% 282_Addendum_MassFormulaProvenance.tex
% OI-279-2 closed: the mass-hierarchy formula was derived in P03 all
% along (Thm beta_qed) and P27 never cited it. 3pi/20 = (1/10)(3pi/2),
% where 3pi/2 is the inverse one-loop QED beta coefficient (textbook)
% — the underived content reduces to the integer 10 and the
% correspondence postulate. P27's dressing 1/(1-x) is a corruption of
% P03's derived (1+x); P03's own form is 7x more accurate (5.5 ppm).
% Verifier: verify_P282.py.
% Copyright Léon Fernando Vlegels -- CC BY 4.0

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\newtheorem{theorem}{Theorem}[section]
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\title{Addendum 282: The Mass-Formula Provenance\\
\large $3\pi/20$ decomposed: one-loop QED $\times$ the integer 10;\\
P27's dressing corrected to P03's (OI-279-2 closed)}
\author{TOE Corpus}
\date{2026-06-10}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} Case B$+$ (provenance found in canon; constant
decomposed to textbook QED $\times$ one integer; dressing corrected;
residual postulate named) \quad|\quad
\textbf{Closes:} OI-279-2 (replaced by OI-282-1) \quad|\quad
\textbf{Verifier:} \texttt{verify\_P282.py} --- all PASS

\begin{abstract}
A279 left the constant $3\pi/20$ in the mass-hierarchy formula
underived and filed OI-279-2 (``derive it or retire the formula as a
fit''). The audit's first finding is archival: \textbf{the formula was
derived in P03 all along} --- Theorem beta\_qed states
$\beta_{\rm geom}/\beta_{\rm QED} = C \cdot \ln(M_{\rm Pl}/m_e)$ with
$C = 10\mu_0^3/(\mu_0^2 + \mu_1)$ --- and P27 reproduced a variant
without citation, which is why the registry believed it underived.
Unpacking the theorem with $\beta_{\rm geom} = \mu_1/\mu_0$ and the
\emph{textbook one-loop QED beta} $\beta_{\rm QED} = 2\alpha^2/3\pi =
2/(3\pi\mu_0^2)$ gives identically
\begin{equation*}
\ln\!\frac{M_{\rm Pl}}{m_e}
\;=\; \frac{3\pi}{20}\,\mu_1\,(1 + \mu_1\alpha^2),
\qquad
\frac{3\pi}{20} \;=\; \frac{1}{10}\cdot\frac{3\pi}{2},
\end{equation*}
so the mysterious constant decomposes as (inverse one-loop QED
coefficient) $\times$ (the integer 10 from $C$): \emph{half of
$3\pi/20$ was never the corpus's to derive --- it is standard QED.}
Second finding: P27's dressing $1/(1-\mu_1\alpha^2)$ is a
\emph{corruption} of P03's derived $(1+\mu_1\alpha^2)$ (they agree only
to first order), and P03's own form is the more accurate: $51.52812$
versus measured $51.52784$ --- $5.5\times10^{-6}$ relative, seven times
better than P27's variant ($3.9\times10^{-5}$). A 22-decade mass ratio
predicted at $5.5$ ppm with one underived integer. What remains
underived is exactly: (i) the integer $10$ in $C$ (decomposition
$10 = 2\times 5$, bulk-coefficient prime times the quintic prime, is
recorded as arithmetic, not provenance), and (ii) the correspondence
postulate itself --- \emph{why} the ratio of geometric to QED flow
should equal $C$ times the hierarchy span, which P03 establishes
numerically, not structurally. \textbf{OI-282-1} replaces OI-279-2 in
this sharper form. P027\_3\_c's registry note is updated; P007\_3
(same complex) inherits the sharpened status.
\end{abstract}

% ────────────────────────────────────────────────────────────────────────────
\section{The Decomposition}

\begin{theorem}[Provenance and decomposition]
\label{thm:prov}
From P03 Thm beta\_qed ($\beta_{\rm geom}/\beta_{\rm QED} = C\,L$,
$C = 10\mu_0^3/(\mu_0^2+\mu_1)$, $L = \ln(M_{\rm Pl}/m_e)$), with
$\beta_{\rm geom} = \mu_1/\mu_0$ and $\beta_{\rm QED} = 2\alpha^2/3\pi$
and $\mu_0 = \alpha^{-1}$:
\begin{equation}
L \;=\; \frac{\beta_{\rm geom}}{\beta_{\rm QED}\,C}
\;=\; \frac{\mu_1}{\mu_0}\cdot\frac{3\pi\mu_0^2}{2}\cdot
\frac{\mu_0^2+\mu_1}{10\,\mu_0^3}
\;=\; \frac{3\pi}{20}\,\mu_1\!\left(1 + \frac{\mu_1}{\mu_0^2}\right)
\;=\; \frac{3\pi}{20}\,\mu_1\,(1+\mu_1\alpha^2),
\end{equation}
identically. The factor $3\pi/2$ enters as the inverse of the one-loop
QED beta coefficient --- externally grounded, not corpus-fitted; the
factor $1/10$ enters from $C$'s algebraic form. Numerically
$L = 51.528124$ versus measured $51.527840$ ($5.5\times10^{-6}$).
\end{theorem}

\begin{remark}[P27's variant corrected]
P27 wrote the dressing as $(1-\mu_1\alpha^2)^{-1}$, agreeing with P03's
derived $(1+\mu_1\alpha^2)$ only to first order in
$\mu_1\alpha^2 = 5.79\times10^{-3}$. The variants differ at
$3.3\times10^{-5}$ --- precisely the scale at which P27's value
($51.52985$) misses the measurement and P03's does not. The corpus
adopts P03's form; P27's is recorded as an uncited, corrupted
transcription. (This also resolves \emph{how} the registry came to
believe the formula underived: P27 never pointed back to P03 \S7.)
\end{remark}

\begin{remark}[What is now underived, exactly --- OI-282-1]
Two items. \emph{The integer 10}: $C = 10\,\mu_0^3/(\mu_0^2+\mu_1)$
fits the measured $C$ at $5.5\times10^{-6}$; the arithmetic
$10 = 2\times5$ (the even prime of the bulk coefficient $16 = 2^4$ and
the quintic prime of P12/P13) is recorded as observation only.
\emph{The correspondence postulate}: P03 verifies
$\beta_{\rm geom}/\beta_{\rm QED} = C\,L$ numerically but supplies no
structural reason why geometric flow per QED flow should equal an
algebraic constant times the hierarchy span. OI-282-1: derive either.
Until then the formula's status is: provenance canonical (P03),
external anchor textbook (one-loop QED), accuracy $5.5$ ppm over 22
decades, free content one integer plus one postulate --- materially
better than ``deep or a fit,'' and honestly short of derived.
\end{remark}

\paragraph{Consequences.} OI-279-2 closed, replaced by OI-282-1.
Registry notes updated: P027\_3\_c (sharpened; dressing corrected),
P007\_3 (route identified: P03 \S7; the ``why'' is the postulate).
The SPARC-paper-adjacent claim inventory gains its second-most
external-facing number ($5.5$ ppm on the mass hierarchy) with an
honest provenance trail.

\paragraph{Verifier.} \texttt{verify\_P282.py}: the identity chain at
40 digits; $\beta_{\rm QED}$ as textbook one-loop; $C$ algebraic vs
measured; both dressings against measurement (P03 wins $7\times$); the
decomposition $3\pi/20 = (1/10)(3\pi/2)$; registry updates. All PASS.

\end{document}
