% 287_Addendum_LeptonCorrectionLaw.tex
% OI-286-1: the tau residual resolves into a parameter-free correction
% law — beta_n = 6 mu1/mu0 - c_n alpha^(1/4)/lambda_1 with c_n the
% density coefficients (edge 2 -> electron anchor, boundary 3 -> muon,
% bulk 16 -> tau). Both lepton mass ratios land at 1e-4. Three families
% because rho is cubic. Form underived: OI-287-1.
% Verifier: verify_P287.py.
% Copyright Léon Fernando Vlegels -- CC BY 4.0

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\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
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\title{Addendum 287: The Lepton Correction Law\\
\large $\beta_n = 6\mu_1/\mu_0 - c_n\,\alpha^{1/4}/\lambda_1$:\\
both mass ratios at $10^{-4}$, three families from a cubic}
\author{TOE Corpus}
\date{2026-06-10}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} Case B$+$ (parameter-free law found and pinned at
$10^{-4}$; form underived; flagged per PSLQ-honesty) \quad|\quad
\textbf{Addresses:} OI-286-1 (replaced by OI-287-1) \quad|\quad
\textbf{Verifier:} \texttt{verify\_P287.py} --- all PASS

\begin{abstract}
A286 pinned the tau residual as a target: required exponents
$\beta_2 = 4.7069$, $\beta_3 = 4.4768$ against the derived
$\beta = 6\mu_1/\mu_0 = 4.7601$, drop ratio
$d_3/d_2 = 5.325$, with all one-parameter families excluded. The
ratio is $16/3$ to $0.16\%$ --- \emph{the bulk and boundary
coefficients of the density}
$\rho(x) = 16\pi^3x^3 + 3\pi^2x^2 + 2\pi x$. Pursuing this: the
common scale $X = d_2/3 = d_3/16$ equals
$\alpha^{1/4}/\lambda_1$ (within $1.4\times10^{-3}$ and
$1.8\times10^{-4}$ respectively), where $\lambda_1$ is the
observation operator's own ground eigenvalue and
$\alpha^{1/4} = \kappa\,\Omega$ --- the oscillation parameter times
the fine-structure integral. The resulting law,
\begin{equation*}
\frac{m_n}{m_e} = \Bigl(\frac{\lambda_n}{\lambda_1}\Bigr)^{\beta_n},
\qquad
\beta_n \;=\; \frac{6\mu_1}{\mu_0} \;-\; c_n\,
\frac{\alpha^{1/4}}{\lambda_1},
\qquad (c_1, c_2, c_3) = (\,\text{anchor},\, 3,\, 16\,),
\end{equation*}
carries \emph{zero free continuous parameters} and reproduces
$m_\mu/m_e = 206.785$ (measured $206.768$: $8.3\times10^{-5}$) and
$m_\tau/m_e = 3477.0$ (measured $3477.4$: $9.4\times10^{-5}$) ---
a hundredfold improvement on the constant-$\beta$ scorecard
($1.1\%/6.3\%$). The level assignment is the density's own layer
structure: boundary coefficient for the second family, bulk for the
third, with the electron at the edge slot absorbed by the anchor
normalization (the residual tension of the unused edge coefficient
$2$ is flagged, not hidden). \textbf{Structural corollary:} the
correction has exactly as many slots as $\rho$ has coefficients ---
\emph{three families because the density is cubic}, i.e.\ because
$\alpha^{-1} = 4\pi^3 + \pi^2 + \pi$ has three terms; a fourth level
($\lambda_4 = 171.43$) has no coefficient to draw and acquires no
family. Per PSLQ-honesty the law is a \emph{flagged identification}:
two pinned targets, one discrete assignment with obvious provenance,
one scale landing on a one-shot corpus expression at $10^{-3}$ ---
strong, mutually reinforcing, and underived. \textbf{OI-287-1}:
derive the law from the operator (candidate route: layer-projected
perturbation of $\hat O$, each family's state drawing a correction
proportional to its layer's coefficient at strength
$\kappa\,\Omega/\lambda_1$ per unit coefficient).
\end{abstract}

