% 298_Addendum_NodalDirection.tex
% OI-287-1. A fifth mechanism class is excluded (no fixed nodal or
% derivative pairing makes the coupling PEAK on the diagonal), but the
% direction is recovered: derivative/nodal functionals flip the
% diagonal from decreasing (A297 position moments, wrong) to
% increasing, the (2,3,16) sign. The degree information lives in the
% nodal structure. The open problem is reframed, not closed.
% Verifier: verify_P298.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 298: The Nodal Direction\\
\large A fifth excluded class, and the recovery of the
$(2,3,16)$ direction}
\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
reframed by a recovered direction) \quad|\quad
\textbf{Addresses:} OI-287-1 \quad|\quad
\textbf{Verifier:} \texttt{verify\_P298.py} --- all PASS

\begin{abstract}
A297 excluded the position-moment reading of the selection rule and
established that the charged-lepton families differ by node count, not
by amplitude at the center. This addendum tests the door A297 opened:
whether a fixed nodal or derivative pairing couples level $n$ to the
degree-$n$ monomial. Four pairings are tried with no free knobs: the
derivative overlap $\langle\psi_n'|x^k|\psi_n'\rangle$, the kinetic
coupling through $-d^2/dx^2$, the density-weighted overlap against the
degree-$k$ Taylor term of $\rho$, and the ladder transition to the
ground state. None makes the coupling peak on the diagonal, so a fifth
mechanism class is excluded. The result is not empty, however. Where
the position moments of A297 gave a diagonal ratio $(1, 0.74, 0.59)$
decreasing with degree, opposite to the target $(c_1,c_2,c_3)/c_1 =
(1, 1.5, 8)$, the three derivative and nodal functionals give
increasing ratios: $(1, 2.82, 5.17)$, $(1, 2.56, 4.78)$, and, for the
density-weighted overlap, $(1, 3.48, 46.35)$. The last reproduces the
target's signature jump at degree three. The sign that the amplitude
got backwards is recovered the moment the functional reads the nodal
structure. The degree information lives where A297 predicted, in the
nodes; what remains open is the mechanism that makes the rising
coupling peak on the diagonal and land on $(2,3,16)$.
\end{abstract}

% ----------------------------------------------------------------------
\section{The Four Pairings}

\begin{proposition}[No diagonal peak from a fixed pairing]
With $H$, $\psi_1,\psi_2,\psi_3$, and the target as in A297, form the
$3\times3$ matrices $M_{nk}$ for: $\langle\psi_n'|x^k|\psi_n'\rangle$;
$\langle\psi_n|(-d^2/dx^2)(x^k\psi_n)\rangle$; $\langle\psi_n|
c_k\pi^k x^k|\psi_n\rangle$; and $\langle\psi_n|x^k|\psi_1\rangle$.
Standardizing each column across the three levels, none attains its
row maximum on the diagonal, and the box control behaves the same.
No fixed nodal or derivative pairing reproduces the index-matching
arg-max pattern. A fifth mechanism class is excluded, joining A292,
A295, A296, and A297.
\end{proposition}

% ----------------------------------------------------------------------
\section{The Recovered Direction}

\begin{proposition}[Derivative functionals carry the right sign]
The normalized diagonal $M_{nn}/M_{11}$ runs $(1, 0.74, 0.59)$,
decreasing, for the A297 position moments, against the target
$(1, 1.5, 8)$. For the derivative overlap it runs $(1, 2.82, 5.17)$;
for the kinetic coupling $(1, 2.56, 4.78)$; for the density-weighted
overlap $(1, 3.48, 46.35)$. All three increase with degree, and the
density-weighted overlap reproduces the target's large jump from the
second to the third level. The direction of $(2,3,16)$ is a property
of the nodal and derivative structure, not of the amplitude.
\end{proposition}

\begin{remark}[What the reframing leaves open]
Two facts now bracket OI-287-1. The amplitude has the wrong sign
(A297); the nodal and derivative functionals have the right sign but
do not peak on the diagonal and do not reach $(2,3,16)$ in magnitude
(this addendum). The missing element is therefore not the location of
the degree information but a localizing factor that makes the rising
coupling concentrate on the diagonal. The surviving strength of A296,
$\kappa\,\Omega/\lambda_1$, names eigenvalue data; the natural next
test, recorded as the next pre-registration and not performed here, is
whether weighting the derivative coupling by the eigenvalue gaps
produces the diagonal peak. This is progress of the kind the registry
is built to keep: the open problem is strictly smaller than it was,
and the direction of its answer is now fixed.
\end{remark}

\paragraph{Status.} OI-287-1 remains open. Its form after this
addendum: \emph{find the factor that makes the increasing
derivative coupling peak on the diagonal at $(2,3,16)$}. No published
number changes; A287's lepton law and its $1\!\times\!10^{-4}$
accuracy stand.

\paragraph{Verifier.} \texttt{verify\_P298.py}: rebuilds the four
pairing matrices and their box controls, asserts the absence of a
diagonal peak, and asserts that the three derivative and nodal
functionals give increasing diagonals while the position moments do
not. All PASS.

\end{document}
