% 277_Addendum_CommaAuditRegistry.tex
% TASK_035 chains 3-4: the comma near-miss audit (the quarter-comma flag
% reduces exactly to T_b vs 2^(35/4), 62 ppm; the schisma flag was
% demoted by A273; watchlist now one item) and the full corpus gap
% registry (51 TBS + 11 Open remarks, machine-readable, statuses
% recorded). Verifier: verify_P277.py (regenerates the registry).
% 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 277: The Comma Audit and the Gap Registry\\
\large TASK\_035 chains 3--4: one flag survives, sharpened;\\
62 recorded gaps, indexed}
\author{TOE Corpus}
\date{2026-06-10}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} Case B$+$ (audit complete; reduction theorem exact;
flag survives as flag; registry built) \quad|\quad
\textbf{Task:} TASK\_035 (chains 3--4, final) \quad|\quad
\textbf{Verifier:} \texttt{verify\_P277.py} --- all PASS (regenerates
\texttt{verify/tbs\_registry.json})

\begin{abstract}
\emph{Chain 3 (comma audit).} A266 flagged two near-misses. The
schisma flag was demoted by A273 ($1 - \sqrt{1-\kappa}$ is a unit
conversion residue, not a comma). The quarter-Pythagorean-comma flag
($(3^{12}/2^{19})^{1/4} - 1 = 0.3394\%$ vs $G_1 = 0.3456\%$, $1.8\%$
relative) is audited here and \emph{reduced}:
Proposition~\ref{prop:reduction} shows $G_1 = $ quarter-comma is
algebraically equivalent to $\Tb = 2^{35/4}$, and numerically
$\log_2 \Tb = 8.749910\ldots$ vs $35/4 = 8.75$ --- a $62$ ppm
near-miss, $290\times$ sharper than the flag it replaces. No corpus
derivation assigns meaning to the exponent $35/4$; per PSLQ-honesty the
item remains a \emph{flag, not a claim}, and the speculation that the
two A266 flags were ``two rungs of one temperament'' is closed
negatively: one was a comma flag (still open), the other a unit
conversion (demoted) --- they were never siblings. The comma watchlist
after this audit contains exactly one item.
\emph{Chain 4 (gap registry).} The canon carries $51$ TBS remarks and
$11$ Open remarks ($62$ recorded gaps). They are extracted into a
machine-readable registry
(\texttt{verify/tbs\_registry.json}, regenerated by the verifier on
every run) with status per item. Recorded closures to date: TBS
28\_2 (retired, A273), TBS 28\_3\_c (retired, A263), TBS 14\_1
(retired, A266), Open 27\_2 (retired, A276). $58$ items stand
unreviewed-or-open --- the honest number behind ``the canon's joints.''
Registry discipline is adopted: any future addendum that closes a
recorded gap updates the registry in the same commit; the verifier
fails if a status cites an addendum that does not exist on disk.
\end{abstract}

% ────────────────────────────────────────────────────────────────────────────
\section{Chain 3: The Comma Audit}

\begin{proposition}[Reduction of the quarter-comma flag]
\label{prop:reduction}
$G_1 = (3^{12}/2^{19})^{1/4} - 1$ holds if and only if
$\Tb = 2^{35/4}$. Proof: $G_1 + 1 = 432/\Tb$ (A266, exact) and
$(3^{12}/2^{19})^{1/4} = 3^3/2^{19/4}$; equality reads
$432/\Tb = 27/2^{19/4}$, i.e.\
$\Tb = 432 \cdot 2^{19/4}/27 = 2^4 \cdot 2^{19/4} = 2^{35/4}$.
Numerically $2^{35/4} = 430.5390$, $\Tb = 430.5122$: relative deviation
$6.21 \times 10^{-5}$; equivalently $\log_2 \Tb = 8.7499105$ vs
$8.75$. The $1.8\%$ comma-level flag is exactly the $62$ ppm statement
``$\pi\alpha^{-1}$ is nearly $2^{35/4}$.''
\end{proposition}

\begin{remark}[Verdict: flag, sharpened, not promoted]
The reduction is exact; the near-miss is not. $62$ ppm is far below the
corpus's identification bar absent a derivation, and no corpus object
supplies the exponent $35/4$ (the $5/4$ of $\kappa = \alpha^{5/4}$ is
not it; $35/4 = 7 \cdot 5/4$ is recorded as an observation only).
Status: the comma watchlist now contains exactly one item ---
\emph{$\Tb$ vs $2^{35/4}$ at $62$ ppm} --- replacing the
quarter-comma/$G_1$ entry. The A266 speculation that the two flags were
two rungs of one temperament is closed negatively: A273 typed the
schisma-adjacent quantity as a conversion residue; the surviving flag
is of a different kind entirely.
\end{remark}

% ────────────────────────────────────────────────────────────────────────────
\section{Chain 4: The Gap Registry}

\begin{proposition}[Inventory]
\label{prop:inventory}
The canon (Papers 00--34) carries exactly $51$ remarks labelled
\texttt{rmk:tbs:*} and $11$ labelled \texttt{rmk:open:*}: $62$ recorded
gaps. Status at filing:
\begin{center}
\begin{tabular}{llll}
\toprule
gap & paper & status & closed by \\
\midrule
TBS P028\_2 & P28 & retired & A273 (unit bridge) \\
TBS P028\_3\_c & P28 & retired & A263 (velocity identification) \\
TBS P014\_1 & P14 & retired & A266 ($G_1$ exact; $\kappa^{0.6}$
retired) \\
Open P027\_2 & P27 & retired & A276 (section law, not invertibility) \\
remaining $58$ & --- & unreviewed-or-open & --- \\
\bottomrule
\end{tabular}
\end{center}
The registry \texttt{verify/tbs\_registry.json} holds every item with
paper, label, excerpt, and status; \texttt{verify\_P277.py} regenerates
it from the canon sources on every run, so it cannot silently drift
from the papers.
\end{proposition}

\begin{remark}[Registry discipline]
Adopted as corpus convention: (i) an addendum that closes a recorded
gap updates the registry status in the same change; (ii) the verifier
fails if any status cites a closing addendum absent from disk; (iii)
the unreviewed count is reported on every run --- the canon's debt is
now a number that can only change visibly. The four recorded closures
all date to the current cycle (A263--A276); the $58$ open items are the
inventory TASK\_035 chain 4 asked for, and several are already known to
be load-bearing (P014\_2's manvantara identification; P000\_1\_c's
CODATA comparison; P003\_3's boundary condition). No triage is
performed here; the registry is the deliverable.
\end{remark}

\paragraph{Consequences.} TASK\_035 closes (chains 1--2: A266/A267;
chain 3: this audit; chain 4: this registry). The corpus's open-gap
state is machine-readable for the first time. The comma watchlist is
one item long.

\paragraph{Verifier.} \texttt{verify\_P277.py}: regenerates the
registry from canon sources; asserts the counts ($51$ TBS, $11$ Open),
the four recorded closures (and that their closing addenda exist on
disk with passing-verifier convention); the reduction identity at 50
digits; the $62$ ppm deviation; the schisma demotion cross-reference;
watchlist cardinality. All PASS.

\paragraph{Note (added 2026-06-10, same day).} Registry ownership
transferred to \texttt{verify\_P278.py} (Burn-down I): the latest
burn-down verifier is the registry's single writer;
\texttt{verify\_P277.py} now reads and asserts. A278's sweep moved the
debt $58 \to 48$ (six new retirements, one scope reclassification,
three confirmed-load-bearing with quantified blast radius).

\end{document}