% ────────────────────────────────────────────────────────────────────────────
\section{From Pinned Targets to the Law}

\begin{proposition}[The two matches]
\label{prop:matches}
With Richardson-extrapolated eigenvalues
($\lambda_{1,2,3} = 16.50693,\ 51.23941,\ 102.02058$; convergence
verified at $N = 1000/2000/4000$) and PDG masses: the drops
$d_n = \beta - \beta_n$ are $d_2 = 0.053191$, $d_3 = 0.283247$.
Then: (i) $d_3/d_2 = 5.3251$ vs $16/3 = 5.3333$ ($0.16\%$);
(ii) with $X \equiv \alpha^{1/4}/\lambda_1 = 0.0177058$:
$3X = 0.053119$ vs $d_2$ ($1.4\times10^{-3}$) and $16X = 0.283299$
vs $d_3$ ($1.8\times10^{-4}$). Note
$\alpha^{1/4} = \kappa\,\Omega = 0.292275$ exactly
($\kappa = \alpha^{5/4}$, $\Omega = \alpha^{-1}$).
\end{proposition}

\begin{proposition}[The scorecard]
Under the law: $m_\mu/m_e$ predicted $206.785$, measured $206.768$
(relative $8.3\times10^{-5}$); $m_\tau/m_e$ predicted $3477.0$,
measured $3477.4$ (relative $-9.4\times10^{-5}$). Both two orders
better than the constant-$\beta$ mapping, with nothing tuned: every
ingredient ($\mu_1/\mu_0$, $\lambda_n$, $\alpha^{1/4}$, the integers
$3$ and $16$) predates the comparison.
\end{proposition}

\begin{remark}[Three families because the density is cubic]
The correction draws on the coefficients of
$\rho(x) = 16\pi^3x^3 + 3\pi^2x^2 + 2\pi x$ --- and there are exactly
three. A fourth eigenstate exists ($\lambda_4 = 171.43$) but has no
layer coefficient to draw: under this law no fourth charged-lepton
family forms. P07 argued three families from $S^3$ topology; the law
restates it at the mass level, and traces it to the three-term
structure of the fine-structure identity itself. The assignment runs
inward: bulk (deepest layer, largest coefficient) to the heaviest
family; the electron sits at the edge --- where observation runs out
(A281) and where the ladder is anchored. The unused edge coefficient
$2$ (which would give $d_1 = 2X = 0.0354$ were the electron not the
anchor) is the law's loose thread, flagged: a derivation must either
produce $c_1 = 0$ from the anchoring or absorb $2X$ into the
normalization.
\end{remark}

\begin{remark}[Look-elsewhere, stated]
The discrete assignment $(3, 16)$ was selected by the measured ratio
among perhaps twenty plausible small-integer pairs --- by itself,
weak. The scale match is the nontrivial event: $X$ had to land on a
corpus expression, and landed on $\alpha^{1/4}/\lambda_1$ ---
both factors already resident in the very mapping being corrected ---
at $10^{-3}$. The joint structure (ratio from $\rho$'s coefficients,
scale from the model's own ground state and the corpus's oscillation
parameter, slot count = family count) is what earns the flag;
no single match would. The law is falsifiable now: it must survive
improved $m_\tau$ ($\pm 0.12\,\mathrm{MeV}$ current) and any future
refinement of $V_{\rm obs}$ that moves $\lambda_n$.
\end{remark}

\paragraph{Consequences.} OI-286-1 closed as a target (the residual
is resolved by the law at $10^{-4}$); \textbf{OI-287-1} replaces it:
derive $\beta_n = \beta - c_n\kappa\Omega/\lambda_1$ from the
operator. P07's honest scorecard updates: electron anchor, muon
$0.008\%$, tau $0.009\%$ --- conditional on a flagged,
parameter-free, underived law.

\paragraph{Verifier.} \texttt{verify\_P287.py}: Richardson
eigenvalues with convergence check; the drops, the ratio vs $16/3$,
both $X$-matches; the mass predictions against PDG; the
$\alpha^{1/4} = \kappa\Omega$ identity; the fourth-level statement;
the edge-slot tension recorded. All PASS.

\end{document}
