% Paper 40: Repairs and Retractions
% PROPOSED CANON (assembled 2026-06-10 from A277-A280, A293, and the
% repair addenda A270/273/274/281/282/294; subject to review/rollback).
% The corpus's record of correcting itself: the gap registry, fourteen
% retirements, the retractions of interpretation, two refutations, the
% standing diagnoses, and the discipline that produced them — filed
% with the same rigor as the theorems, because a corpus that can only
% confirm itself measures nothing.
% Verifiers: verify_P277/278/279/280/293.py + tbs_registry_gen.py
% (registry single source; all green at filing).
% 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{\Tbe}{T_{\rm breath}}

\title{Paper 40: Repairs and Retractions\\
\large The gap registry, the corrections, the refutations,\\
and the discipline that produced them}
\author{L\'{e}on Fernando Vlegels}
\date{2026-06-11}

\begin{document}
\maketitle

\begin{abstract}
A corpus that can only confirm itself measures nothing. This paper is
the canonical record of the corpus measuring itself. The canon carries
$65$ recorded gaps: $54$ remarks typed \emph{TBS} (to be shown) and
$11$ typed \emph{Open}, written into the papers themselves as
first-class, machine-findable objects. Addendum 277 extracted them
into a registry (\texttt{verify/tbs\_registry.json}), regenerated from
the \LaTeX\ sources on every verifier run by a single writer
(\texttt{tbs\_registry\_gen.py}), under one hard rule: a status that
cites a closing document fails the build if that document is absent
from disk. A burn-down followed, on the record: A277 created the
registry with $58$ items unreviewed; A278 swept the prior addenda and
ran computational kills ($58 \to 48$); A279 settled the named
load-bearing items ($48 \to 44$); A280 forced the P07 decision and
delivered the corpus's first formal refutation; A293 settled the
severe items ($44 \to 38$); A295 classified everything that remained
($38 \to 0$ unreviewed). The final classification, quoted from the
registry: $15$ retired, $1$ superseded, $21$ scope, $4$ refuted,
$21$ confirmed-load-bearing, $0$ unreviewed; alongside these, $327$
verifier-documented items record every expected-fail check in the
paper-series verifiers, the registry in executable form. A later
addendum (A344) added three flavor-sector refutations from Paper 01,
bringing the refuted count to $7$. The refuted items are told here as
full case studies: P016\_3 (the
Mersenne shell statistics, irreproducible under all four natural
readings of their own definition), P020\_1 (the electroweak VEV
conjecture, off by fourteen orders of magnitude as stated),
P017\_2 with P017\_3\_c (the Dirac spectra: the printed system is
ill-posed, and the natural repair produces box spectra with no
factor of seven and no $0.891$), and P001\_4/P001\_5/P001\_6 (the
tri-bimaximal $\theta_{13} = 0$, the CP phase $\delta_{\mathrm{CP}}
\approx 240^\circ$, and the topological strong-CP argument, none of
which survive current data). The standing diagnoses, the
distinction between retired, refuted, and scope, and the
living-ledger property of this paper close the record. No claim in
the corpus, including the seed identity itself, is protected from
its own registry.
\end{abstract}

% ════════════════════════════════════════════════════════════════════
\section{Introduction: Why a Corpus Should Audit Itself}

A body of claims earns trust by what it is able to lose. A theory
that confirms everything it touches has not been tested by anything
it touches; its agreement with the world is a property of its
bookkeeping, not of the world. This is an old point, and it is
usually applied from outside: referees, replications, adversarial
collaborations. The corpus applies it from inside. It carries its own
list of weak points, written into the papers at the places where the
weakness lives, in a fixed form that a program can find; it maintains
a machine-readable ledger of those weak points and their fates; and
it accepts, as a matter of standing policy, that some of its own
headlines will die when the ledger is worked through. Several did.
This paper records how, and what the verdicts were.

The need is structural, not decorative. The corpus makes three kinds
of statement: derivations, which close or do not; identifications of
corpus quantities with measured constants, which carry residuals; and
conjectures, which are scaffolding until derived. All three kinds
were written down, in places, with known soft spots: a boundary
condition asserted and not derived, a constant matched and not
explained, a table published and not recomputed. A corpus that kept
such spots in margins or memory would slowly confuse the proven with
the familiar. The discipline described here exists to make that
confusion impossible: every soft spot is a tracked object, every
tracked object has a status, and every status change cites the
document that caused it or the build fails.

Two commitments shape this paper. First, it contains no new claims.
Every number is quoted from the registry
(\texttt{addenda/verify/tbs\_registry.json}) or from the addendum
that produced it; the paper describes the discipline and records the
verdicts, exactly as the registry holds them. Second, the ledger is
allowed its dates. The corpus's prose elsewhere avoids chronicle;
here the dated entries and addendum numbers are the content, because
a record of self-correction is precisely a record of when, what, and
by which document.

