% 280_Addendum_RegistryBurndownIII.tex
% Run close-out: the P07 rework (mass mapping fails under claimed BCs;
% the de facto Dirichlet fit survives only with a retracted boundary
% interpretation — decision filed as OI-280-1) and the P16 verdict
% (headline refuted under all natural readings; the corpus's first
% formally refuted claim). Verifier: verify_P280.py.
% Copyright Léon Fernando Vlegels -- CC BY 4.0

\documentclass[11pt]{article}
\usepackage{amsmath,amssymb,amsthm}
\usepackage[margin=1in]{geometry}
\usepackage{booktabs}
\usepackage{hyperref}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}

\title{Addendum 280: Registry Burn-down III\\
\large The P07 rework and the P16 verdict}
\author{TOE Corpus}
\date{2026-06-10}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} Case B (one decision forced and filed; one claim
formally refuted) \quad|\quad
\textbf{Verifier:} \texttt{verify\_P280.py} --- all PASS

\begin{abstract}
The two quantified debts from A278, finished.
\emph{P07 rework.} The observation-operator spectrum is recomputed
under the paper's \emph{claimed} boundary conditions ($\psi(0)=0$,
$\psi'(1)=0$) and its \emph{de facto} ones ($\psi(0)=\psi(1)=0$), with
the derived exponent $\beta = 6\mu_1/\mu_0 = 4.7600$ --- no refit is
permitted by the paper's own no-free-parameters claim. Under the de
facto Dirichlet conditions the lepton predictions are
$m_\mu/m_e = 219.6$ (log-error $1.1\%$) and $m_\tau/m_e = 5825$
(log-error $6.3\%$), log-accuracy $96.3\%$, consistent with the
published $94.8\%$. Under the \emph{claimed} conditions the muon
prediction collapses to $119.4$ (log-error $10.3\%$, a tenfold
degradation) while tau barely moves. Conclusion: \textbf{P07's headline
accuracy and its stated boundary physics cannot both be true.} Either
the ``observation is free at the universe boundary'' interpretation is
retracted and $\psi(1) = 0$ adopted (the mass data select Dirichlet:
observation \emph{vanishes} at the universe boundary), or the accuracy
claim falls. The decision is physics, not bookkeeping; filed as
\textbf{OI-280-1} with the data's preference recorded. P007\_1/P007\_2
move to status \emph{reworked}.
\emph{P16 verdict.} The Mersenne shell-preference claim is tested under
four natural readings of its own definition (cumulative $S_n$ as
stated; per-prime $p\kappa$; modular form; exponent variant). All four
fail: best counts $2$/$10$ against claimed $14$/$19$, with means at or
near the uniform null. One forensic note: the per-prime reading
reproduces the claimed \emph{mean} ($0.193$ vs $0.187$) within $3\%$
while its counts still fail --- suggesting the published table mixed
readings or thresholds. Under every reading the headline is
unsupported: \textbf{P016\_3 is the corpus's first formally refuted
claim} (status \emph{refuted}). Honest accounting: refutation of a
canon headline by the canon's own registry discipline is the system
working, not failing. Registry after this addendum: $11$ retired, $1$
superseded, $1$ scope, $2$ reworked, $1$ refuted, $2$
confirmed-load-bearing, $44$ unreviewed-or-open.
\end{abstract}

% ────────────────────────────────────────────────────────────────────────────
\section{The P07 Rework}

\begin{proposition}[Spectrum and predictions under both boundary
conditions]
\label{prop:p07}
With $\hat O = -d^2/dx^2 + \rho'(x)^2/2\mu_0^2 + E_{\rm self}
x^2(1-x)^2$, $N = 1000$, and the derived $\beta = 6\mu_1/\mu_0 =
4.7600$:
\begin{center}
\begin{tabular}{lcccc}
\toprule
BCs & $\lambda_{1,2,3}$ & $m_\mu/m_e$ (206.77) & $m_\tau/m_e$ (3477.4)
& log-acc \\
\midrule
de facto (DD) & $16.51,\ 51.24,\ 102.02$ & $219.6$ ($1.1\%$) & $5825$
($6.3\%$) & $96.3\%$ \\
claimed (DN) & $15.73,\ 42.95,\ 79.18$ & $119.4$ ($10.3\%$) & $2196$
($5.6\%$) & $92.0\%$ \\
\bottomrule
\end{tabular}
\end{center}
(Errors are log-space relative, the metric consistent with P07's
published $94.8\%$.) The muon prediction degrades tenfold under the
paper's own claimed boundary conditions, and $\beta$ is derived, so no
refit can absorb the change.
\end{proposition}

