% 311_Addendum_GeometricMeasure.tex
% Copyright Léon Fernando Vlegels -- CC BY 4.0
% OI-287-1. A311 asks whether the inner-product measure A307 located is FORCED
% by the B^4/S^3/Hopf radial geometry rather than searched. The base metric of
% P18 S2.1 is ds^2 = dr^2 + r^2 dOmega_3^2, so the 4-volume element is r^3 dr
% dOmega_3 and the geometrically forced radial measure is r^3 dr, not flat dr.
% P18 S2.2 gives the radial operator with the -(3/r) d/dr term, self-adjoint in
% r^3 dr. Five declared candidates run without fitting: two forced by the arena
% (the volume weight r^3 on the canonical operator, and the genuine 4D-radial
% operator in its r^3 measure), three for comparison (flat control, A307's
% density rho, a Hopf-lapse weight). The forced volume r^3 lifts mu to 1.337,
% into the band, by essentially the same amount as A307's rho at 1.340. The
% coincidence is analytic: rho = 16 pi^3 r^3 + lower, the leading r^3 term
% carrying 0.933 of rho(1), so A307's density measure is the arena volume at
% leading order. The 4D-radial operator gives the closest fit at L2 = 0.399. No
% measure restores the diagonal peak. The verdict is strengthened-partial: the
% measure is the forced 4D arena volume, not a fit parameter, short of full
% closure. Verifier: verify_P311.py.

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

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

\title{Addendum 311: The Geometric Measure\\
\large The inner product is the forced volume of the arena, not a searched weight}
\author{TOE Corpus}
\date{2026-06-13}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} Case C (strengthened partial; the measure is forced by the
arena, OI-287-1 lead grounded, not closed) \quad|\quad
\textbf{Addresses:} OI-287-1 \quad|\quad
\textbf{Verifier:} \texttt{verify\_P311.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, lifting $\mu$ from $1.12$ to
$1.34$ with the density weight $\rho\,dx$. A309 found that this weight is not
fixed by fit: the closest member of a declared density-type family was the
non-canonical power $x\,dx$, and $\rho\,dx$ sat farther from the target. The
question left open was therefore whether any measure is forced by the geometry
rather than found by search. This addendum tests the natural candidate. Paper 18
S2 places the master operator on sections over the four-ball $B^4$ with boundary
$S^3$, Hopf-fibered, and gives the base metric $ds^2 = dr^2 + r^2\,d\Omega_3^2$.
The four-volume element is then $r^3\,dr\,d\Omega_3$, and the radial reduction of
a rotationally symmetric problem carries the measure $r^3\,dr$, the surface of
$S^3$ at radius $r$ scaling as $r^3$. This is the geometrically forced weight,
fixed by the arena and not chosen. Paper 18 S2.2 gives the matching radial
operator $-d^2/dr^2 - (3/r)\,d/dr$, self-adjoint in $r^3\,dr$ in the
Sturm-Liouville form $-r^{-3}\,d/dr(r^3\,d/dr)$. Five candidates run without
fitting: two forced by the arena, the volume weight $r^3$ on the canonical
operator and the genuine four-dimensional radial operator in its $r^3$ measure,
and three for comparison, the flat control, A307's density $\rho$, and an $S^1$
Hopf-lapse weight $r^3\sqrt{1-r^2}$. The forced volume $r^3$ lifts $\mu$ to
$1.337$, into the band $[1.3, 1.7]$, by essentially the same amount as A307's
density at $1.340$. The coincidence is analytic. The density is
$\rho(r) = 16\pi^3 r^3 + 3\pi^2 r^2 + 2\pi r$, with leading term $16\pi^3 r^3$
carrying $0.933$ of $\rho(1)$, so on $[0,1]$ the density measure is the arena
volume $r^3$ times a constant to leading order, and A307's lift is the geometric
lift in disguise. The four-dimensional radial operator in its own measure gives
the closest fit of all candidates at $L^2 = 0.399$. No measure restores a valid
diagonal peak, so the pre-registered HIT criterion is met by none. The verdict is
strengthened-partial: the measure that carries the lift is the forced volume of
the $B^4$ arena, established as geometry rather than fit, and the magnitude piece
of OI-287-1 reduces to using the right arena measure, short of the diagonal-peak
recovery that would close it.
\end{abstract}

