% 309_Addendum_MeasureFamily.tex
% Copyright Léon Fernando Vlegels -- CC BY 4.0
% OI-287-1. A309 sweeps a declared principled density-measure family on the
% lead A307 returned. A307 localized the missing factor of the A299 selection
% rule to the inner-product measure: the density weight rho dx lifts mu from
% the flat-measure base 1.12 to 1.34, and a change to the eigenstates with the
% overlap held flat does not move mu. A307 left two quantities open: the exact
% weight, and the recovery of the diagonal peak. A309 declares a small family
% of density-type weights, the power weights x^a, the density rho and its
% powers rho^p, and the moment-measures x^m rho, nine measures in all with the
% flat control, and solves the generalized eigenproblem H psi = lambda W psi
% for each by the numpy Cholesky reduction. No measure restores a valid
% diagonal peak, so the HIT criterion is met by none. The closest member by L2
% is the non-canonical power x dx at 0.3252, the same value A307 returned, not
% improved; the corpus-canonical density-type members all sit farther. The
% density rho dx is therefore not forced by the L2 fit. The honest verdict is
% NULL-ish: the family does not improve on A307. One structural fact is
% recorded, that mu over the rho-power sub-family peaks at the unit power
% W = rho, a local maximum rather than a monotone crossing, which leaves rho
% distinguished as a stationary point of the lift but not as the fit minimum.
% Verifier: verify_P309.py.

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

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

\title{Addendum 309: The Measure Family\\
\large A declared sweep of density-type weights leaves the lift located but
the weight unforced}
\author{TOE Corpus}
\date{2026-06-13}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} Case D (null-ish; the declared family does not improve on the
A307 lead, and no canonical measure is distinguished by fit) \quad|\quad
\textbf{Addresses:} OI-287-1 \quad|\quad
\textbf{Verifier:} \texttt{verify\_P309.py} --- all PASS

\begin{abstract}
A307 located the factor missing from the A299 base diagonal $(1, 1.12, 7.50)$
in the inner-product measure used to form the overlap. The density weight
$\rho\,dx$, with the eigenstates drawn from the generalized eigenproblem
$H\psi = \lambda\rho\psi$ and the overlap formed in that measure, lifts $\mu$
from $1.12$ to $1.34$, while a change to the eigenstates alone with the overlap
held flat does not move $\mu$. A307 left open the exact weight and the recovery
of the diagonal peak. This addendum declares a principled family of
density-type weights and sweeps it without fitting. The family has nine
members: the flat control $W = 1$; the power weights $W = x^a$ for
$a \in \{1, 2, 3\}$; the density $W = \rho$ and its powers $W = \rho^{1/2}$ and
$W = \rho^2$; and the moment measures $W = x\rho$ and $W = x^2\rho$, the natural
weights of P18's moment hierarchy $\mu_n = \int x^n\rho$. Every member is
strictly positive on the open interval and so a valid Sturm-Liouville weight;
each generalized eigenproblem is solved by the Cholesky reduction
$C = L^{-1}HL^{-T}$ with $W = LL^{T}$. No member restores a valid diagonal peak
under the column-standardized criterion paired with the sine-box control, so
the pre-registered HIT criterion, a valid peak together with all three entries
within twenty percent of $(1, 1.5, 8)$, is met by none. The closest member by
$L^2$ is the linear power $W = x$ at $0.3252$, the same value A307 returned and
not an improvement; every corpus-canonical density-type member sits farther,
$\rho\,dx$ itself at $0.7406$. The density measure is therefore not forced by
the fit. One structural fact survives the null reading. Over the rho-power
sub-family the lift $\mu$ peaks at the unit power, $\mu = 1.31$ at
$W = \rho^{1/2}$, $\mu = 1.34$ at $W = \rho$, and $\mu = 1.28$ at $W = \rho^2$.
The density is a local maximum of the lift, not a monotone crossing of the
target. The verdict is null-ish: the declared family does not improve on A307,
and the exact weight remains the open quantity.
\end{abstract}