% ════════════════════════════════════════════════════════════════════
\section{The Discipline}

\subsection{The remark convention}

Gaps are recorded in the canon papers themselves as theorem-class
objects: a \texttt{remark} environment typed \texttt{[TBS]} (to be
shown) or \texttt{[Open]}, carrying a label of the form
\texttt{rmk:tbs:PNNN\_k} or \texttt{rmk:open:PNNN\_k}, where
\texttt{PNNN} names the paper and \texttt{k} indexes the gap within
it. A TBS remark marks a statement the paper uses but does not
establish; an Open remark marks a question the paper raises and
leaves. Both sit in the body of the paper, adjacent to the claim they
qualify, in the same typographic register as the theorems. The
convention makes the gap a first-class citizen of the document: it
cannot be lost in a footnote, it survives every edit that preserves
the label, and it is findable by a regular expression. Papers 00--34
carry $54$ TBS remarks and $11$ Open remarks: $65$ recorded
gaps. (Of the TBS remarks, the three latest --- P001\_4, P001\_5,
P001\_6 --- were added to Paper 01 by A344 to record three
flavor-sector predictions as refuted.)

\subsection{The registry and its single writer}

Addendum 277 (2026-06-10) extracted the $62$ remarks into
\texttt{verify/tbs\_registry.json}: one record per gap, holding the
paper, the label, a $200$-character excerpt, a status, the closing
document where one exists, and a note. The registry is not edited by
hand. It is regenerated from the \LaTeX\ sources on every verifier
run, so it cannot silently drift from the papers: if a remark is
added, removed, or relabelled, the next run reflects it.

Ownership was deliberately narrowed as the registry matured. A277's
verifier wrote the file first. The same day, ownership transferred to
the Burn-down~I verifier (\texttt{verify\_P278.py}), with A277's
verifier demoted to reading and asserting. A279 moved the status map
and the generation logic into a shared module,
\texttt{tbs\_registry\_gen.py}, which has been the single writer
since: burn-down verifiers import it, regenerate the registry, and
assert only their own contributions. No verifier owns the global
counts, so two verifiers can never fight over them, and a later
correction never breaks an earlier verifier's record. The
expectations are evolution-tolerant by construction.

\subsection{The closure rule}

\begin{proposition}[Registry discipline, adopted as canon (A277)]
(i) An addendum that closes a recorded gap updates the registry
status in the same change. (ii) A status that cites a closing
document fails the build if the cited document does not exist on
disk. (iii) The unreviewed count is reported on every run. The
canon's debt is therefore a number that can only change visibly:
nothing is closed by forgetting, and nothing is closed by assertion
without a citable document behind the assertion.
\end{proposition}

The second clause is the load-bearing one. A ledger whose entries can
point at nothing is a list of moods. Requiring the closing document
to exist, and requiring the build to fail otherwise, converts every
closure into a checkable cross-reference: the reader who doubts a
retirement can open the addendum it cites, and the build system has
already verified that there is an addendum to open.

% ════════════════════════════════════════════════════════════════════
\section{The Burn-Down}

The registry was created with $58$ of its $62$ items unreviewed. It
was driven to zero in seven documented stages over two days,
listed here by addendum number. The
trajectory of the unreviewed count: $58 \to 48 \to 44 \to 44 \to 38
\to 0$.

\textbf{A277 (2026-06-10), creation.} The inventory: $51$ TBS, $11$
Open. Four closures already on record were registered retroactively:
P028\_2 (closed by A273, the unit bridge), P028\_3\_c (A263, the
velocity identification), P014\_1 (A266, the exact comma $G_1$
replacing $\kappa^{0.6}$), and Open P027\_2 (A276, the section law
replacing the invertibility objection). Unreviewed: $58$.

\textbf{A278 (2026-06-10), Burn-down I.} Stage 1 swept addenda 35--255
for closures that had never been registered: P031\_1\_c had been
explicitly closed by A237 on 2026-05-24; P015\_3\_c was retired by
A266/A267 (the asserted $432$ now derived, the $\pi/2$ correction
identified as a $5.6\%$-wrong proxy for the exact comma gap
$432 - \Tbe$); P034\_3\_c was retired by A265/A273/A275 (the quoted
``$430.5$ s'' replaced by the dimensionless $\Tbe$ with the SI
duration derived separately); Open P019\_3 was superseded by P32's
corner-residual route, with the equivalence of routes noted as
unproved; and P021\_2\_c was reclassified \emph{scope}, a permanent
honesty marker rather than fixable debt. Stage 2 ran computational
kills: P003\_3 retired (the Neumann misstatement is real, the proof
is Dirichlet, the spectrum is reproduced to five digits, downstream
effect zero); P029\_1 retired (the corrected resonance peak
$\sqrt{\omega_0^2 - \gamma^2/2}$ confirmed to seven digits);
P007\_1/P007\_2 confirmed load-bearing with quantified teeth
(rebuilding the observation operator under the claimed versus de
facto boundary conditions shifts $\lambda_{1,2,3}$ by $-4.7\%$,
$-16.2\%$, $-22.4\%$); and P016\_3 confirmed failing as stated (the
recomputed Mersenne counts are $1$ and $9$ against the claimed $14$
and $19$). Unreviewed: $48$.

