titled \emph{The two-moment certificate is robust under
Rudnick--Sarnak-range cubic augmentation with capacity control}, stated
broader exact absorption claims than its proofs and computations supported.
Those full-class conclusions are withdrawn. Its printed clean-grid witness
missed the implemented cubic band, its displayed cubic tolerance differed
from the code's, and selected row values are irrational algebraic numbers,
not rational numbers. This revision replaces the witness and states its
budgets exactly. Section~\ref{sec:limits} separates the theorem proved here,
the earlier numerical observations, and the open analytic bridges.

\section{The finite model}\label{sec:model}
Fix a circle of circumference $N$ and a finite symmetric integer band $B$,
with uniform harmonic weights $1/M$, $M=|B|$. A configuration consists of
real positions $\theta_a$ with positive integer marks $m_a$, and optionally
conjugate pairs $\theta_p\pm id_p$ of common positive integer mark $m_p$.
Positions are taken modulo $N$. Its total mass is
$\sum_a m_a+2\sum_p m_p=N$.
Put
\[
 c_s=\sum_a m_a e^{-2\pi i s\theta_a/N}
       +2\sum_p m_p\cosh(2\pi s d_p/N)e^{-2\pi i s\theta_p/N},
 \qquad H_B[j,k]=\frac{c_{j-k}}{M}\quad(j,k\in B).
\]
The identity $c_{-s}=\overline{c_s}$ makes $H_B$ Hermitian and
$\tr H_B=N$. At the two bands used here,
\[
 B_1=\{-32,\ldots,32\},\quad M_1=65,
 \qquad B'=\{-16,\ldots,16\},\quad M'=33,
\]
write $F_1=\tr H_{B_1}^2$, $F'=\tr H_{B'}^2$, $C'=\tr H_{B'}^3$, and
$n(V)=\#\{\lambda\in\operatorname{Spec}(H_{B'}):|\lambda|\ge V\}$.
An \emph{atom-only} configuration has no conjugate pairs; its matrices are
positive semidefinite. Marks at coincident real positions are combined.
Its objectives are $\Nd=\#\{a\}$ and $p_1=\#\{a:m_a=1\}/N$.
A law is a probability distribution on configurations, with finite expected
$F_1$; each constraint below is an expectation constraint.

\subsection{Exact budgets and the meaning of the cubic slack}
All decimal constants in the following display mean exact rational numbers:
\begin{align*}
 N&=64,&\varepsilon&=0.05=1/20,&C_{\mathrm{led}}&=10,\\
 B_1^{\mathrm{cen}}&=84.71214548308222,
 &B_2^{\mathrm{cen}}&=135.09544203420174,
 &B_3^{\mathrm{cen}}&=304.5549106659001.
\end{align*}
The superscript distinguishes a budget center from a harmonic band. Below,
write $b_j=B_j^{\mathrm{cen}}$ and $U=b_1(1+\varepsilon)$.
These are fixed model parameters inherited from the primary finite cell of
version 1. No equality to a zeta statistic, or to an unrounded random-matrix
expectation, is assumed.

The baseline atom problem imposes mass $64$ and
\begin{equation}\label{eq:base}
 b_1(1-\varepsilon)\le E[F_1]\le b_1(1+\varepsilon).
\end{equation}
The augmented problem adds
\begin{align}
 b_2(1-\varepsilon)&\le E[F']\le b_2(1+\varepsilon),\label{eq:fp}\\
 |E[C']-b_3|&\le\varepsilon N,\label{eq:cp}\\
 E[n(V)]&\le C_{\mathrm{led}}N V^{-4}\quad\hbox{for every }V>0.\label{eq:ladder}
\end{align}
Thus the cubic half-width is the \emph{absolute} amount $\varepsilon N=3.2$.
Version 1's code used the different relative amount $\varepsilon b_3$.
The witness below has $E[C']=b_3$ exactly and satisfies both conventions.
No auxiliary garnish variable is needed for this theorem.

