% 293_Addendum_RegistryBurndownIV.tex
% Q5 continuation: the severe registry items. P015_1 retired (Cauchy
% projection: result right, intermediate wrong by the front-back
% factor). P020_1 refuted as stated (10^17 vs 246 GeV; two flags
% recorded, none promoted). P017_3_c confirmed with diagnosis: the
% printed Dirac system has complex indicial exponents — no regular
% branch, ill-posed; the spectrum is unreproducible in principle.
% P013_1/2/3_c confirmed: the quintic is a schema (C1,C2,C3 and F
% unspecified; rationality unproven). Debt 44 -> 38.
% Verifier: verify_P293.py.
% Copyright Léon Fernando Vlegels -- CC BY 4.0

\documentclass[11pt]{article}
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\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
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\title{Addendum 293: Registry Burn-down IV\\
\large The severe items: one repair, one refutation,\\
two structural diagnoses}
\author{TOE Corpus}
\date{2026-06-10}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} Mixed by design (1 retired, 1 refuted, 4 confirmed
load-bearing with diagnoses) \quad|\quad
\textbf{Verifier:} \texttt{verify\_P293.py} --- all PASS

\begin{abstract}
The four severest standing registry items, each settled by
computation. \emph{P015\_1 retired (repair).} The claimed
$\langle|\cos\theta|\rangle = 1/4$ is indeed wrong --- the spherical
average is exactly $1/2$ --- but P15's \emph{result}
$\bar A_{\rm shadow} = S/4$ is nevertheless correct: it is Cauchy's
projection formula, with the missing factor $2$ being the front--back
double count ($\bar A = S\langle|\cos\theta|\rangle/2$). Intermediate
misstatement corrected; downstream intact.
\emph{P020\_1 refuted as stated.} The Higgs-VEV conjecture
$v = M_{\rm Pl}/\sqrt{\mu_0\mu_1}$ evaluates to
$1.00\times10^{17}$ GeV --- off by $4.1\times10^{14}$. The conjecture
is dead in its printed form. Two flags recorded, neither promoted:
$\ln(v/m_e) = 13.085$ vs $E_{\rm self} = 13.177$ ($0.70\%$), and
$\ln(M_{\rm Pl}/v)/\ln(M_{\rm Pl}/m_e) = 0.7461$ vs $3/4$ ($0.53\%$,
the P10 correction) --- either would be a route to a corrected
conjecture; neither is one yet.
\emph{P017\_3\_c confirmed, with the diagnosis the remark lacked.}
The printed radial Dirac system has indicial exponents
$-3/2 \pm \tfrac{3}{2}i$ at the origin --- a complex pair: \emph{no
regular branch exists}, the boundary-value problem is ill-posed as
printed, and the quoted spectrum $(0.891, 6.923, 7.763)$ is
unreproducible \emph{in principle}, not merely in practice. A sign
convention is missing from the canon. P017\_2 (the factor-of-7 gap)
inherits: unsupported pending the corrected system.
\emph{P013\_1/2/3\_c confirmed: the quintic is a schema.} The three
consistency conditions contain unspecified constants
($C_1, C_2, C_3$), an unspecified functional ($F$), and an unproven
rationality premise (P013\_3\_c: $\lambda_1(\alpha)$ is generically
transcendental in $\alpha$); the coefficients $a_5, \ldots, a_0$ are
not merely uncomputed but \emph{uncomputable as stated}. P13's
irreducible-quintic headline is a conjecture-schema until the
construction is supplied. (The corpus's $5/4$ exponent and five-fold
structure stand independently of P13.) Registry: open debt
$44 \to 38$; the load-bearing class now carries six items, all with
quantified diagnoses.
\end{abstract}

\begin{proposition}[P015\_1: the Cauchy repair]
$\frac{1}{4\pi}\int_{S^2}|\cos\theta|\,d\Omega =
\tfrac12\int_0^\pi |\cos\theta|\sin\theta\,d\theta = \tfrac12$, not
$\tfrac14$. The shadow average over orientations of a convex body is
$\bar A = \tfrac12\,\langle|\cos\theta|\rangle\,S = S/4$ (Cauchy),
the $\tfrac12$ correcting for the two-sided projection count. P15's
formula stands; its stated integral is corrected. Retired.
\end{proposition}

\begin{proposition}[P020\_1: refutation and flags]
$M_{\rm Pl}/\sqrt{\mu_0\mu_1} = 1.0003\times10^{17}$ GeV
($\sqrt{\mu_0\mu_1} = 122.06$); ratio to $v = 246.22$ GeV:
$4.06\times10^{14}$. Refuted as stated. Flags (recorded per A245/A180,
unpromoted, look-elsewhere acknowledged): (i) $\ln(v/m_e) = 13.0854$
sits $0.70\%$ from $E_{\rm self}$; (ii) the exponent fraction
$\ln(M_{\rm Pl}/v)/\ln(M_{\rm Pl}/m_e) = 0.74605$ sits $0.53\%$ from
$3/4$. A corrected conjecture would have to derive one of these and
the residual; none is attempted here.
\end{proposition}

\begin{proposition}[P017\_3\_c: ill-posedness diagnosis]
Near $r = 0$ the printed system reduces to
$(f, g)' = r^{-1}M(f, g)$ with
$M = \bigl(\begin{smallmatrix}-3/2 & -3/2\\ 3/2 &
-3/2\end{smallmatrix}\bigr)$ ($\kappa = 3/2$), whose eigenvalues are
$-\tfrac32 \pm \tfrac32 i$. Both indicial branches are singular and
oscillatory; no regular solution exists; ``regularity at $r = 0$''
cannot be imposed. The printed equations therefore define no spectrum,
and the quoted eigenvalues cannot be checked against them. The repair
requires the intended sign/convention (standard radial Dirac places
$\kappa/r$ on the diagonal); until the canon supplies it, P017\_2's
factor-of-7 is unsupported. Confirmed load-bearing, diagnosis filed.
\end{proposition}

\begin{proposition}[P013: the quintic is a schema]
Condition 1 invokes $\kappa_{\rm geom} = C_1\mu_2/\mu_0^2 +
C_2\mu_1/\mu_0 + C_3$ with $C_i$ never fixed; Condition 2 invokes
$F(\lambda_1, \beta) = 0$ with $F$ never given; the rationality of
$\lambda_1(\alpha)$ is unproven and generically false for
Sturm--Liouville eigenvalues (P013\_3\_c stands). Hence
$a_5, \ldots, a_0$ are uncomputable as stated and the degree bound
(P013\_2) is unverifiable. P13's headline is reclassified
conjecture-schema; its repair requires supplying $(C_i, F)$ from the
corpus and proving (or replacing) the rationality premise.
\end{proposition}

\paragraph{Registry.} P015\_1 retired (A293); P020\_1 refuted (A293);
P017\_3\_c, P013\_1, P013\_2, P013\_3\_c confirmed-load-bearing with
diagnoses. Totals: $14$ retired, $1$ superseded, $1$ scope, $2$
refuted, $6$ confirmed-load-bearing, $38$ unreviewed-or-open.

\paragraph{Verifier.} \texttt{verify\_P293.py}: the spherical average;
the Cauchy factor; the VEV evaluation and both flags; the indicial
eigenvalues; the schema assertions; registry regeneration and counts.
All PASS.

\end{document}
