% 307_Addendum_Measure.tex
% Copyright Léon Fernando Vlegels -- CC BY 4.0
% OI-287-1. A positive localization. A305 established that the missing
% factor in the A299 base cannot be a kernel acting on the overlap matrix,
% because off-diagonal suppression leaves the diagonal entry fixed and so
% cannot lift mu from 1.12 toward 1.5; the localizing information is not
% present in the overlap matrix to be recovered by reweighting. This
% addendum tests the only remaining locus: a factor that enters earlier
% than the overlap, in the measure used to form it or in the eigenstates
% themselves. Six declared candidates run without fitting. The flat-measure
% control recovers (1, 1.12, 7.50). A density-weighted measure rho dx, and
% the simple power measures x^2 dx and x^3 dx, move mu into [1.3, 1.7]. A
% change of the eigenstates alone, through the Sturm-Liouville weight with
% the overlap held flat, does not move mu. The lift lives in the measure,
% not in the bare eigenstates. No candidate produces a valid diagonal peak,
% so the result is a partial localization, not a closure.
% Verifier: verify_P307.py.

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

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

\title{Addendum 307: The Measure\\
\large A positive localization of the missing factor to the inner product}
\author{TOE Corpus}
\date{2026-06-13}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} Case C (partial localization; OI-287-1 lead refined,
not closed) \quad|\quad
\textbf{Addresses:} OI-287-1 \quad|\quad
\textbf{Verifier:} \texttt{verify\_P307.py} --- all PASS

\begin{abstract}
A305 closed the space of post-overlap reweightings. The factor missing from
the A299 base $B_{nk} = \langle\psi_n|c_k\pi^k x^k|\psi_n\rangle / \lambda_n$,
diagonal $(1, 1.12, 7.50)$ against the target $(1, 1.5, 8)$, is not a function
of the family index alone (A300), not a function of the monomial degree alone
(A303), and not a kernel acting on the overlap matrix (A305). The reason A305
gave is structural: a kernel that suppresses off-diagonal entries equals one
on the matched pair and so multiplies the diagonal entry by one, leaving it
fixed; the lift the target demands raises a diagonal entry, and the
localizing information is not present in the overlap matrix to be recovered by
any reweighting of it. One locus remains. The factor must enter earlier than
the overlap, in the inner product that forms it or in the eigenstates
themselves. This addendum tests that locus with six declared candidates and no
fitting. The flat measure $dx$ is run as a control and recovers
$(1, 1.12, 7.50)$. A density-weighted measure $\rho\,dx$, with the eigenstates
drawn from the corresponding generalized eigenproblem $H\psi = \lambda\rho\psi$
and the overlap and normalization both formed in that measure, lifts $\mu$ to
$1.34$. The power measures $x^2\,dx$ and $x^3\,dx$ lift $\mu$ to $1.33$ and
$1.34$. The linear measure $x\,dx$ lifts $\mu$ to $1.28$ and gives the closest
overall fit at $L^2 = 0.33$, half the control distance. A change of the
eigenstates alone, the Sturm-Liouville weight $\rho$ applied to the operator
with the overlap held in the flat measure, leaves $\mu$ at $1.11$ and does not
move it. The dressing of the operator by $\rho/\Omega$ moves $\mu$ only to
$1.14$. The signal is therefore located. The lift toward $1.5$ comes from the
measure in which the overlap is formed, not from a change to the eigenstates
left in the flat measure. No candidate produces a valid diagonal peak under
the column-standardized criterion paired with the sine-box control, so this is
a partial localization and not the closure of OI-287-1. The next probe should
search measures rather than eigenstate deformations.
\end{abstract}