\section{Exact atom-only absorption}\label{sec:main}
\subsection{The analytic lower bound}
For any atom-only configuration and a symmetric band, the position kernel
$D_B(x)=M^{-1}\sum_{j\in B}e^{-2\pi ijx/N}$ is real. Therefore
\[
 \tr H_B^2=\sum_{a,a'}m_am_{a'}D_B(\theta_a-\theta_{a'})^2
       \ge\sum_a m_a^2.
\]
The terms with distinct positions are nonnegative; equality requires their
kernels to vanish when both marks are positive. For integer $m\ge1$,
\[
 1\ge(3m-m^2)/2,\qquad \one_{m=1}\ge2m-m^2,
\]
with equality in both precisely for $m\in\{1,2\}$. Summing gives
\begin{equation}\label{eq:corners}
 E[\Nd]/N\ge\frac{3N-U}{2N},\qquad
 E[p_1]\ge2-\frac{U}{N}
\end{equation}
under the upper baseline constraint. This reasoning covers arbitrary real
positions and unbounded integer mark alphabets, not merely a grid dictionary.
The integrality device is classical; see \cite{CGG98}.

\begin{lemma}[grid Parseval]\label{lem:grid}
Let $B$ consist of $M$ consecutive integers and put $\theta_k=kN/M$,
$0\le k<M$. For any nonnegative marks on these grid sites,
$\tr H_B^2=\sum_k m_k^2$.
\end{lemma}
\begin{proof}
Let $A[j,k]=M^{-1/2}e^{-2\pi ijk/M}$, $j\in B$, $0\le k<M$.
Orthogonality of the $M$ characters makes $A$ unitary. The frequency
matrix is $A\operatorname{diag}(m_k)A^*$, so its eigenvalues are exactly
the marks, including zero marks at unused sites. Taking the squared trace
proves the statement. This proof also covers a translated consecutive band.
\end{proof}

\subsection{Three exact columns}
For $N=64$, use the $65$ grid sites $\theta_k=64k/65$.
Table~\ref{tab:marks} specifies each column completely: each digit is the
mark at one site, and zero denotes an unused site. The two strings in a row
are concatenated in the displayed order, beginning at $k=0$.
\begin{table}[ht]\centering\small
\begin{tabular}{ccl}
\toprule
column & sites & marks in increasing site order\\\midrule
$A_0$ & $0\ldots32$ & \texttt{111111111111111111111111111110111}\\
      & $33\ldots64$ & \texttt{11111111111111111111111111111111}\\
$A_1$ & $0\ldots32$ & \texttt{210012110211210102101210121012101}\\
      & $33\ldots64$ & \texttt{20112101210120112101201121012101}\\
$A_2$ & $0\ldots32$ & \texttt{200202020202020202020202020202020}\\
      & $33\ldots64$ & \texttt{20202020020202020202020202020202}\\\bottomrule
\end{tabular}
\caption{Exact integer data. Each column has mass $64$.}\label{tab:marks}
\end{table}
Their $(\Nd,\#\{m=1\},F_1)$ values are respectively
$(64,64,64)$, $(48,32,96)$ and $(32,0,128)$.
Let $f_i=F'(A_i)$ and $q_i=C'(A_i)$, defined by the exact matrices of
Section~\ref{sec:model}; they are real algebraic numbers in the real
subfield of the $65$th cyclotomic field. In particular they are not defined
by rounding their displayed decimal approximations.

Put
\begin{equation}\label{eq:weights}
 r=\frac{U-64}{32},\qquad
 w_2=\frac{b_3-q_0(1-r)-q_1r}{q_0-2q_1+q_2},\qquad
 w_1=r-2w_2,\qquad w_0=1-r+w_2.
\end{equation}
These equations impose
\begin{equation}\label{eq:equalities}
 \sum_i w_i=1,\qquad\sum_i w_iF_1(A_i)=U,
 \qquad\sum_i w_iq_i=b_3
\end{equation}
identically, provided the denominator is nonzero.

