% 295_Addendum_PrePublicationClosure.tex
% Pre-publication closure run. (i) P17 repaired-and-refuted: the
% corrected radial Dirac system (regular branch restored) yields evenly
% spaced spectra in all channels and boundary choices — no 0.891 ground
% state, no factor 7; the printed spectrum is unreproducible under both
% the printed and the corrected system. (ii) Selection rule: second
% mechanism class excluded (oscillation dressing is O(kappa^2); the
% lepton correction is O(kappa)). (iii) Registry driven to ZERO
% unreviewed: every remaining item classified, with P018_2 retired and
% P017_2 refuted en route. Verifier: verify_P295.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 295: Pre-Publication Closure\\
\large The P17 verdict, a second exclusion for the selection rule,\\
and the registry at zero unreviewed}
\author{TOE Corpus}
\date{2026-06-11}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} Case B (one refutation completed; one exclusion
added; registry classification complete) \quad|\quad
\textbf{Verifier:} \texttt{verify\_P295.py} --- all PASS

\begin{abstract}
Three closures before the package ships.
\emph{(i) P17, repaired and refuted.} A293 diagnosed the printed
radial Dirac system as ill-posed (no regular branch). The natural
repair is performed here: 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.
Shooting on $[0,1]$ across both channels ($\kappa = 3/2, 5/2$) and
both admissible boundary conditions yields evenly spaced spectra
(level spacing $\approx \pi$): $(2.27, 5.90, 9.12)$,
$(4.00, 7.40, 10.62)$, $(3.40, 7.20, 10.52)$, $(5.10, 8.67)$. No
reading produces the printed $(0.891, 6.923, 7.763)$; no reading
produces a factor-of-7 gap (ratios $1.8$--$2.6$). \textbf{P017\_3\_c
is upgraded from diagnosed to refuted-as-stated} (the spectrum is
unreproducible under the printed and the corrected system), and
\textbf{P017\_2 (the factor of 7) is refuted} with it. The corrected
spectra are filed as the repaired operator's actual output, available
to any future successor of P17's program.
\emph{(ii) Selection rule, second exclusion.} A292 excluded universal
first-order perturbations. Here the oscillation-dressing class is
excluded: averaging over the breath's $\kappa$-oscillation produces
corrections at order $\kappa^2$ (as in the $\alpha$-comma), while the
lepton correction $X = \kappa\Omega/\lambda_1$ is first order in
$\kappa$. The two mechanism classes are disjoint and both are now
closed; OI-287-1's derivation must come from elsewhere (the
level-selective coupling form remains the only open route).
\emph{(iii) Registry: zero unreviewed.} Every remaining item is
classified. New statuses: P018\_2 retired (its normalization conflict
is the $\mu_0$-factor documented in A279/A282); P017\_2 refuted (this
addendum); the seventeen honesty remarks of the Millennium
application papers (P21--P26) classified \emph{scope} (the papers are
outside the load-bearing spine per the package design; their recorded
defects stand without scheduled repair); the remainder classified
confirmed-load-bearing with notes (formalization gaps: P000\_3,
P003\_2, P008\_1/2\_c, P009\_3\_c, P010\_3, P015\_2, P030\_3\_c,
P031\_2\_c/3\_c, P033\_3\_c, P034\_2\_c; technical/scope: P008\_3\_c,
P029\_2\_c/3\_c, P030\_1; testable: P030\_2) and two quantified
diagnoses (P002\_1: the ratio $\Delta E/E_{\rm self} = 0.0464$ vs
$22\kappa = 0.0469$ agrees at $1.2\%$, the conflation flagged is
real but numerically benign; P033\_2: the formula
${\rm Vol}(S^{d-1}) = \pi^d$ is confirmed false at every $d$, correct
values supplied). \textbf{The registry now contains no unreviewed
items}: $62$ gaps, every one carrying a status, a note, and where
applicable a closing or refuting document.
\end{abstract}

% ────────────────────────────────────────────────────────────────────────────
\section{The P17 Verdict}

\begin{proposition}[The corrected system and its spectra]
The repair replaces the printed off-diagonal $\kappa/r$ placement
(complex indicial exponents, A293) with the standard radial Dirac
form, restoring a regular branch ($f \sim {\rm const}$,
$g \sim M(0^+) f\, r/(\kappa + 5/2)$ at the origin). Numerical
shooting (RK4, $N = 4000$--$20000$, convergence checked) gives, for
(channel, boundary condition): $(3/2, f(1){=}0)$:
$2.27, 5.90, 9.12$; $(3/2, g(1){=}0)$: $4.00, 7.40, 10.62$;
$(5/2, f)$: $3.40, 7.20, 10.52$; $(5/2, g)$: $5.10, 8.67$. The
spectra are box-like with spacing near $\pi$, qualitatively unlike
the printed claim, whose small ground state and close upper pair
appear in no reading. The printed eigenvalues, already underivable
from the printed equations, are not recovered by the natural repair:
refuted as stated. Any successor program inherits the corrected
spectra as its starting point.
\end{proposition}

% ────────────────────────────────────────────────────────────────────────────
\section{The Second Exclusion}

\begin{proposition}[Dressing mechanisms cannot produce the lepton law]
Averages of observables over the breath oscillation produce
corrections beginning at order $\kappa^2$ (the $\alpha$-comma's
$\kappa^2/2 \approx 2.3\times10^{-6}$ is the prototype; symmetric
averaging cancels first order). The lepton correction strength is
$X = \kappa\,\Omega/\lambda_1 = 0.0177$, first order in $\kappa$ and
four orders larger than any dressing. Together with A292's exclusion
of universal first-order perturbations, both generic mechanism
classes are closed; OI-287-1 stands, with its search space reduced to
mechanisms that couple each level to its own layer at first order in
$\kappa$.
\end{proposition}

% ────────────────────────────────────────────────────────────────────────────
\section{Registry: Zero Unreviewed}

\begin{remark}[Classification, complete]
Statuses assigned per the abstract;
\texttt{tbs\_registry\_gen.py} carries the full map and the build
fails on any unclassified item. Final counts: $15$ retired, $1$
superseded, $19$ scope, $3$ refuted, $24$ confirmed-load-bearing
(each with a quantified or structural note), $0$ unreviewed. The
canon's debt is now fully itemized: nothing in the corpus is both
open and unexamined.
\end{remark}

\paragraph{Verifier.} \texttt{verify\_P295.py}: the corrected-system
spectra (re-shot at verification time, all four readings); the
absence of the printed eigenvalues and of any factor-7 ratio; the
order-counting for the dressing exclusion; the P002\_1 and P033\_2
numerics; registry regeneration with zero-unreviewed assertion.
All PASS.

\end{document}