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

\begin{proposition}[The missing lift lives in the measure, not the eigenstates]
Let $B_{nk} = \langle\psi_n|c_k\pi^k x^k|\psi_n\rangle / \lambda_n$ be the A299
base, diagonal $(1, 1.12, 7.50)$, formed in the flat measure $dx$ from the
ground states of $H = -d^2/dx^2 + \rho'(x)^2/(2\Omega^2) + E_{\mathrm{self}}\,
x^2(1-x)^2$. For a strictly positive weight $W(x)$ define the $W$-overlap
$B^W_{nk} = \langle\psi^W_n|c_k\pi^k x^k|\psi^W_n\rangle_W /
\langle\psi^W_n|\psi^W_n\rangle_W / \lambda^W_n$, with $\psi^W_n$ and
$\lambda^W_n$ drawn from the generalized eigenproblem $H\psi = \lambda W\psi$
and the inner product $\langle f\rangle_W = \int f\,W\,dx$. Evaluated on the
declared weights, the flat control $W = 1$ returns $\mu = 1.12$; the density
weight $W = \rho$ returns $\mu = 1.34$; the power weights $W = x^2$ and
$W = x^3$ return $\mu = 1.33$ and $\mu = 1.34$; and $W = x$ returns
$\mu = 1.28$ with the smallest distance $L^2 = 0.33$. When instead the
eigenstates are changed by the Sturm-Liouville weight $W = \rho$ but the
overlap is formed in the flat measure, $\mu$ remains at $1.11$ and does not
move toward $1.5$; when the operator is dressed by $\rho/\Omega$ with the
overlap flat, $\mu$ reaches only $1.14$. The lift therefore tracks the measure
in which the overlap is formed and not the change to the eigenstates. No
candidate satisfies the column-standardized diagonal-peak criterion paired
with the failing sine-box control, so the pre-registered HIT criterion, a
valid peak together with all three entries within $20\%$, is met by none. The
result is a partial localization of the missing factor to the inner product.
\end{proposition}

\begin{remark}[Why the measure carries the lift, and what is left]
The contrast between the density measure $\rho\,dx$ and the Sturm-Liouville
weight is the decisive reading. Both draw their states from the same
generalized eigenproblem $H\psi = \lambda\rho\psi$, so the eigenstates are
identical between them. They differ only in the measure used to form the
overlap. The measure that carries $\rho$ into the integral lifts $\mu$ to
$1.34$; the one that forms the overlap flat leaves $\mu$ at $1.11$. The lift is
attached to the measure, not to the states. This is consistent with A305 read
forward. A305 found that the localizing information is absent from the flat
overlap matrix, and a reweighting after the fact cannot supply it. The present
result names where it is instead present: in the geometric density inner
product, the integral against $\rho\,dx$, which is the natural pairing of
P18's density-coupling sector. The lift is real and points the right way, from
$1.12$ toward $1.5$, but it stops short. The largest single move reaches
$1.34$, and the closest fit by $L^2$ comes from the linear measure $x\,dx$
rather than from $\rho$ itself, which leaves the precise weight underdetermined
by this small declared set. No measure in the set restores the diagonal peak,
the second half of the criterion, so the column structure that distinguishes
the operator's eigenfunctions from a generic basis is not yet recovered. The
honest statement is that the locus is now fixed and the magnitude is not. The
factor enters in the inner product; the exact weight that carries $\mu$ the
full way to $1.5$ while also restoring the peak is the open quantity.
\end{remark}

\paragraph{Status.} OI-287-1 remains open, with its lead refined rather than
unchanged. The missing factor is now located in the measure used to form the
overlap, not in the eigenstates and not in a post-overlap correction. The
density inner product $\rho\,dx$ lifts $\mu$ from $1.12$ to $1.34$ and the
power measures lift it comparably, while a change to the eigenstates with the
overlap held flat does not move $\mu$. The remaining quantities are the exact
weight and the recovery of the diagonal peak, neither supplied by the six
declared candidates. No published number changes.

\paragraph{Verifier.} \texttt{verify\_P307.py}: recomputes the flat
eigenstructure (eigenvalues ordered $(16.51, 51.24, 102.02)$, node counts
$(0,1,2)$ for both the flat and the $\rho$-weighted states), confirms the flat
control diagonal reproduces $(1, 1.12, 7.50)$, confirms that the density
measure and the $x^2$ and $x^3$ measures carry $\mu$ into the band
$[1.3, 1.7]$ while the Sturm-Liouville eigenstate change with a flat overlap
does not move $\mu$, and confirms the partial-signal conditions, that no
candidate yields a valid diagonal peak and the closest non-control candidate
sits at $L^2 = 0.33$ below the control distance $0.63$. All PASS.

\end{document}