\begin{theorem}[finite atom-only absorption]\label{thm:main}
For the exact model parameters in Section~\ref{sec:model}, the law on
$A_0,A_1,A_2$ with weights \eqref{eq:weights} is feasible for
\eqref{eq:base}--\eqref{eq:ladder}. For both objectives, the baseline and
augmented infima over \emph{all atom-only configurations at arbitrary
positions} are equal, with values
\begin{align*}
 \inf E[\Nd]/N&=\frac{3N-U}{2N}
 =\frac{103052247242763669}{128000000000000000},\\
 \inf E[p_1]&=2-U/N
 =\frac{39052247242763669}{64000000000000000}.
\end{align*}
The same law works for every $C_{\mathrm{led}}\ge10$ and for the broader
relative cubic band $|E[C']-b_3|\le\varepsilon b_3$.
\end{theorem}

\begin{proof}
The following finite inequalities are certified by the companion Arb
computation, at $256$ bits. The displayed decimal intervals are deliberately
wider than its output enclosures:
\begin{align*}
 q_0&\in(245.45087235,245.45087237),\\
 q_1&\in(411.08651303,411.08651305),\\
 q_2&\in(301.35412844,301.35412847),\\
 w_0&\in(0.47469100426,0.47469100428),\\
 w_1&\in(0.27100071778,0.27100071782),\\
 w_2&\in(0.25430827791,0.25430827796),\\
 \sum_iw_if_i&\in(135.43015807,135.43015809).
\end{align*}
The denominator in \eqref{eq:weights} is strictly negative. In particular
all weights are positive. The last interval lies strictly inside
$[b_2(1-\varepsilon),b_2(1+\varepsilon)]$.
The exact identities \eqref{eq:equalities} prove the baseline and cubic
constraints without a numerical feasibility tolerance.

For each of the three $33\times33$ half-band matrices, certified Hermitian
$LDL^*$ pivots for $vI-H$, $v=2,3,4$, exclude zero. Their signs give the
counts above these thresholds:
\[
\begin{array}{c|ccc}
 &2&3&4\\\hline
 A_0&0&0&0\\ A_1&16&2&0\\ A_2&3&2&0
\end{array}
\]
No eigenvalue equals a tested threshold, and all three matrices are positive
semidefinite. Write $\bar n_v=\sum_iw_i n_{A_i}(v)$.
For $0<V\le2$, $E[n(V)]\le33<640/16$.
For $2<V\le3$, $E[n(V)]\le\bar n_2<640/81$, the strict margin exceeding
$2.8022$. For $3<V<4$, $E[n(V)]\le\bar n_3<640/256$, the margin exceeding
$1.4493$. At $V\ge4$ every count is zero. Monotonicity therefore proves
\eqref{eq:ladder} at every threshold.

Each column has only marks $1,2$, and Lemma~\ref{lem:grid} gives equality
in the analytic lower-bound chain. Hence its mixture attains both bounds
\eqref{eq:corners}. The augmented infimum cannot be below the baseline
infimum; the same law attains both, proving the theorem. Increasing the
ladder constant or widening the cubic band preserves feasibility.
\end{proof}

\paragraph{Certificate and trust base.}
The companion \texttt{certificate.py} takes the integer mark vectors in
\texttt{witness-data.json}, forms the Fourier matrices directly with Arb
balls, computes their traces, solves the algebraic weight formulas, and
checks every strict margin. It never rationalizes a floating-point row or
adds feasibility grace. A Hermitian $LDL^*$ decomposition without pivoting
is accepted only when every real pivot excludes zero; congruence then
certifies the inertia. The exact equalities are algebraic consequences of
\eqref{eq:weights}, rather than interval equality tests. The trust base is
Python's integer/rational arithmetic, the displayed finite algorithms, and
Arb's inclusion contract through python-flint 0.6.0 \cite{Arb}.
This is a computer-assisted finite proof, not a new Lean formalization.

\section{Structural explanation}\label{sec:structure}
\subsection{The true conjugate-pair spectrum}
For a pair of depth $d$ and mark $m$, set
\[
p_j=M^{-1/2}e^{-2\pi ij\theta/N}e^{2\pi jd/N},\qquad
q_j=M^{-1/2}e^{-2\pi ij\theta/N}e^{-2\pi jd/N}.
\]
Its contribution is $m(pq^*+qp^*)$. Symmetry of the band gives
$\langle p,q\rangle=1$ and $\|p\|^2=\|q\|^2=A$, where
\[
 A=M^{-1}\sum_{j\in B}\cosh(4\pi jd/N)\ge1.
\]
The vectors $p+q$ and $p-q$ are orthogonal and give eigenvalues
$m(1+A)$ and $m(1-A)$. At $d=0$ the latter vanishes. Thus the normalized
pair charges are
\[
 (\mathrm{mass},F,C)=
 (2m,\ 2m^2(1+A^2),\ 2m^3(1+3A^2)).
\]
In the shallow limit they become $(2m,4m^2,8m^3)$, with spectrum $(2m,0)$,
not a symmetric pair $(+2m,-2m)$. For a flat continuum band,
$A=\sinh(2\pi\lambda d)/(2\pi\lambda d)$.

