% 300_Addendum_NodeCountBracket.tex
% OI-287-1. A seventh mechanism class is excluded: the node count of
% the eigenfunctions (0,1,2) does bracket the target -- A299's base
% undershoots, a linear node gain overshoots -- but no single power
% (1+nu)^p reconciles the mu and tau entries (mu wants p=0.42, tau
% wants p=0.06). A uniform node-power gain is ruled out. The localizing
% factor lifts mu and tau by different amounts, so it is not a function
% of the node count alone.
% Verifier: verify_P300.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 300: The Node-Count Bracket\\
\large A seventh excluded class, and a two-sided bound on the
localizing factor}
\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
bounded from both sides) \quad|\quad
\textbf{Addresses:} OI-287-1 \quad|\quad
\textbf{Verifier:} \texttt{verify\_P300.py} --- all PASS

\begin{abstract}
A299 left OI-287-1 in a sharp state: the density-Taylor overlap divided
by $\lambda_n$ reproduces the target diagonal $(1, 1.5, 8)$ to
$(1, 1.12, 7.50)$, missing only by a factor that lifts the $\mu$ entry
and sharpens the diagonal. A297 established that the three families are
distinguished by their node count, $0$, $1$, $2$ interior zeros. This
addendum asks the obvious next question: is that node count the missing
localizing factor? It is not, but the way it fails is informative. The
node count brackets the target. The A299 base, with no node weighting,
undershoots ($\mu = 1.12 < 1.5$, $\tau = 7.50 < 8$); a linear node gain
$(1+\nu_n)$ overshoots ($\mu = 2.24$, $\tau = 22.50$). The target lies
strictly between. But no single power $(1+\nu_n)^p$ reconciles the two
entries: hitting $\mu = 1.5$ requires $p \approx 0.42$, while hitting
$\tau = 8$ requires $p \approx 0.06$. A uniform node-power gain is
therefore excluded as a seventh mechanism class. The localizing factor
lifts the $\mu$ and $\tau$ entries by different relative amounts, so it
is not a function of the node count alone.
\end{abstract}

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

\begin{proposition}[The node count brackets but does not reconcile]
Let $B_{nk} = \langle\psi_n|c_k\pi^k x^k|\psi_n\rangle / \lambda_n$ be
the A299 base, whose diagonal normalizes to $(1, 1.12, 7.50)$. Let
$\nu_n$ be the node count of $\psi_n$, equal to $(0, 1, 2)$. Weighting
row $n$ by $(1+\nu_n)$ gives the diagonal $(1, 2.24, 22.50)$; by
$(1+\nu_n)^2$, $(1, 4.48, 67.50)$. Neither peaks on the diagonal under
the column-standardized criterion, so a seventh mechanism class is
excluded. The target $(1, 1.5, 8)$ is bracketed: each of its nontrivial
entries lies between the unweighted value ($p = 0$) and the linear
value ($p = 1$). But the single power that places $\mu$ at $1.5$,
$p = \log(1.5/1.12)/\log 2 \approx 0.42$, is not the single power that
places $\tau$ at $8$, $p = \log(8/7.50)/\log 3 \approx 0.06$. No uniform
$(1+\nu_n)^p$ satisfies both.
\end{proposition}

\begin{remark}[What the bracket says, and does not say]
The result is two-sided, which is more than the prior nulls gave. The
selection rule's missing factor is now bounded: it must lift the $\mu$
row by more than nothing and less than a full factor of the node count,
and the $\tau$ row by far less in relative terms. The disparity is the
content. A factor that depended only on $\nu_n$ would lift both rows by
the same power and could be tuned to either entry but not both; the data
refuse that. So the missing factor is either jointly indexed by family
and degree, not by family alone, or it is a function of a quantity that
happens to coincide with the node count for the $\mu$ row but not the
$\tau$ row. The honest narrowing: the localizing factor is not a power
of the node count. It distinguishes the second and third families by
more than their zero counts differ.
\end{remark}

\paragraph{Status.} OI-287-1 remains open. Its form after this
addendum: the missing factor in
$\langle\psi_n|c_k\pi^k x^k|\psi_n\rangle / \lambda_n$ lifts the $\mu$
entry by roughly $34\%$ and the $\tau$ entry by roughly $7\%$, lies
between the identity and a linear node gain, and is not a uniform power
of the node count. No published number changes.

\paragraph{Verifier.} \texttt{verify\_P300.py}: recomputes the node
counts $(0,1,2)$, the A299 base and its linear and quadratic node
gains, confirms none peaks on the diagonal, confirms the target is
bracketed entry by entry, and confirms the $\mu$ and $\tau$ entries
demand different node powers ($0.42$ versus $0.06$). All PASS.

\end{document}