\begin{remark}[The forced decision --- OI-280-1]
The accuracy claim and the boundary interpretation are incompatible.
The mass data select the Dirichlet condition $\psi(1) = 0$:
observation \emph{vanishes} at the universe boundary rather than being
``free'' there --- which is, note, the more natural reading of the
corpus's own edge layer (the boundary is where observation runs out,
P04's edge fraction). But adopting it requires retracting P07's stated
interpretation, and that retraction is a physics decision the canon
must make explicitly, not a patch an addendum may apply silently
(Butlerian line). \textbf{OI-280-1}: derive the boundary condition of
the observation operator from corpus primitives; the data's preference
(DD) is recorded, the prose's claim (DN) is quantified as untenable
with the published accuracy.
\end{remark}

% ────────────────────────────────────────────────────────────────────────────
\section{The P16 Verdict}

\begin{proposition}[Refutation under all natural readings]
\label{prop:p16}
For the 51 known Mersenne prime exponents, with $\kappa =
\alpha^{5/4}$, the counts $(\#\,d_n<0.01,\ \#\,d_n<0.05,\
\overline{d_n})$:
\begin{center}
\begin{tabular}{lccc}
\toprule
reading & $<0.01$ & $<0.05$ & mean \\
\midrule
cumulative $S_n = \sum p_i\kappa$ (as stated) & 1 & 9 & 0.221 \\
per-prime $p\kappa$ & 2 & 10 & 0.193 \\
modular $p \bmod \kappa^{-1}$ & 2 & 10 & 0.193 \\
exponent variant $p\kappa^{4/5}$ & 1 & 9 & 0.261 \\
\midrule
claimed (P16 Table) & 14 & 19 & 0.187 \\
uniform null & $\approx 1.0$ & $\approx 5.1$ & 0.25 \\
\bottomrule
\end{tabular}
\end{center}
No reading reproduces the claimed counts; the strongest excess
($10$ vs $5.1$ expected at the $0.05$ threshold) has $p \approx 0.03$
before any look-elsewhere correction across readings and thresholds,
below the corpus bar. Forensic note: the per-prime mean matches the
claimed $0.187$ within $3\%$, suggesting the published table mixed a
per-prime mean with counts from an unstated computation. The headline
claim --- Mersenne exponents prefer integer shells --- is refuted as
stated. P16's downstream uses (the ``3 measurements of $\kappa$''
analogy table in P07 \S2) inherit a caveat, not a collapse: nothing
else in the corpus cites the shell counts quantitatively.
\end{proposition}

\begin{remark}[On refuting one's own canon]
This is the registry discipline functioning as designed: a claim was
recorded, recomputed from its own definition, scanned across charitable
readings, and refused. A corpus that can only confirm itself measures
nothing (A272's never-empty corollary, applied to the corpus itself:
a school whose students never stall has stopped measuring). The
refutation is filed with the same verifier rigor as the cycle's
theorems.
\end{remark}

\paragraph{Registry.} P007\_1, P007\_2 $\to$ \emph{reworked}
(OI-280-1 pending); P016\_3 $\to$ \emph{refuted}. Totals: $11$
retired, $1$ superseded, $1$ scope, $2$ reworked, $1$ refuted, $2$
confirmed-load-bearing (P000\_1\_c, P027\_3\_c --- the $\alpha$-comma
and the mass-hierarchy constant, both with OIs), $44$
unreviewed-or-open. The burn-down run (A277 registry $\to$ A278 stages
1--2 $\to$ A279 stage 3 $\to$ this addendum) closes.

\paragraph{Verifier.} \texttt{verify\_P280.py}: both spectra and all
four predictions with log-errors; the $\beta$ derivation; the tenfold
muon degradation; all four P16 readings with counts and means; the
uniform-null comparison; registry regeneration and totals. All PASS.

\end{document}