\textbf{A279 (2026-06-10), Burn-down II.} The three named
load-bearing targets. P000\_1\_c: the deviation of
$\Omega = 4\pi^3 + \pi^2 + \pi$ from the measured $\alpha^{-1}$ was
computed precisely ($2.2234$ ppm), named the $\alpha$-comma, and kept
open with an objective. P014\_2: retired by exact decomposition,
$432/\pi^3 = (1+G_1)\,\Omega/\pi^2$ identically; the claimed
identification with $E_{\rm self}$ is structurally false (the
additive excesses differ by a factor $2.16$). P027\_3\_c: the
mass-hierarchy formula exact-restated and sharpened, with the
constant $3\pi/20$ isolated as the underived part. Bonus: P018\_1
retired, its factor-$\sim$138 mystery resolved exactly as the
$\mu_0 = \Omega$ normalization ($0.37385 \times \Omega = 51.23$,
dressed $51.53$). The status map moved to the shared generator in
this addendum. Unreviewed: $44$.

\textbf{A280 (2026-06-10), Burn-down III.} The two quantified debts
from A278, finished. P007\_1/P007\_2 moved to \emph{reworked}: the
mass predictions and the stated boundary physics cannot both be true,
and the decision was filed as OI-280-1 rather than patched silently.
P016\_3 moved to \emph{refuted}: the corpus's first formally refuted
canon claim (Section~\ref{sec:p16}). Unreviewed: $44$
(reclassifications only).

\textbf{A281--A283 (2026-06-10), repairs between burn-downs.} A281
derived the observation operator's boundary condition from corpus
primitives, resolving OI-280-1; P007\_1 and P007\_2 moved from
reworked to retired, with the prose retraction recorded (observation
vanishes at the universe boundary rather than being free there; the
mass data select the corrected reading). A282 found the provenance of
the mass-hierarchy formula in P03 and sharpened P027\_3\_c's note.
A283 typed the $\alpha$-comma's mechanism and sharpened P000\_1\_c's
note.

\textbf{A293 (2026-06-10), Burn-down IV.} The severe items. P015\_1
retired by repair: the claimed $\langle|\cos\theta|\rangle = 1/4$ is
wrong (the spherical average is exactly $1/2$), but the paper's
result $\bar A_{\rm shadow} = S/4$ is Cauchy's projection formula,
the missing factor of $2$ being the front-back double count.
P020\_1 refuted (Section~\ref{sec:p20}). P017\_3\_c confirmed with
the diagnosis the remark lacked: the printed radial Dirac system has
complex indicial exponents at the origin and is ill-posed as printed.
P013\_1, P013\_2, P013\_3\_c confirmed: the quintic is a
conjecture-schema (Section~\ref{sec:diagnoses}). Unreviewed: $38$.

\textbf{A295 (2026-06-11), pre-publication closure.} The P17 repair
was performed and the printed spectrum was not recovered: P017\_3\_c
upgraded from diagnosed to refuted-as-stated, and P017\_2 (the factor
of seven) refuted with it (Section~\ref{sec:p17}). P018\_2 retired
(its normalization conflict is the $\mu_0$ factor documented in
A279/A282). Every remaining item was classified: the application
papers' honesty remarks to \emph{scope}, the formalization gaps to
\emph{confirmed-load-bearing} with notes, two items settled with
quantified diagnoses (P002\_1: the flagged conflation is real and
numerically benign at $1.2\%$; P033\_2: the formula
${\rm Vol}(S^{d-1}) = \pi^d$ is confirmed false at every $d$, with
correct values supplied). Unreviewed: $0$. Nothing in the corpus is
both open and unexamined.

\textbf{A344 (2026-06-15), flavor-sector refutations.} After the
burn-down closed, three predictions of Paper 01 were tested against
current data and recorded as refuted (Section~\ref{sec:flavor}): the
tri-bimaximal reactor angle $\theta_{13} = 0$ (P001\_4), excluded by
the measured $\theta_{13} \approx 8.5^\circ$; the CP phase
$\delta_{\mathrm{CP}} \approx 240^\circ$ (P001\_5), now $\sim 3\sigma$
above the NuFIT-6.0 central value $177^\circ$; and the topological
strong-CP argument (P001\_6), which constrains the instanton winding
number, not the Lagrangian $\theta$-coupling. Three TBS remarks were
added to Paper 01, raising the recorded-gap total to $65$ and the
refuted count to $7$.

% ════════════════════════════════════════════════════════════════════
\section{The Classification}

The registry's final counts, quoted directly from
\texttt{tbs\_registry.json}:

\begin{center}
\begin{tabular}{lr}
\toprule
status & count \\
\midrule
retired & $15$ \\
superseded & $1$ \\
scope & $21$ \\
refuted & $7$ \\
confirmed-load-bearing & $21$ \\
unreviewed & $0$ \\
\midrule
total recorded gaps & $65$ \\
\midrule
verifier-documented (separate class) & $327$ \\
\bottomrule
\end{tabular}
\end{center}

The statuses are not interchangeable shades of ``closed.'' Each makes
a different assertion, and the differences carry the paper's meaning.

