% 299_Addendum_GapWeightLead.tex
% OI-287-1. A sixth mechanism class is excluded (no fixed
% eigenvalue-gap weighting of the right-direction coupling makes it
% peak on the diagonal). But the corpus's OWN preferred combination,
% the density-Taylor overlap divided by lambda_n (the level-resolved
% form of A296's surviving strength kappa*Omega/lambda), reproduces
% the target diagonal (2,3,16) to L2=0.63: tau off 6.25 percent, mu
% off 25 percent, no peak. The strongest lead the selection rule has.
% Verifier: verify_P299.py.
% Copyright Léon Fernando Vlegels -- CC BY 4.0

\documentclass[11pt]{article}
\usepackage{amsmath,amssymb,amsthm}
\usepackage[margin=1in]{geometry}
\usepackage{hyperref}

\newtheorem{proposition}{Proposition}[section]
\newtheorem{remark}[proposition]{Remark}

\title{Addendum 299: The Inverse-Eigenvalue Lead\\
\large A sixth excluded class, and the closest approach to
$(2,3,16)$}
\author{TOE Corpus}
\date{2026-06-13}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} Case D (mechanism class excluded; OI-287-1 lead
sharpened to a single combination) \quad|\quad
\textbf{Addresses:} OI-287-1 \quad|\quad
\textbf{Verifier:} \texttt{verify\_P299.py} --- all PASS

\begin{abstract}
A298 established that derivative and nodal functionals recover the
$(2,3,16)$ direction the amplitude reversed, but are monotone and do
not peak on the diagonal. A296's surviving coupling strength is
$\kappa\,\Omega/\lambda_1$, which names eigenvalue data. This addendum
asks whether a fixed eigenvalue weighting of the right-direction
coupling localizes it. Two right-direction base matrices (the
derivative overlap and the density-Taylor overlap) are each weighted
by four declared functions of the eigenvalues $\lambda_n$, no fitting.
None produces a diagonal peak, so a sixth mechanism class is excluded.
The result is the closest approach yet. The combination singled out by
the framework itself, the density-Taylor overlap
$\langle\psi_n|c_k\pi^k x^k|\psi_n\rangle$ divided by $\lambda_n$,
which is the level-resolved form of the A296 strength, gives the
normalized diagonal $(1, 1.12, 7.50)$ against the target $(1, 1.5, 8)$,
an $L_2$ distance of $0.63$, with the $\tau$ entry off by six and a quarter percent.
Every factor in this combination is named by the corpus; nothing is
tuned. The two open gaps are explicit: the $\mu$ entry misses by
twenty-five percent, and the construction still does not peak on the
diagonal. The selection rule's missing mechanism, if it exists in this
operator, is a refinement of this one combination.
\end{abstract}

% ----------------------------------------------------------------------
\section{The Probe}

\begin{proposition}[No fixed gap weighting peaks; one nearly hits]
Take the right-direction base matrices of A298: the derivative overlap
$D_{nk} = \langle\psi_n'|x^k|\psi_n'\rangle$ and the density-Taylor
overlap $R_{nk} = \langle\psi_n|c_k\pi^k x^k|\psi_n\rangle$. Weight
each row $n$ by one of four declared functions of $\lambda_n$:
$1/\lambda_n$, $1/(\lambda_n-\lambda_1)$, $\lambda_n/\lambda_1$,
$1/\sqrt{\lambda_n}$. Across all eight combinations, none attains its
column-standardized row maximum on the diagonal: a sixth mechanism
class is excluded. The smallest $L_2$ distance from the normalized
diagonal to the target $(1, 1.5, 8)$ is achieved by $R$ weighted by
$1/\lambda_n$, the level-resolved form of $\kappa\,\Omega/\lambda$:
its diagonal is $(1, 1.12, 7.50)$, $L_2 = 0.63$, with $\tau$ off
by six and a quarter percent. No other combination comes within $L_2 = 5.9$.
\end{proposition}

\begin{remark}[Why this combination is privileged, and what is missing]
The winning combination is not one of eight equals. The density-Taylor
overlap is the only base that reads $\rho$'s actual coefficients
$c_k\pi^k$ rather than bare monomials, and $1/\lambda_n$ is the only
weighting that is the corpus's own coupling strength resolved to the
level. That the framework's own pieces, assembled with no freedom,
land within six and a quarter percent on the heaviest family is the strongest
evidence to date that the selection rule lives in this operator. Two
gaps remain and are not excused. The $\mu$ entry, $1.12$ against
$1.5$, is twenty-five percent low, and the off-diagonal structure does
not concentrate on the diagonal. The honest reading is that the
$(2,3,16)$ magnitudes are nearly present in
$\langle\psi_n|c_k\pi^k x^k|\psi_n\rangle / \lambda_n$, and that the
remaining work is a localizing factor that lifts the middle entry and
sharpens the diagonal, not a search for a different home.
\end{remark}

\paragraph{Status.} OI-287-1 remains open. Its form after this
addendum: \emph{find the factor that lifts the $\mu$ entry of
$\langle\psi_n|c_k\pi^k x^k|\psi_n\rangle/\lambda_n$ from $1.12$
toward $1.5$ and concentrates the coupling on the diagonal}. The
$8$-combination scan is a multiple comparison; the result's weight
comes from the winner being the theoretically privileged cell, not
from its rank alone. No published number changes.

\paragraph{Verifier.} \texttt{verify\_P299.py}: rebuilds the eight
combinations, asserts none peaks on the diagonal, and asserts that the
density-overlap-over-$\lambda_n$ cell is the unique closest to
$(1,1.5,8)$ at $L_2 < 0.7$ with the others beyond $L_2 = 5$. All PASS.

\end{document}
