% 279_Addendum_RegistryBurndownII.tex
% Stage 3 of the registry burn-down: the three load-bearing items.
% P000_1_c: the alpha-comma named and typed (2.2234 ppm; exactness would
% contradict A267). P014_2: retired — 432/pi^3 = (1+G1) Omega/pi^2
% exactly; the E_self identification is structurally false. P027_3_c:
% formula exact-restated; P018_1's factor-138 identified as mu0 = Omega
% (retired). Verifier: verify_P279.py (registry via tbs_registry_gen).
% Copyright Léon Fernando Vlegels -- CC BY 4.0

\documentclass[11pt]{article}
\usepackage{amsmath,amssymb,amsthm}
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\usepackage{booktabs}
\usepackage{hyperref}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newcommand{\Tb}{T_{\rm breath}}

\title{Addendum 279: Registry Burn-down II\\
\large Stage 3: the $\alpha$-comma, the manvantara decomposition,\\
and the mass-hierarchy formula audit}
\author{TOE Corpus}
\date{2026-06-10}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} Mixed by design (2 retired with exact results; 2
typed and sharpened with OIs; 1 flag recorded fragile) \quad|\quad
\textbf{Verifier:} \texttt{verify\_P279.py} --- all PASS (registry via
\texttt{tbs\_registry\_gen.py}, the new single source)

\begin{abstract}
The three Stage-3 targets named in A277/A278, plus one bonus closure.
\emph{P000\_1\_c (CODATA) --- typed, named, kept open with teeth.}
$\Omega = 4\pi^3+\pi^2+\pi = 137.0363038$ exceeds CODATA-2018
$\alpha^{-1} = 137.035999084(21)$ by $3.0469\times10^{-4}$ ---
$2.2234$ ppm, roughly $1.45\times10^4$ times the experimental
uncertainty. The identity $\alpha^{-1} = \Omega$ is therefore
\emph{not} exact against the measured constant, and the corpus's own
theorem says it should not be: an exact closure here would be the
unique exact physical closure in a canon whose A267 proves closures do
not happen. The residual is hereby named the \emph{$\alpha$-comma};
its magnitude is underived (\textbf{OI-279-1}). A QED-running reading
fails directionally ($\Omega$ exceeds the Thomson value; running below
$m_e$ cannot raise $\alpha^{-1}$). Small-combination scan: $\kappa/7$
matches the 2018 deviation at $8$ ppm but degrades to $300$ ppm against
CODATA-2022 --- the fragility is itself evidence; flag recorded, not
promoted ($\kappa^2/2$ off $2.3\%$, same verdict).
\emph{P014\_2 (manvantara) --- retired, Case A.} The exact decomposition
\begin{equation*}
\frac{432}{\pi^3} \;=\; (1+G_1)\,\frac{\Omega}{\pi^2}
\;=\; \Bigl(4\pi + 1 + \tfrac{1}{\pi}\Bigr)(1+G_1) \;=\; 13.93266
\end{equation*}
holds identically (both steps algebraic). The claimed identification
with $E_{\rm self} = 4\pi + \Delta E \approx 13.177$ is structurally
false, not approximately true: the additive parts differ by the factor
$(1+1/\pi)/\Delta E = 2.16$. The $\approx 14$ proximity that fed the
manvantara reading belongs to $\Omega/\pi^2$, not to $E_{\rm self}$.
\emph{P027\_3\_c ($M_{\rm Pl}/m_e$) --- sharpened.} The formula is
exact-restated as $\ln(M_{\rm Pl}/m_e) = (3\pi/20)\,(\mu_1/\mu_0)\,
\Omega\,(1-\mu_1\alpha^2)^{-1}$: a dimensionless moment ratio times
$\alpha^{-1}$, dressed. It evaluates to $51.5299$ against measured
$51.5278$ ($3.9\times10^{-5}$). What remains underived is the constant
$3\pi/20$; the decomposition $3\pi/20 = (3/4)\cdot(\pi/5)$ (P10's
three-fourths correction times the quintic fifth) is recorded as a flag
(\textbf{OI-279-2}). \emph{Bonus: P018\_1 retired.} Its
factor-$\sim$138 mystery (integral evaluates to $0.374$, claim says
$51.53$) is resolved exactly: the integral computes
$(3\pi/20)(\mu_1/\mu_0) = 0.37385$ and the published number carries the
$\mu_0 = \Omega$ normalization --- $0.37385 \times \Omega = 51.23$,
dressed $51.53$. The numerical inconsistency was a missing
normalization factor, now documented. Debt: $48 \to 44$ unreviewed,
with the registry's status map moved to a shared module
(\texttt{tbs\_registry\_gen.py}) so burn-down verifiers assert their
own contributions and never fight over global counts.
\end{abstract}

