% 278_Addendum_RegistryBurndownI.tex
% Registry burn-down, stages 1-2: retroactive sweep of addenda 35-255
% plus computational quick kills. Six retirements, one scope
% reclassification, three confirmed-load-bearing quantifications.
% Open debt: 58 -> 48. Registry ownership transfers to verify_P278.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}
\newcommand{\Tb}{T_{\rm breath}}

\title{Addendum 278: Registry Burn-down I\\
\large Retroactive sweep and quick kills:\\
six retirements, three confirmed load-bearing, $58 \to 48$}
\author{TOE Corpus}
\date{2026-06-10}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} Mixed by design (6 retired, 1 rescoped, 3 confirmed
load-bearing with quantified blast radius) \quad|\quad
\textbf{Verifier:} \texttt{verify\_P278.py} --- all PASS (now owns
\texttt{verify/tbs\_registry.json})

\begin{abstract}
Stages 1--2 of the registry burn-down (A277). \emph{Stage 1
(retroactive sweep):} addenda 35--255 are swept for unrecorded
closures. Found: A237 explicitly closed TBS P031\_1\_c (recorded
2026-05-24, never registered). Supersessions: TBS P015\_3\_c is retired
by A266/A267 --- the asserted-without-derivation $432$ is now derived
(A267 Thm~1.2), and the $\pi/2$ ``bifurcation correction'' is a
$5.6\%$-wrong proxy for the exact comma gap $432 - \Tb = G_1\Tb =
1.4878$ (the master equation $(432-\pi/2)/\pi = 137.0099$ misses
$\alpha^{-1}$ at $1.9\times10^{-4}$; the exact identity replaces it).
TBS P034\_3\_c (BREATH\_PERIOD ``$\approx 430.5$ s'') is retired by
A265/A273/A275: $\Tb$ is dimensionless (430.51 system units) and its SI
duration is derived, $1.374$ Myr conditional. Open P019\_3 is
superseded by P32's corner-residual route (canonical since; equivalence
of routes remains unproved and is so noted). TBS P021\_2\_c is
reclassified \emph{scope} --- a permanent honesty marker (the paper
states it does not resolve the formal Clay problem), not fixable debt.
\emph{Stage 2 (computational kills):} (i)~TBS P003\_3 retired:
$\rho'(0) = 2\pi \neq 0$ confirms the Neumann misstatement, but P03's
stability proof uses Dirichlet perturbation conditions and its spectrum
$\omega_n = n\pi\sqrt{1-\kappa}$ is reproduced by finite differences to
five digits --- misstatement, zero downstream effect. (ii)~TBS
P029\_1 retired: the remark's corrected peak
$\omega_{\rm peak} = \sqrt{\omega_0^2 - \gamma^2/2}$ is confirmed to
seven digits by direct maximization; the resonance analogy's conclusion
is unaffected. (iii)~TBS P007\_1/P007\_2 \emph{confirmed load-bearing}:
rebuilding the observation operator with the paper's claimed mixed
boundary conditions versus its actual Dirichlet-imposing discretization
shifts $\lambda_1$ by $-4.7\%$ and $\lambda_{2,3}$ by $-16\%/-22\%$ ---
P07's mass-spectrum claims require rework; the blast radius is bounded
(downstream $\mu_1/\mu_0$ uses, including P027\_3\_c, are moments of
$\rho$, not eigenvalues, and are untouched). (iv)~TBS P016\_3
\emph{confirmed, claim fails as stated}: recomputing the cumulative
shell distances for all 51 Mersenne exponents from the paper's own
definition gives counts $1$ ($d_n < 0.01$) and $9$ ($d_n < 0.05$), mean
$0.221$, against claimed $14$, $19$, $0.187$ --- and the recomputed
values show no significant excess over the uniform null. Open debt:
$58 \to 48$, with three of the $48$ now carrying quantified teeth.
\end{abstract}

% ────────────────────────────────────────────────────────────────────────────
\section{Stage 1: Retroactive Sweep}

\begin{proposition}[Status changes by citation]
\label{prop:sweep}
\begin{center}
\begin{tabular}{llll}
\toprule
gap & change & by & note \\
\midrule
TBS P031\_1\_c & retired & A237 & explicit closure, 2026-05-24,
unregistered \\
TBS P015\_3\_c & retired & A266/A267 & 432 derived; $\pi/2$ $\to$ exact
comma $G_1\Tb$ \\
TBS P034\_3\_c & retired & A265/A273/A275 & ``430.5 s'' $\to$
dimensionless $\Tb$; SI derived \\
Open P019\_3 & superseded & P32 & corner-residual route canonical;
equivalence unproved \\
TBS P021\_2\_c & scope & --- & permanent marker, not fixable debt \\
\bottomrule
\end{tabular}
\end{center}
The sweep found no other explicit closures in addenda 35--255; A237
also states P031\_2\_c and P031\_3\_c remain open (registry agrees).
\end{proposition}