% ----------------------------------------------------------------------
\section{The Sweep}

\begin{proposition}[No declared density-type weight restores the peak, and the
canonical density is not the fit minimum]
Let $B^W_{nk} = \langle\psi^W_n|c_k\pi^k x^k|\psi^W_n\rangle_W /
\langle\psi^W_n|\psi^W_n\rangle_W / \lambda^W_n$ be the A299-style diagonal
overlap formed in the measure $W\,dx$, with $\psi^W_n$ and $\lambda^W_n$ drawn
from the generalized eigenproblem $H\psi = \lambda W\psi$ for the operator
$H = -d^2/dx^2 + \rho'(x)^2/(2\Omega^2) + E_{\mathrm{self}}\,x^2(1-x)^2$.
Evaluated on the nine declared weights, the flat control $W = 1$ returns the
A299 diagonal $(1, 1.12, 7.50)$ at $L^2 = 0.6281$. The power weights return
$\mu = 1.28, 1.33, 1.34$ for $a = 1, 2, 3$, with $W = x$ closest at
$L^2 = 0.3252$. The density and its powers return $\mu = 1.31, 1.34, 1.28$ for
$W = \rho^{1/2}, \rho, \rho^2$, the density itself at $L^2 = 0.7406$. The moment
weights $W = x\rho$ and $W = x^2\rho$ return $\mu = 1.32$ and $\mu = 1.31$ at
$L^2 = 0.75$ and $L^2 = 3.89$. No member satisfies the column-standardized
diagonal-peak criterion paired with the failing sine-box control, so the HIT
criterion is met by none. The closest member by $L^2$ is the linear power
$W = x$, which is not a corpus-canonical density object, and its distance
$0.3252$ equals the A307 value and does not improve it. Every canonical
density-type member sits farther from the target than this non-canonical power.
The density measure $\rho\,dx$ is therefore not forced by the fit over the
declared family.
\end{proposition}

\begin{remark}[The density is a stationary point of the lift, not its minimum]
The sweep separates two readings of "distinguished." The fit reading asks which
weight is closest to the target by $L^2$, and answers with the non-canonical
power $W = x$, leaving the canonical density unforced. The lift reading asks how
$\mu$ depends on the weight within a one-parameter canonical sub-family, and
answers differently. Over the rho-power family $W = \rho^p$ the lift is
$\mu = 1.31, 1.34, 1.28$ at $p = \tfrac12, 1, 2$. The value is not monotone in
the exponent; it rises to the unit power and falls past it. The density $\rho$,
the corpus-named member, sits at the local maximum of the lift over its own
power family. This is a weaker statement than the pre-registered
strengthened-partial condition, which required either that the fit minimum be
canonical or that $\mu$ cross the target monotonically, and neither holds. It is
recorded because it is the one place where $\rho$ stands apart from its
neighbours by an intrinsic property of the lift rather than by proximity to the
target. The honest summary is that A307's locus stands on its own argument, the
contrast between the density measure and the Sturm-Liouville eigenstate change,
and not on $\rho$ being the closest weight. The exact weight that would carry
$\mu$ the full way to $1.5$ and restore the diagonal peak is not in this family.
Neither the power weights, which approach the target in $\mu$ but never peak,
nor the canonical density weights, which lift $\mu$ comparably but fit worse,
recover the column structure. The open quantity is unchanged from A307 and the
search is not narrowed by this set.
\end{remark}

\paragraph{Status.} OI-287-1 remains open. The A307 reading is unchanged: the
missing factor lives in the measure used to form the overlap, established there
by the contrast between the density measure and the eigenstate-only change, not
by any fit. A309 adds that the declared density-type family does not improve on
that lead. No member restores the diagonal peak, and the closest member by
$L^2$ is the non-canonical power $W = x$ at $0.3252$, the A307 value, so the
density $\rho\,dx$ is not forced as the weight. The single structural fact
recorded is that $\mu$ over the rho-power sub-family peaks at the unit power, so
$\rho$ is a local maximum of the lift but not the fit minimum. No published
number changes.

\paragraph{Verifier.} \texttt{verify\_P309.py}: recomputes the flat
eigenstructure (eigenvalues ordered $(16.51, 51.24, 102.02)$, node counts
$(0,1,2)$), confirms the flat control diagonal reproduces $(1, 1.12, 7.50)$,
confirms the density measure $W = \rho$ lifts $\mu$ to $1.34$ reproducing the
A307 lead, confirms that no declared measure yields a valid diagonal peak and
the HIT criterion is met by none, confirms that the closest non-control member
is the non-canonical power $W = x$ at $L^2 = 0.3252$ which does not improve the
A307 value so $\rho\,dx$ is not forced by fit, and confirms that the rho-power
lift is not monotone in the exponent but peaks at the unit power. Eight checks,
all PASS.

\end{document}