% ────────────────────────────────────────────────────────────────────────────
\section{P000\_1\_c: The $\alpha$-Comma}

\begin{proposition}[The deviation, stated precisely]
$\Omega - \alpha^{-1}_{\rm CODATA18} = 3.04692\times10^{-4}$
($2.2234$ ppm relative; $\approx 1.45\times10^4\,\sigma_{\rm exp}$).
Against CODATA-2022 ($137.035999177(21)$): $3.04599\times10^{-4}$
($2.2228$ ppm). The geometric identity is accurate to $2.2$ ppm and is
not exact against the measured constant at any current edition.
\end{proposition}

\begin{remark}[Why exactness would be wrong]
A267 (Necessity of Detuning) makes every identification of a corpus
quantity with an externally fixed value carry a nonzero residual. Had
$\Omega$ matched CODATA to experimental precision, the corpus would
contain exactly one exact physical closure --- and its own central
theorem would indict it. The $2.2$ ppm residual is thus typed: it is
the \emph{$\alpha$-comma}, the detuning of the corpus's most
external-facing chord. What A267 does not supply is the magnitude;
\textbf{OI-279-1}: derive the $\alpha$-comma's size from a closure map
of the A276 family (candidate: the measurement itself as the closure
--- the Thomson-limit dressing of the bare geometric coupling ---
but a QED-running reading fails directionally, since $\alpha^{-1}$
cannot run upward below $m_e$; whatever dressing applies is not
textbook running).
\end{remark}

\begin{remark}[Scan results --- flags only]
$\kappa/7 = 3.04689\times10^{-4}$ matches the 2018 deviation to $8$ ppm
--- and the 2022 deviation only to $3\times10^{-4}$. A match that
cannot survive a CODATA revision is not physics; recorded as a fragile
flag, with the fragility as the verdict. $\kappa^2/2$ misses by
$2.3\%$. Neither is promoted; the look-elsewhere budget for
one-small-constant-over-small-integer matches makes $8$ ppm unexciting.
\end{remark}

% ────────────────────────────────────────────────────────────────────────────
\section{P014\_2: The Manvantara Decomposition (Retired)}

\begin{theorem}[Exact decomposition; the identification is false]
$432/\pi^3 = (1+G_1)\,\Omega/\pi^2$ identically (substitute
$1+G_1 = 432/\Tb$, $\Tb = \pi\Omega$), and
$\Omega/\pi^2 = 4\pi + 1 + 1/\pi$ identically. Hence
$432/\pi^3 = 13.93266$, of which $13.88468$ is the pure geometric part
and the factor $(1+G_1)$ is the 432-comma. The claimed identification
$432/\pi^3 = E_{\rm self}$ fails structurally: writing both as $4\pi +
(\text{excess})$, the excesses are $1 + 1/\pi = 1.3183$ versus
$\Delta E = 0.6106$ --- a factor $2.16$, not a residual. P14's $5.7\%$
``discrepancy'' was the distance between two different objects, not an
error in one. The $\approx 14$ proximity ($-0.48\%$) belongs to
$\Omega/\pi^2$; whatever the manvantara count encodes, it is the
corpus's $\Omega/\pi^2$, not its self-lensing energy. Retired.
\end{theorem}

% ────────────────────────────────────────────────────────────────────────────
\section{P027\_3\_c and P018\_1: The Mass-Hierarchy Formula}

\begin{proposition}[Exact restatement; P018\_1 resolved]
The P27 formula is identically
\begin{equation}
\ln\!\frac{M_{\rm Pl}}{m_e} \;\stackrel{?}{=}\;
\frac{3\pi}{20}\,\frac{\mu_1}{\mu_0}\,\Omega\,
\frac{1}{1-\mu_1\alpha^2},
\qquad \mu_0 = \Omega,\quad
\mu_1 = \tfrac{16\pi^3}{5}+\tfrac{3\pi^2}{4}+\tfrac{2\pi}{3},
\end{equation}
evaluating to $51.5299$ versus measured $51.5278$
($3.9\times10^{-5}$ relative). P018\_1's factor-$\sim$138 discrepancy
is resolved exactly: the normalisation integral computes
$(3\pi/20)(\mu_1/\mu_0) = 0.37385$ (the reported $0.374$), and the
published $51.53$ carries the $\mu_0 = \Omega$ factor; the
inconsistency was an undocumented normalization, not a wrong integral.
P018\_1 retired.
\end{proposition}

\begin{remark}[What remains underived]
The constant $3\pi/20$. The decomposition $3\pi/20 = (3/4)\cdot
(\pi/5)$ --- P10's three-fourths correction times a quintic fifth
(P12/P13) --- is exact as arithmetic and unproven as provenance;
recorded as a flag. \textbf{OI-279-2}: derive $3\pi/20$ from the
observation-operator moment structure, or retire the formula as a fit.
Until then P027\_3\_c stands confirmed-load-bearing: one underived
constant buying a $3.9\times10^{-5}$ hit on a 22-decade mass ratio is
either deep or a two-parameter fit, and the corpus does not yet know
which. (P007\_3, the same complex, inherits this status; note that
A278 bounded the blast radius --- these are moment quantities,
untouched by P07's BC defect.)
\end{remark}

\paragraph{Registry.} After this addendum: $11$ retired, $1$
superseded, $1$ scope, $5$ confirmed-load-bearing, $44$
unreviewed-or-open (debt $48 \to 44$). Status map and generation moved
to \texttt{verify/tbs\_registry\_gen.py} (single source); burn-down
verifiers assert their own contributions only.

\paragraph{Verifier.} \texttt{verify\_P279.py}: the deviation against
both CODATA editions; the $\kappa/7$ fragility; the exact decomposition
at 50 digits and the factor-$2.16$ failure; the formula evaluation, the
measured value, and the $0.374 \times \Omega$ resolution; the
$(3/4)(\pi/5)$ arithmetic; registry regeneration and counts. All PASS.

\end{document}