\begin{proposition}[isolated-block cubic identity]\label{prop:cubic}
For a direct sum of isolated atom and pair blocks with total mass $L$, squared
trace $F$ and signed cubic trace $C$,
\[
 C-3F+2L=\sum_{\mathrm{zeros}}m(m-1)(m-2)
        +6\sum_{\mathrm{pairs}}m_p^2(m_p-1)A_p^2.
\]
The first sum counts each atom once and each pair twice, using the mark on
each zero. In particular it vanishes on mark-$\{1,2\}$ atoms and
multiplicity-one pairs, where $C=3F-2L$.
\end{proposition}
\begin{proof}
An atom contributes $m^3-3m^2+2m=m(m-1)(m-2)$. A pair contributes
$2m^3(1+3A^2)-6m^2(1+A^2)+4m
=2m(m-1)(m-2)+6m^2(m-1)A^2$. Sum the identities.
\end{proof}
The cubic row is redundant with mass and Frobenius \emph{on this isolated
block class}. A deep pair's raw cubic value is larger than a double's;
there is no assertion that the two raw values agree at all depths.
Interactions between different blocks are outside the identity.

\subsection{Why a second bandwidth can read arrangement}
At the $65$-site grid let $c_s=\sum_km_k e^{-2\pi isk/65}$. The half-band row is
\[
 F'=\frac1{33^2}\sum_{s=-32}^{32}(33-|s|)|c_s|^2.
\]
Unlike the full-band matrix, it is not a complete unitary conjugation of the
mark diagonal. Its rows can therefore change with relative positions.
For example two simple atoms at separations $64/65$ and $128/65$ have
$F'=2+2D'(64/65)^2$ and $2+2D'(128/65)^2$, respectively. Here
\[
 D'(64/65)=\frac{\sin(33\pi/65)}{33\sin(\pi/65)},\qquad
 |D'(128/65)|=\frac1{66\cos(\pi/65)},
\]
and the first magnitude exceeds $0.6$ while the second is below $0.02$.
Both full-band Frobenius values are $2$.

A common translation, however, changes neither trace. In particular the
vacancy column $A_0$ has $F'=4128/33$ for \emph{every} vacancy location:
$c_0=64$ and $|c_s|=1$ for $s\not\equiv0\pmod{65}$. Version 1's contrary
sentence about detecting the vacancy location was incorrect.

\subsection{Sine-process budgets and their limited role}
For the intensity-one determinantal sine process and a flat continuum
window $\psi_\lambda(x)=\sin(\pi\lambda x)/(\pi\lambda x)$, direct integration
of the two- and three-point correlation determinants gives
\[
 m_2(\lambda)=\lambda^{-1}+\lambda/3,\qquad
 m_3(\lambda)=1+\lambda^{-2}\quad(0<\lambda\le1).
\]
For clarity, put $u=\lambda^{-1}\one_{[-\lambda/2,\lambda/2]}$,
$b=\one_{[-1/2,1/2]}$ and $t(\alpha)=(1-|\alpha|)_+$. Expanding the
determinants and applying Plancherel yields
\begin{align*}
 m_2&=1+\int u^2-\int t(u*u),\\
 m_3&=1+3\left(\int u^2-\int t(u*u)\right)
       +\int u^3-3\int(t*u)u^2+2\int(b*u)^3.
\end{align*}
For this flat $u$, the five integrals are, in order,
$\lambda^{-1}$, $1-\lambda/3$, $\lambda^{-2}$,
$\lambda^{-1}-1/3$ and $1-\lambda/2$. Substitution proves the formulas.
In particular the centers $4/3,13/6,5$ arise naturally as continuum model
values. These formulas do not identify a finite model's cubic row with a
zeta statistic. Theorem~\ref{thm:main} uses its displayed exact parameters
and does not depend on such an identification.