\subsection{Retired: the gap is closed, with a citable document}

\emph{Retired} means the defect was examined and resolved: either
repaired (the statement corrected, the result preserved) or shown to
have no downstream effect. Every retirement cites the document that
closed it, and the build verifies the document exists. Retirement is
the strongest closure the registry grants, and it subdivides in
practice into two kinds.

The first kind is \emph{repair}: the printed statement was wrong, the
underlying result survives the correction. The repair table:

\begin{center}
\small
\begin{tabular}{lll}
\toprule
item & defect & repair \\
\midrule
P003\_3 & Neumann claim, $\rho'(0) = 2\pi \neq 0$ & proof used
Dirichlet; spectrum intact (A278) \\
P007\_1/2 & discretization vs claimed BCs & Dirichlet derived twice
over (A281) \\
P015\_1 & $\langle|\cos\theta|\rangle = 1/4$ & true value $1/2$;
$S/4$ is Cauchy's formula (A293) \\
P029\_1 & peak frequency formula & $\sqrt{\omega_0^2 - \gamma^2/2}$
confirmed (A278) \\
P018\_1 & factor-$\sim$138 discrepancy & the factor is
$\mu_0 = \Omega$; normalization documented (A279) \\
P018\_2 & normalization conflict & the same $\mu_0$ factor
(A295, via A279/A282) \\
P014\_1 & $1 + \kappa^{0.6}$ & exact comma $G_1 = 432/\Tbe - 1$
(A266) \\
P014\_2 & $432/\pi^3 = E_{\rm self}$ & exact:
$(4\pi + 1 + 1/\pi)(1 + G_1)$; identification false (A279) \\
P015\_3\_c & the $\pi/2$ correction & a $5.6\%$-wrong proxy for the
comma gap $432 - \Tbe$ (A278) \\
P028\_2 & $\sqrt{1-\kappa}$ vs $2\pi$ & a unit mismatch; bridge
derived (A273/P35) \\
P028\_3\_c & lapse identification & closed by the null-projection
identification (A263/P39) \\
P027\_2 & 2-to-1 invertibility objection & $C \circ P = I$ is a
section law (A276/P38) \\
P031\_1\_c & monodromy base case & closed by A237 (2026-05-24,
registered by A278) \\
P034\_3\_c & ``$430.5$ s'' & dimensionless $\Tbe$; SI derived
(A265/P35) \\
\bottomrule
\end{tabular}
\end{center}

Each repair preserves the downstream result while correcting the
statement; none required protecting a claim from data.

The second kind is \emph{retraction of interpretation}: the
mathematics stands, the prose around it is withdrawn. The recorded
cases: ``observation is free at the universe boundary'' (P07),
retracted because observation vanishes there, by substrate
inheritance and by the nesting definition, with the mass data
selecting the corrected reading at $9.1\times$ (A280/A281); P27's
mass-formula dressing $1/(1-\mu_1\alpha^2)$, a corrupted and uncited
transcription of P03's derived $(1+\mu_1\alpha^2)$, with the original
adopted as seven times closer to measurement (A282/P36); the
Fatou/Julia dictionary, retired because the breath map is elliptic
and its Julia set empty, the canonical/dark dichotomy being carried
instead by Arnold tongues and kernel typing (A274/P37/P38); the
tempered branch hypothesis, overturned within a day by the derived
unit bridge (A273/P35), the overturning itself part of the record;
and the one-scale halo's apparent rejection (the $7.6\sigma$
core-disk trend), a measurement artifact, quantified and collapsed by
masking (A270/P39).

\subsection{Superseded: replaced, not resolved}

\emph{Superseded} marks a gap whose question was overtaken by a
different route rather than answered on its own terms. The single
case: Open P019\_3, superseded by P32's corner-residual route, which
has been canonical since. The registry records, deliberately, that
the equivalence of the two routes remains unproved. Supersession is
weaker than retirement and the status says so.

\subsection{Scope: a recorded limit, not fixable debt}

\emph{Scope} marks a defect that is real, acknowledged, and outside
the corpus's repair obligations: a boundary of the claim rather than
a hole in it. The $21$ scope items divide into three groups. One is
P021\_2\_c, a permanent honesty marker: the paper itself states that
it does not resolve the formal Clay problem, and no repair could
change that without changing the paper's ambition. Sixteen are the
recorded defects of the Millennium application papers P21--P26,
classified by A295: those papers sit outside the load-bearing spine
by package design, their defects stand on the record, and repair is
not scheduled. Four are technical or scope notes (P008\_3\_c,
P029\_2\_c, P029\_3\_c, P030\_1): external anchors or
theorem-hypothesis caveats whose corpus use is unaffected. A scope
item is never silently promoted to ``fine''; it stays in the ledger
as a named limit.

