% 286_Addendum_TauResidual.tex
% The tau residual: constant derived-beta excluded; the required
% exponent run quantified (beta_2 = 4.7069, beta_3 = 4.4768); all
% examined one-parameter extensions fail; the offset family fits
% trivially but lands on no corpus constants. Verdict: new structure
% required; targets pinned for any future derivation.
% Verifier: verify_P286.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 286: The Tau Residual\\
\large Constant $\beta$ excluded; the exponent run pinned;\\
one-parameter rescues refused}
\author{TOE Corpus}
\date{2026-06-10}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} Case B$-$ (exclusion theorem $+$ pinned targets; no
rescue found, none invented) \quad|\quad
\textbf{Files:} OI-286-1 \quad|\quad
\textbf{Verifier:} \texttt{verify\_P286.py} --- all PASS

\begin{abstract}
A281 settled P07's boundaries and left the tau as the operator's open
question. This audit determines what kind of question it is.
\emph{Exclusion:} with the settled spectrum
($\lambda_{1,2,3} = 16.507, 51.239, 102.020$, $N{=}2000$) and the
derived exponent $\beta = 6\mu_1/\mu_0 = 4.7601$, the required
per-level exponents are $\beta_2 = 4.7069$ (muon, $-1.12\%$) and
$\beta_3 = 4.4768$ (tau, $-5.95\%$): \textbf{no constant exponent fits
the three leptons}; the derived one overshoots both, increasingly with
level. \emph{The run is super-quadratic:} the drops
$(0.053, 0.283)$ scale by $5.3\times$ from level 2 to 3, excluding
corrections linear ($2\times$) or quadratic ($4\times$) in level index,
and log-eigenvalue, $\sqrt{\lambda}$, and index-exponential forms fare
no better --- \emph{no one-parameter corpus-natural extension fits muon
and tau simultaneously} (eight families tested). \emph{The trivial
rescue, refused:} the offset family $m_n/m_e =
((\lambda_n-\lambda_0)/(\lambda_1-\lambda_0))^{\beta'}$ has two
parameters and two targets, so it fits exactly --- at $\lambda_0 =
5.203$, $\beta' = 3.797$. Neither value lands on a corpus constant at
bar precision (nearest: $\beta' \approx 6(\mu_1/\mu_0)^2 = 3.776$ at
$0.5\%$; $\lambda_0 \approx \lambda_1/\pi = 5.254$ at $1.0\%$ ---
recorded as flags, worthless until one is derived, since a
two-parameter fit to two numbers identifies nothing).
\emph{Verdict:} the tau residual is not noise and not a tuning error;
it is \textbf{structure the mapping lacks} --- either a derived
running $\beta(n)$ or a $V_{\rm obs}$ correction that moves
$\lambda_3$ by $-12\%$ while holding $\lambda_2/\lambda_1$ nearly
fixed. Any future derivation now has pinned targets:
$(\beta_2, \beta_3) = (4.7069, 4.4768)$, equivalently
$(\lambda_0, \beta') = (5.203, 3.797)$. Filed as \textbf{OI-286-1}.
The honest scorecard of P07 after this cycle: electron exact (anchor),
muon $1.1\%$, tau $6.0\%$ in exponent --- a real partial success whose
failure mode is now precisely characterized, which is more useful to a
future derivation than the optimistic ``$94.8\%$'' ever was.
\end{abstract}

\begin{proposition}[Exclusion and pinning]
With the A281 spectrum and derived $\beta$: the per-level required
exponents $\beta_n = \ln(m_n/m_e)/\ln(\lambda_n/\lambda_1)$ are
$\beta_2 = 4.7069$, $\beta_3 = 4.4768$, against derived $4.7601$.
The drop ratio $(\beta-\beta_3)/(\beta-\beta_2) = 5.34$ excludes
level-linear ($2$) and level-quadratic ($4$) corrections; tested and
failing likewise: corrections proportional to $\ln(\lambda_n/\lambda_1)$
($1.61$), $\sqrt{\lambda_n}-\sqrt{\lambda_1}$ ($1.95$),
$\lambda_n-\lambda_1$ ($2.46$), $e^{n-2}$ ($2.72$), and the
multiplicative dressings $(1\pm\mu_1\alpha^2)^{n}$ ($\sim10^{-3}$,
two orders too small). Eight families, zero fits.
\end{proposition}

\begin{remark}[What a rescue must look like]
The pinned pair $(\lambda_0, \beta') = (5.203, 3.797)$ is a complete
reparametrization of the residual: any proposal can be checked against
it in one line. A proposal earns filing only by \emph{deriving} one of
the pinned values; matching them by construction is the trivial fit
this addendum already performed and refused. The nearest-flag pair
($6(\mu_1/\mu_0)^2$, $\lambda_1/\pi$) is recorded for the day a
derivation produces either from the operator's structure --- at which
point the other becomes a prediction, and the tau becomes a test
instead of a residual.
\end{remark}

\paragraph{Verifier.} \texttt{verify\_P286.py}: the spectrum
($N{=}2000$); the required exponents; the exclusion table (eight
families with their ratios against the required $5.34$); the offset
solve by bisection and the pinned pair; the nearest-flag distances.
All PASS.

\end{document}