\section{Two spectral-capacity facts}\label{sec:capacity}
\subsection{Sharp capacity conditional on a genuine ladder}
Let $\mu$ be a positive measure on $[1,\infty)$ representing absolute
escaped eigenvalue heights per unit mass. Suppose
\[
 \int h^2d\mu\le\eta,\qquad
 \mu([h,\infty))\le C h^{-4}\quad(h\ge1).
\]
\begin{theorem}[sharp continuum capacity]\label{thm:capacity}
For $C>0$ and $0<\eta\le2C$,
\[
 \sup_\mu\int h^3d\mu=2\sqrt{2C\eta}.
\]
Equality is attained by density $4Ch^{-5}$ on $(a,\infty)$ and no mass
below $a$, where $a=\sqrt{2C/\eta}$.
\end{theorem}
\begin{proof}
For $a\ge1$, $h^3\le ah^2+(h^3-ah^2)_+$. Tonelli and the tail bound give
\[
 \int h^3d\mu\le a\eta+
 C\int_a^\infty(3h^2-2ah)h^{-4}dh
 =a\eta+2C/a.
\]
The minimizing $a=\sqrt{2C/\eta}$ is at least $1$ and gives the asserted
bound. The displayed density has second moment $2C/a^2=\eta$, third moment
$4C/a=2\sqrt{2C\eta}$, and tail
$C\min(a^{-4},h^{-4})$, proving attainment.
\end{proof}
At fixed $C$ the allowed cubic shift is $o(N)$ if $\eta=o(1)$. The theorem
assumes a bound at every threshold; seven threshold inequalities alone
do not imply its tail estimate. Theorem~\ref{thm:main} verifies its own
finite witness ladder directly, not a ladder for actual zeta matrices.

\subsection{A divergent-cutoff observation in the isolated-spectrum model}
Suppose $N\to\infty$, $V_0\to\infty$, and an isolated-spectrum family has
mass $N$, $F=\kappa N+o(N)$, bounded cubic deficit $\Delta_NN$ with
$0\le\Delta_N\le D$, and a fixed positive fraction of simple positive
atoms. Choose
\[
 h=\min(\lfloor V_0/2\rfloor,\lfloor N^{1/4}\rfloor),
 \qquad k=\lceil\Delta_NN/h^3\rceil.
\]
Add $k$ isolated positive atoms of integer mark $h$ and delete $kh$ simple
atoms. There are enough simples for large $N$, since $kh=o(N)$.
The mass is unchanged, the Frobenius change is $o(N)$, and the cubic
change is $\Delta_NN+o(N)$. Each added eigenvalue is below $V_0$, so rows
seeing only $|\lambda|\ge V_0$ do not see the additions. The distinct and
simple objectives do not increase.

This elementary construction explains why a single divergent-cutoff cubic
tail constraint can be ineffective in that isolated-spectrum setting. It is
not a realization theorem for arbitrary interacting configurations on a
fixed circle; the direct-sum and deficit assumptions are part of the claim.

\section{Pair-containing classes and unresolved scope}\label{sec:limits}
The analytic lower bound in Section~\ref{sec:main} uses atom-only positivity.
For pairs the mixed position couplings can be negative, so that proof cannot
be applied unchanged. A weaker, uniform spectral bound remains available.
For a configuration containing atoms and pairs define
$\Nd=\#\mathrm{atoms}+2\#\mathrm{pairs}$.
Each atom block has positive inertia at most one, and each pair block
$m(pq^*+qp^*)$ also has positive inertia at most one.
Subadditivity gives $n_+(H)\le\#\mathrm{atoms}+\#\mathrm{pairs}\le\Nd$.
Since $3x-x^2\le(9/4)\one_{x>0}$ for real $x$,
\[
 \Nd\ge\frac49(3N-F_1).
\]
At the ideal center $F_1\le(4/3)N$ this gives $\Nd/N\ge20/27$,
which is weaker than the atom-only corner $5/6$. It does not establish
full pair-class absorption.