\subsection{Refuted: the claim is dead}

\emph{Refuted} means the claim was recomputed from its own definition
and failed, under charitable readings, with the failure quantified
and filed with the same verifier rigor as a theorem. The registry
holds four refuted items, covering three cases (the two P17 items
fall together). They are told in full in
Section~\ref{sec:refutations}. Refutation is not retirement: nothing
survives to be preserved. It is also not scope: the claim was not
beyond the corpus's obligations, it was inside them and lost.

\subsection{Confirmed-load-bearing: the gap is real and standing}

\emph{Confirmed-load-bearing} means the gap was examined and found to
matter: it is a genuine dependency of something the corpus asserts,
it is not yet closed, and the registry carries a quantified or
structural note saying exactly what is missing. The $21$ items
include the corpus's two open hearts (the $\alpha$-comma P000\_1\_c
and the mass-hierarchy constant P027\_3\_c, both with named
objectives), the three quintic items of P13, and a documented class
of formalization gaps (Section~\ref{sec:diagnoses}). Load-bearing
status is a debt acknowledgment, not a deferral: each note states
what a closing document would have to contain.

% ════════════════════════════════════════════════════════════════════
\section{The Refutations}
\label{sec:refutations}

Seven registry items are refuted: four from the burn-down, and three
flavor-sector predictions of Paper 01 recorded by A344. Each case is
recorded here in the same shape: the claim, the test, the verdict,
and what survives.

\subsection{P016\_3: the Mersenne shell statistics}
\label{sec:p16}

\emph{The claim.} P16 asserted that Mersenne prime exponents prefer
integer shells: that the cumulative quantities $S_n = \sum_{i \le n}
p_i \kappa$, with $\kappa = \alpha^{5/4}$, land near integers far
more often than chance allows. The published table reported $14$
exponents with shell distance $d_n < 0.01$ and $19$ with $d_n <
0.05$ out of $51$, with mean distance $0.187$.

\emph{The test.} A278 recomputed the counts from the paper's own
definition over all $51$ known Mersenne prime exponents and obtained
$1$ and $9$, mean $0.221$, against expectations of approximately
$1.0$ and $5.1$ under the uniform null. A280 then widened the test to
four natural readings of the definition, on the principle that a
claim should be given every charitable construction before a verdict:
cumulative $S_n$ as stated; per-prime $p\kappa$; the modular form
$p \bmod \kappa^{-1}$; and the exponent variant $p\kappa^{4/5}$. The
results: best counts $2$ and $10$ against the claimed $14$ and $19$,
with means at or near the uniform null in every reading. The
strongest excess anywhere ($10$ observed against $5.1$ expected at
the $0.05$ threshold) carries $p \approx 0.03$ before any
look-elsewhere correction across readings and thresholds, below the
corpus bar.

\emph{The verdict.} No reading reproduces the published counts; under
every reading the headline is unsupported. P016\_3 is the corpus's
first formally refuted canon claim, filed by A280 with full verifier
rigor. One forensic note is on the record: the per-prime reading
reproduces the claimed \emph{mean} ($0.193$ against $0.187$, within
$3\%$) while its counts still fail, which suggests the published
table mixed a per-prime mean with counts from an unstated
computation.

\emph{What survives.} The downstream exposure was checked: nothing
else in the corpus cites the shell counts quantitatively. P07's
``three measurements of $\kappa$'' analogy table inherits a caveat,
not a collapse. The exponent $5/4$ and the constant $\kappa$
themselves are established elsewhere and are untouched.

\subsection{P020\_1: the electroweak VEV conjecture}
\label{sec:p20}

\emph{The claim.} P20 conjectured that the electroweak vacuum
expectation value is $v = M_{\rm Pl}/\sqrt{\mu_0\mu_1}$, tying the
electroweak scale to the corpus's moment constants.

\emph{The test.} A293 evaluated it. With $\sqrt{\mu_0\mu_1} =
122.06$, the formula gives $1.0003 \times 10^{17}$ GeV. The measured
value is $246.22$ GeV. The ratio is $4.06 \times 10^{14}$: the
conjecture misses by more than fourteen orders of magnitude.

\emph{The verdict.} Refuted as stated. There is no reading under
which a factor of $10^{14}$ is a residual; the printed formula is
dead.

\emph{What survives.} Two flags were recorded in the same addendum,
per the corpus's flag discipline (recorded, unpromoted,
look-elsewhere acknowledged): $\ln(v/m_e) = 13.0854$ sits $0.70\%$
from $E_{\rm self} = 13.177$, and the exponent fraction
$\ln(M_{\rm Pl}/v)/\ln(M_{\rm Pl}/m_e) = 0.74605$ sits $0.53\%$ from
$3/4$, the P10 correction. Either could seed a corrected conjecture;
neither is one, and the registry says so in those words. A corrected
conjecture would have to derive one of these and the residual. The
refutation of P020\_1 does not touch the mass-hierarchy formula of
P27, which uses the moment constants differently and is tracked
separately under P027\_3\_c.

