% 297_Addendum_SelectionEigenstructure.tex
% OI-287-1 (the selection rule, "why level n reads degree n"). A
% fourth mechanism class is excluded: index-matching is NOT carried by
% the position-moment structure of the unperturbed observation
% operator's eigenfunctions. By-product: the Dirichlet boundary forces
% every level linear at the center, so the distinguishing feature of
% level n is its node count, not a degree-n amplitude at the origin.
% The open problem is sharpened, not closed. Verifier: verify_P297.py.
% Copyright Léon Fernando Vlegels -- CC BY 4.0

\documentclass[11pt]{article}
\usepackage{amsmath,amssymb,amsthm}
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\usepackage{hyperref}

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

\title{Addendum 297: The Selection Rule Is Not in the Amplitude\\
\large A fourth excluded mechanism class for OI-287-1, and a
sharpening}
\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
sharpened, not closed) \quad|\quad \textbf{Addresses:} OI-287-1
\quad|\quad \textbf{Verifier:} \texttt{verify\_P297.py} --- all PASS

\begin{abstract}
The selection rule (A296, OI-287-1) states that charged-lepton family
$n$ draws the degree-$n$ coefficient of the density $\rho$. Three
mechanism classes for this index matching are already excluded: the
universal first-order coupling (A292), the time-average dressing at
the wrong order in $\kappa$ (A295), and the bare diagonal Taylor probe
(A296). Each perturbed the observation operator from outside. This
addendum tests the remaining inside reading: that the eigenfunctions
$\psi_n$ of the unperturbed operator already carry degree-$n$
character, so that index matching is a property of the bare spectrum
rather than an inserted coupling. The reading is excluded. The
position-moment matrix elements $\langle\psi_n|x^k|\psi_n\rangle$
carry no diagonal preference: they are monotone in both indices and
indistinguishable from the structureless box, and their normalized
diagonal ratios $(1, 0.74, 0.59)$ run opposite to the target
$(1, 1.5, 8)$. The probe leaves one positive structural fact. Under
the Dirichlet condition derived in A281, every eigenfunction has
leading power one at the center; the families therefore cannot be
distinguished by amplitude at the origin, only by node count, the
$n$-th level having $n-1$ interior zeros. OI-287-1's surviving form
is correspondingly narrower: whatever couples level $n$ to degree $n$
acts through the nodal or derivative structure of the spectrum, not
through position-moment overlap.
\end{abstract}

% ----------------------------------------------------------------------
\section{The Test and Its Result}

\begin{proposition}[No index matching in the position moments]
Let $H = -\,d^2/dx^2 + V(x)$ with
$V = [\rho'(x)]^2/(2\Omega^2) + E_{\rm self}\,x^2(1-x)^2$ and
$\rho'(x) = 48\pi^3 x^2 + 6\pi^2 x + 2\pi$, Dirichlet on $[0,1]$, as
fixed in P03 and used by \texttt{verify\_P296.py}. Let $\psi_1,
\psi_2, \psi_3$ be its lowest three eigenfunctions, identified with
the families $e, \mu, \tau$. Form $A_{nk} = \langle\psi_n|x^k|\psi_n
\rangle$ for $k = 1,2,3$. After standardizing each column across the
three levels, no row attains its maximum on the diagonal: the
arg-max pattern is $(k_3, k_1, k_3)$, not $(k_1, k_2, k_3)$. The same
construction on the structureless box $\psi_n = \sqrt{2}\sin(n\pi x)$
shows the same absence of diagonal preference. The normalized
diagonal ratios $\langle\psi_n|x^n|\psi_n\rangle /
\langle\psi_1|x|\psi_1\rangle = (1, 0.74, 0.59)$ are monotone
decreasing, opposite to the target $(c_1,c_2,c_3)/c_1 = (1, 1.5, 8)$.
The index matching of A296 is therefore not present in the
position-moment structure of the bare eigenfunctions.
\end{proposition}

\noindent This closes a fourth mechanism class. With A292
(universal first order), A295 (dressing, wrong $\kappa$-order), A296
(external diagonal probe), and the present position-moment reading
all excluded, the surviving requirement of A296 stands unaltered: the
coupling must join level $n$ to degree $n$ at first order in $\kappa$
with strength $\kappa\,\Omega/\lambda_1$.

% ----------------------------------------------------------------------
\section{Why the Amplitude Cannot Carry It}

\begin{proposition}[Linear onset at the center]
Each eigenfunction has leading power one as $x \to 0$: the fitted
log-log slopes are $(0.998, 0.992, 0.985)$ for the three levels. This
is forced, not fitted. The center is the point where observation
vanishes (A281), and the boundary condition there is Dirichlet,
$\psi(0) = 0$. A smooth potential then gives every level the same
linear onset, so the degree of the leading term at the origin is one
for all $n$ and cannot encode $n$.
\end{proposition}

\begin{remark}[The feature is nodal]
What distinguishes level $n$ is not its shape at the center but the
number of times it crosses zero: $\psi_n$ has $n-1$ interior nodes.
The author's reading, recorded as a reading and not a result, is that
this is the natural home of the rule. The center must be a zero for
every family, so the families cannot differ there; they can differ
only in how many further zeros they carry, and "family $n$ draws
degree $n$" should then be a statement about the $n$-th nodal sector
rather than about the amplitude. The next probe (a separate
pre-registration) tests whether a nodal or derivative pairing
reproduces the coefficients $(2,3,16)$ that the position moments do
not.
\end{remark}

\paragraph{Status.} OI-287-1 remains open. Its surviving form, after
this addendum, is one sentence: \emph{find the nodal or derivative
pairing that couples the $n$-th level to the degree-$n$ monomial at
strength $\kappa\,\Omega/\lambda_1$}. Nothing here changes a published
number; the lepton law and its $1\!\times\!10^{-4}$ accuracy (A287)
are untouched.

\paragraph{Verifier.} \texttt{verify\_P297.py}: rebuilds $H$,
recomputes $A_{nk}$ and the box control, and asserts the absence of
diagonal preference, the opposed diagonal ratios, and the
unit leading powers. All PASS.

\end{document}