\paragraph{What numerical dictionary equality means.}
Version 1 reported $940$ small marginal differences on accumulated column
dictionaries under its implemented relative cubic slack, together with
multistart pricing and finite pair-interference tests. Those are numerical
observations about the searched dictionaries. Adding a new column can lower
the base optimum without lowering the augmented optimum: the new column
may violate an added row. An already feasible atom witness does not exclude
this possibility when the true baseline optimum over a larger class is
unknown. Likewise, a finite collection of near-CUE witnesses and heuristic
pricing does not prove global optimality in a pinned class. Neither
full-class nor pinned-class exact absorption is asserted in this revision.

\paragraph{Analytic licensing.}
The signed cubic budget's identification with a sine-process value and the
upgrade from one spectral cutoff to an all-threshold ladder were left open
in version 1. They remain open here. The classical Rudnick--Sarnak support
range does not on its own identify every finite row or justify the required
uniformity. The unconditional pair-correlation theorem of
Baluyot--Goldston--Suriajaya--Turnage-Butterbaugh \cite{BGSTB24} is a
second-order result and supplies no third-order identification. These are
separate from the exact finite atom-only theorem proved above.

\paragraph{Formalization.}
The earlier Lean files at repository tag \texttt{paper-2026-08-28} certify
selected grid Parseval and integer-corner statements. They do not certify
the full LP, the revised algebraic witness, or any unresolved analytic
bridge. The new witness is certified by the accompanying Arb computation;
no fresh Lean build or additional kernel-checked theorem is claimed.

\paragraph{Algebraic versus rational rows.}
The vacancy row $F'=4128/33$ is rational, but the selected columns $A_1,A_2$
have irrational half-band Frobenius values. The companion exact polynomial
check reduces their Fourier numerators modulo $\Phi_{65}$ and obtains
nonconstant remainders. Thus rounding rows to decimal rationals and checking
Fraction arithmetic is not an exact check of the corresponding geometric
configurations. Version 1's checker also allowed positive feasibility grace;
neither device is used for Theorem~\ref{thm:main}.

\section*{Data, code and revision record}
The accompanying \texttt{cubic-augmentation-v2-}\allowbreak\texttt{source-and-checks.zip} contains
this source, exact input data, the portable Arb certificate and output,
an exact cyclotomic row check, requirements, build instructions and a change
record. Run \texttt{python3 certificate.py} in the archive's \texttt{anc/} folder.
The input vectors in Table~\ref{tab:marks} match that file exactly. Historical
version-1 numerical data remain available at the original publication record
and the public repository tag \texttt{paper-2026-08-28}. Those historical
computations are not invoked as exact proof certificates in this revision.
The concept DOI is
\href{https://doi.org/10.5281/zenodo.22171688}{10.5281/zenodo.22171688}.

\begin{thebibliography}{99}
\bibitem{AF26} L. Alp\"oge and R. Furman, \emph{More than two thirds of the
zeta zeros are simple and on the critical line}, arXiv:2608.13637, 2026.
\bibitem{RS96} Z. Rudnick and P. Sarnak, \emph{Zeros of principal
$L$-functions and random matrix theory}, Duke Math. J. \textbf{81} (1996),
269--322; \href{https://doi.org/10.1215/S0012-7094-96-08115-6}{doi:10.1215/S0012-7094-96-08115-6}.
\bibitem{CGG98} J. B. Conrey, A. Ghosh and S. M. Gonek, \emph{Simple zeros
of the Riemann zeta-function}, Proc. London Math. Soc. (3) \textbf{76}
(1998), 497--522; \href{https://doi.org/10.1112/S0024611598000306}{doi:10.1112/S0024611598000306}.
\bibitem{BGSTB24} S. A. C. Baluyot, D. A. Goldston, A. I. Suriajaya and
C. L. Turnage-Butterbaugh, \emph{An unconditional Montgomery theorem for
pair correlation of zeros of the Riemann zeta function}, Acta Arith.
\textbf{214} (2024), 357--376; arXiv:2306.04799.
\bibitem{Arb} F. Johansson, \emph{Arb: efficient arbitrary-precision
midpoint-radius interval arithmetic}, IEEE Trans. Comput. \textbf{66}
(2017), 1281--1292; \href{https://doi.org/10.1109/TC.2017.2690633}{doi:10.1109/TC.2017.2690633}.
\end{thebibliography}
\end{document}