\subsection{P017\_2 and P017\_3\_c: the Dirac spectra}
\label{sec:p17}

\emph{The claim.} P17 printed a radial Dirac system on the density
background, quoted its spectrum as $(0.891,\ 6.923,\ 7.763)$, and
built on a factor-of-seven gap between levels (P017\_2). P017\_3\_c
recorded, originally as a TBS item, that the spectrum had not been
independently reproduced.

\emph{The test, stage one (A293): the printed system.} Near the
origin the printed system reduces to $(f,g)' = r^{-1} M (f,g)$ with
$M$ having eigenvalues $-\tfrac32 \pm \tfrac32 i$ at $\kappa = 3/2$:
a complex indicial pair. Both branches are singular and oscillatory;
no regular solution exists; regularity at $r = 0$ cannot be imposed.
The printed equations define no spectrum, so the quoted eigenvalues
cannot be checked against them even in principle. The diagnosis
identified the defect precisely: the printed system places
$\kappa/r$ off the diagonal, where the standard radial Dirac form
places it on the diagonal. At this stage the item was confirmed
load-bearing with the diagnosis filed, and P017\_2 was marked as
inheriting: unsupported pending the corrected system.

\emph{The test, stage two (A295): the natural repair.} The
standard-form system $f' = ((\kappa - \tfrac32)/r) f - M g$,
$g' = -((\kappa + \tfrac32)/r) g + M f$ with $M(r) = m +
\alpha\rho(r)$ restores a regular branch at the origin. Numerical
shooting on $[0,1]$ (RK4, $N = 4000$ to $20000$, convergence
checked), across both channels ($\kappa = 3/2,\ 5/2$) and both
admissible boundary conditions, yields evenly spaced, box-like
spectra with level spacing near $\pi$: $(2.27,\ 5.90,\ 9.12)$,
$(4.00,\ 7.40,\ 10.62)$, $(3.40,\ 7.20,\ 10.52)$, $(5.10,\ 8.67)$.
No reading produces the printed small ground state $0.891$; no
reading produces the close upper pair; the level ratios run
$1.8$ to $2.6$, and no factor of seven appears anywhere.

\emph{The verdict.} P017\_3\_c is refuted as stated on two
independent grounds: the printed system is ill-posed, and the printed
spectrum is not recovered by the natural repair. P017\_2, the factor
of seven, is refuted with it.

\emph{What survives.} The corrected spectra are filed as the repaired
operator's actual output, available as the starting point for any
successor of P17's program. The diagnosis itself (where the sign
convention went wrong, and what the corrected system is) is a
permanent gain: the refutation leaves behind a working operator,
just not the claimed numbers.

\subsection{P001\_4, P001\_5, P001\_6: the flavor-sector predictions}
\label{sec:flavor}

\emph{The claims.} Paper 01 made three flavor-sector predictions from
its $A_4$ tetrahedral structure: a tri-bimaximal reactor angle
$\theta_{13} = 0$; a leptonic CP phase $\delta_{\mathrm{CP}} \approx
240^\circ$, claimed within $2\sigma$ of an experimental $197^\circ \pm
25^\circ$; and a topological resolution of the strong-CP problem,
$\theta_{\mathrm{QCD}} = 0$ from boundary conditions on $B^4$, with
``no axion needed.''

\emph{The test.} Each was checked against current data. The reactor
angle is measured at $\sin^2\theta_{13} = 0.02224$, i.e.\
$\theta_{13} \approx 8.5^\circ$ (NuFIT-6.0, 2024, normal ordering),
nonzero at more than $5\sigma$ since Daya Bay (2012); $\theta_{13} =
0$ sits more than $30\sigma$ from the measured value. The CP phase
global fit has moved to $\delta_{\mathrm{CP}} \approx 177^\circ\
{}^{+19}_{-20}$ degrees, consistent with CP conservation within
$1\sigma$, placing the predicted $240^\circ$ about $3.3\sigma$ above
the central value; the paper's claimed agreement used the superseded
$197^\circ \pm 25^\circ$ fit. The strong-CP argument fixes the
instanton winding number $n = 0$, but the physical strong-CP
parameter is the Lagrangian coupling $\bar\theta$, an independent
input: the action enters as $\bar\theta\, n$, so a configuration with
$n = 0$ contributes at every $\bar\theta$, and fixing the winding does
not fix the coupling.

\emph{The verdict.} All three are refuted. P001\_4 ($\theta_{13} =
0$) is excluded by the data; P001\_5 ($\delta_{\mathrm{CP}} \approx
240^\circ$) fails on the current central value; P001\_6 (topological
strong CP) constrains the wrong object and does not solve strong CP,
which remains an open problem of the Standard Model. The three are
filed by A344, with TBS remarks added to Paper 01 adjacent to the
claims they qualify.