% ────────────────────────────────────────────────────────────────────────────
\section{Stage 2: Computational Kills and Confirmations}

\begin{proposition}[P003\_3 retired: misstatement, spectrum intact]
$\rho'(x) = 48\pi^3x^2 + 6\pi^2x + 2\pi$ gives $\rho'(0) = 2\pi \neq 0$:
the stated Neumann condition is indeed false for the background density.
But P03's stability theorem imposes its boundary conditions on the
\emph{perturbation} $\eta$, and its own proof solves
$X'' + \lambda(1-\kappa)^{-1}X = 0$ with $\sin(n\pi x)$ modes ---
Dirichlet. Finite-difference recomputation reproduces
$\lambda_n = (1-\kappa)(n\pi)^2$ to five digits. The remark flagged a
misstatement about the background, not an error in the proof; the
spectrum every downstream addendum uses (A269, A273) is unaffected.
\end{proposition}

\begin{proposition}[P029\_1 retired: correction confirmed]
Maximizing $A(\omega) = F_0/\sqrt{(\omega_0^2-\omega^2)^2 +
\gamma^2\omega^2}$ gives $\omega_{\rm peak} =
\sqrt{\omega_0^2 - \gamma^2/2}$ (confirmed numerically to $10^{-7}$ at
$\gamma/\omega_0 = 0.37$); the remark's correction stands and the
analogy's conclusion (peak at $\approx\omega_0$ for small damping,
architecture not training) is unaffected.
\end{proposition}

\begin{proposition}[P007\_1/P007\_2 confirmed load-bearing; blast radius
bounded]
\label{prop:p07}
The observation operator $\hat O = -d^2/dx^2 + \rho'(x)^2/2\mu_0^2 +
E_{\rm self}\,x^2(1-x)^2$ discretized at $N = 1000$: with the paper's
de facto Dirichlet--Dirichlet conditions, $\lambda_{1,2,3} = (16.51,\
51.24,\ 102.02)$; with its claimed Dirichlet--Neumann conditions,
$(15.73,\ 42.95,\ 79.18)$ --- shifts of $-4.7\%$, $-16.2\%$, $-22.4\%$.
The boundary-condition defect is therefore material to P07's
mass-spectrum claims, which require rework under the correct
discretization. Bounded: the moment ratio $\mu_1/\mu_0 = 0.7933$ and
every downstream use of it (including P027\_3\_c's
$\log(M_{\rm Pl}/m_e)$ formula) involve integrals of $\rho$, not
eigenvalues of $\hat O$, and are untouched by this defect.
\end{proposition}

\begin{proposition}[P016\_3 confirmed: the claim fails as stated]
\label{prop:p16}
With $S_n = \sum_{i\le n} p_i\kappa$, $\kappa = \alpha^{5/4}$, over the
51 known Mersenne prime exponents: $\#\{d_n < 0.01\} = 1$ (claimed 14),
$\#\{d_n < 0.05\} = 9$ (claimed 19), mean $d_n = 0.221$ (claimed
0.187). Under the uniform null the expectations are $\approx 1.0$ and
$\approx 5.1$; the observed values carry no significant excess
($P(\ge 9\,|\,5.1) \approx 0.07$, below the corpus bar). P16's shell
preference claim is unsupported by its own definition; either the
definition in the paper is not the one computed, or the claim is dead.
Status: confirmed load-bearing for P16's headline.
\end{proposition}

\paragraph{Registry state and ownership.} After this addendum: $62$
recorded gaps $=$ $10$ retired, $1$ scope, $3$ confirmed-load-bearing
(P007\_1, P007\_2, P016\_3), $48$ unreviewed-or-open.
\texttt{verify\_P278.py} regenerates the registry (single-writer
discipline; \texttt{verify\_P277.py} now reads and asserts rather than
writes). Stage 3 targets remain: P000\_1\_c (CODATA), P014\_2
(manvantara), P027\_3\_c ($M_{\rm Pl}/m_e$), now joined by the P07
rework and the P16 verdict as named debts with numbers attached.

\paragraph{Verifier.} \texttt{verify\_P278.py}: re-runs every
computation above (P03 spectrum, P29 peak, P07 both spectra and shifts,
P16 recount, P15 master-equation numbers); regenerates the registry
with the full status map; asserts the counts and that every closing
addendum exists on disk. All PASS.

\end{document}