\emph{What survives.} The $A_4$ family structure and the lens-space
account of three generations are untouched by these refutations; what
falls is the specific quantitative flavor output --- the zero reactor
angle, the $240^\circ$ phase, and the axion-free strong-CP claim.
Paper 01's abstract was reframed to mark all three as not having
survived current data and to flag them for this registry; the
recording here makes that citation true.

% ════════════════════════════════════════════════════════════════════
\section{The Standing Diagnoses}
\label{sec:diagnoses}

The $21$ confirmed-load-bearing items are not undifferentiated debt;
each carries a note stating exactly what is missing. The deepest is a
structural diagnosis rather than a numerical one.

\textbf{The P13 quintic is a conjecture-schema (P013\_1, P013\_2,
P013\_3\_c; A293).} P13's headline asserts that a certain quantity
satisfies an irreducible quintic with computable coefficients. The
diagnosis: Condition 1 invokes $\kappa_{\rm geom} = C_1\mu_2/\mu_0^2
+ C_2\mu_1/\mu_0 + C_3$ with the constants $C_1, C_2, C_3$ never
fixed; Condition 2 invokes $F(\lambda_1, \beta) = 0$ with the
functional $F$ never given; and the rationality premise
(P013\_3\_c) is unproven and generically false, since Sturm-Liouville
eigenvalues are generically transcendental in the parameter. The
coefficients $a_5, \ldots, a_0$ are therefore not merely uncomputed
but uncomputable as stated, and the degree bound (P013\_2) is
unverifiable until the construction is supplied. The headline stands
reclassified as a schema: a shape a future derivation could fill,
not a result. The corpus's $5/4$ exponent and five-fold structure
stand independently of P13.

\textbf{The $\alpha$-comma (P000\_1\_c; A279/A283/A284).} The
identity $\alpha^{-1} = \Omega$ is accurate to $2.2234$ ppm against
CODATA and is not exact at any edition, roughly $1.45 \times 10^4$
experimental standard deviations away. The registry's note records
the trajectory: typed as the time-average dressing of P03's
oscillation, with deficit $1 - \sqrt{1 - A^2}$; the reading
$A = \kappa$ predicts $2.2745$ ppm, the right sign and $2.3\%$ high;
inverted, the residual becomes the first measurement of the
oscillation amplitude, $A = 0.9887(2)\,\kappa$, stable across CODATA
editions. A284 bracketed the correction ($1 - \pi/\Omega$ versus
$1 - 1/44$, identification below bar) and filed the conditional edge
prediction $\alpha^{-1} = 137.035999236$, $0.43$ ppb from
CODATA-2022, with a $0.1$-ppb discriminator. The exact amplitude
remains the objective (OI-283-1). This item is the corpus owning a
residual in its most external-facing identity and converting it into
a pre-registered prediction rather than absorbing it.

\textbf{The mass-hierarchy constant (P027\_3\_c; A279/A282).} The
formula $\ln(M_{\rm Pl}/m_e)$ evaluates to $51.5299$ against the
measured $51.5278$ ($3.9 \times 10^{-5}$ relative). Its provenance
was found: derived in P03, uncited by P27, with P27's dressing a
corrupted transcription of P03's form (the original is seven times
more accurate and is adopted). What remains underived is the integer
$10$ in $3\pi/20 = (1/10)(3\pi/2)$, where $3\pi/2$ is the textbook
one-loop QED coefficient, together with the correspondence postulate
itself (OI-282-1). One underived integer buying a $3.9 \times
10^{-5}$ hit on a 22-decade mass ratio is either deep or a fit, and
the registry says the corpus does not yet know which.

\textbf{The formalization-gap class (A295).} The remaining
load-bearing items are documented gaps of formalization or
verification rather than suspected errors, each with a one-line note
in the registry: the R-operator never formally specified (P000\_3);
the $5/4$ global minimality unproven, local result only (P003\_2);
the moments-to-hierarchy correspondence postulate (P007\_3);
dimension not derived from transversality (P008\_1); three recorded
circularities (P008\_2\_c, P009\_3\_c, P010\_3); the unexplained
$4/3$ step (P015\_2); a drift-bound inconsistency that is testable
against the GON implementation and not yet run (P030\_2); two
structural claims unverified (P030\_3\_c, P033\_3\_c); two open
monodromy steps (P031\_2\_c, P031\_3\_c); an associativity proof gap
(P034\_2\_c); the conflation in P002\_1, confirmed real and
numerically benign at $1.2\%$; and P033\_2, where the printed formula
${\rm Vol}(S^{d-1}) = \pi^d$ is confirmed false at every $d$, the
correct values are supplied, and the downstream argument is flagged
for rework.

% ════════════════════════════════════════════════════════════════════
\section{The Verifiers: the Registry in Executable Form}

Every canon paper has a verifier, \texttt{verify\_P000.py} through
\texttt{verify\_P034.py} and onward, which recomputes the paper's
checkable claims at run time. The suite carries a convention that is
the registry's executable counterpart: the \emph{expected fail}. A
check that a paper's claim does not survive recomputation is not
deleted and not inverted; it is kept in the verifier, printed as a
\texttt{[FAIL]} line with the claim and the reason, and documented as
failing. The suite treats a documented failure as a recorded fact,
not a build error: the verifier passes because it correctly reports
the claim's status, which is the verifier's actual job.

These lines are harvested into the registry by the single writer's
second function, \texttt{generate\_vdc()}, which runs each
paper-series verifier, parses its \texttt{[FAIL]} output, and files
one \emph{verifier-documented} item per check, with the verifier
name, the claim, and the note. The registry currently carries $327$
such items, in a section separate from the $62$ remark-derived gaps
so that existing consumers of the remark counts are undisturbed.

The point of the convention is that the corpus's defects are
reproducible by execution, not only by reading. A reader who doubts
the prose does not have to trust this paper or any addendum: running
the suite reproduces the confirmations and the failures alike, and
the registry's verifier-documented section is, at any moment, exactly
what the suite reports. The closure rule binds in the same direction:
a status citing a closing addendum is checked against the disk on
every regeneration. The ledger is not an account of the code; it is
output of the code.

% ════════════════════════════════════════════════════════════════════
\section{The Living Ledger}

This paper is the prose face of a generated file, and the
relationship runs in one direction: the registry is the source of
truth, and the paper quotes it. Three consequences follow.

First, every count printed in an addendum is a dated snapshot. The
burn-down addenda each printed the totals as they stood at filing,
and the snapshots disagree with one another by construction, because
the ledger moved between them. They also disagree, in places, with
the final map: A295's printed totals predate the last status
adjustments, under which P017\_3\_c moved from the load-bearing class
to refuted (its repair having failed to recover the printed spectrum)
and four technical notes settled as scope. Where a printed snapshot
and the registry differ, the registry is current. The snapshots are
not errors; they are the ledger's history, and the single-writer
discipline exists precisely so that the history and the present state
cannot be confused.

Second, this paper updates as the registry does. Its numbers are
quoted from the registry at the date in its header; a future status
change obligates a revision here, in the same change, under the same
rule that binds every other closure. A repairs-and-retractions paper
that froze while its registry moved would be an instance of the
disease it documents.

Third, the zero matters but is not a terminus. Zero unreviewed means
every recorded gap has been examined and classified; it does not mean
the debt is paid. Twenty-one items stand confirmed-load-bearing with
named objectives, and the registry will reopen, renumber, and
reclassify as the corpus grows. The invariant is not that the ledger
is ever finished; it is that the ledger is never silent.

% ════════════════════════════════════════════════════════════════════
\section{Conclusion}

A framework that cannot lose measures nothing. The statement is not
rhetoric; it is the operating principle this paper documents. A
measurement is a procedure that could have come out otherwise. A
corpus whose every recorded check confirms its claims has either
stopped checking or stopped recording, and in both cases its
agreement with anything has ceased to carry information. The registry
is the corpus's refusal of that state: $65$ gaps recorded in the
papers themselves, driven to zero unreviewed through six documented
stages, with seven claims refuted (four from the burn-down and three
flavor-sector predictions of Paper 01 recorded by A344), fifteen
retired on citable repairs, twenty-one classified as standing debt
with named objectives, and $327$ expected-fail checks kept executable
so that the defects reproduce on demand.

\begin{remark}[The discipline, adopted as canon]
(1) Every recorded gap lives in the registry; closing one updates the
registry in the same change, and a closure must cite its closing
document or the build fails. (2) Refutations and deflations receive
the same verifier rigor as theorems: the registry has refuted canon
headlines, deflated a hoped-for selection (A290), and overturned a
working hypothesis within a day, and each is cited here as evidence
the instrument works. (3) Same-day erratum and upgrade notes are
appended to filed addenda rather than silently edited. (4) Flags are
never promoted without a derivation (A245/A180); fragile fits are
recorded as fragile (the $\kappa/7$ case); chains outrank fits
because they falsify in more places. (5) No claim, including the
seed identity itself, is protected from its own registry: the seed's
$2.2$ ppm deviation was named, typed, and turned into a
pre-registered prediction (P36). The corpus corrects itself in
public, on the record, by machine-checked rules; this paper exists so
that the record is canon.
\end{remark}

\paragraph{Provenance and verification.} Assembled without new
claims from A277 (registry), A278--A280, A293, and A295 (burn-downs
and closure), the flavor-sector refutations of A344, and the repair
addenda A263, A266, A270, A273--A276,
A279, A281--A284, A294. Verifiers
\texttt{verify\_P277/278/279/280/293/295/344.py} and the single-writer
generator \texttt{tbs\_registry\_gen.py}, all green at filing; the
registry JSON ships with the package as its quality certificate. All
counts in this paper are quoted from
\texttt{addenda/verify/tbs\_registry.json}: $15$ retired, $1$
superseded, $21$ scope, $4$ refuted, $21$ confirmed-load-bearing,
$0$ unreviewed, $327$ verifier-documented.

\end{document}