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

\begin{proposition}[The forced arena volume $r^3$ carries the A307 lift]
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 canonical
operator $H = -d^2/dx^2 + \rho'(x)^2/(2\Omega^2) + E_{\mathrm{self}}\,x^2(1-x)^2$.
The four-volume element of the base $B^4$ with metric $ds^2 = dr^2 +
r^2\,d\Omega_3^2$ fixes the radial measure $W_{\mathrm{geom}} = r^3$, not the
flat $dx$. Evaluated on the five declared weights, the flat control $W = 1$
returns the A299 diagonal $(1, 1.12, 7.50)$ at $L^2 = 0.628$. The forced volume
$W = r^3$ returns $(1, 1.337, 8.706)$ with $\mu = 1.337$ in the band
$[1.3, 1.7]$. The genuine four-dimensional radial operator
$-d^2/dr^2 - (3/r)\,d/dr + V$, self-adjoint in $r^3\,dr$ and solved in its own
measure, returns $(1, 1.222, 8.286)$ at $L^2 = 0.399$, the smallest distance of
any candidate. The comparison density measure $W = \rho$ returns $\mu = 1.340$,
reproducing A307. The Hopf-lapse weight $W = r^3\sqrt{1-r^2}$ returns
$\mu = 1.179$, below the band. The forced volume and the density agree to within
$0.003$ in $\mu$ because $\rho(r) = 16\pi^3 r^3 + 3\pi^2 r^2 + 2\pi r$ has leading
term $16\pi^3 r^3$, which carries $0.933$ of $\rho(1)$; the density measure is
the arena volume to leading order. 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 twenty percent of $(1, 1.5, 8)$, is met by none.
\end{proposition}

\begin{remark}[The measure is forced, not fitted]
A309 left the measure underdetermined by fit, the closest weight being the
non-canonical power $x\,dx$. A311 changes the question from which weight fits to
which weight the arena requires. The four-ball $B^4$ with its round-$S^3$ slices
has one radial volume element, $r^3\,dr$, and it is not a parameter. That this
forced weight lifts $\mu$ to $1.337$, the same lift A307 read from the density
$\rho$, is the substantive result: the two are the same measure at leading order,
and the density coupling of P18 S3.5 carries the geometry of the bulk through its
cubic head. The flat $dx$ of A299, by contrast, is the one un-geometric choice,
the radial measure of a line rather than of a four-ball, and it is exactly the
choice that leaves $\mu$ at $1.12$. The reading of A307 is thereby grounded. The
factor lives in the measure, and the measure is the volume of the arena. Two
limits remain. The lift stops at $1.34$ rather than reaching $1.5$, and no
measure in the set restores the diagonal peak, so the column structure that
distinguishes the operator's eigenfunctions from a generic basis is not yet
recovered. The four-dimensional radial operator improves the fit to $L^2 = 0.399$
by sharpening $\tau$ toward $8$ while holding $\mu$ near $1.22$, which suggests
the residual gap is shared between the measure and the operator's angular and
fiber sectors not carried by the radial reduction alone. The honest statement is
that the measure is now forced and the magnitude is not yet complete. The
geometry fixes the weight; the remaining lift and the peak are the open
quantities.
\end{remark}

\paragraph{Status.} OI-287-1 remains open, with its A307 lead grounded in
geometry rather than left as a fit. The inner-product measure that carries the
lift is the four-volume element $r^3\,dr$ of the $B^4$ arena, fixed by the base
metric $ds^2 = dr^2 + r^2\,d\Omega_3^2$ of P18 S2.1 and matched to the radial
operator $-d^2/dr^2 - (3/r)\,d/dr$ of P18 S2.2. The forced volume lifts $\mu$ to
$1.337$, into the partial band, by the same amount as A307's density $\rho$,
which equals the arena volume at leading order since $16\pi^3 r^3$ carries $0.933$
of $\rho(1)$. The four-dimensional radial operator in its own measure gives the
closest fit at $L^2 = 0.399$. The remaining quantities are the lift from $1.34$
to $1.5$ and the recovery of the diagonal peak, neither supplied here. No
published number changes.

\paragraph{Verifier.} \texttt{verify\_P311.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 forced four-volume weight $W = r^3$ lifts $\mu$ to $1.337$ inside
the band $[1.3, 1.7]$, confirms the four-dimensional radial operator in its $r^3$
measure moves $\mu$ to $1.222$ toward $1.5$ and gives the closest fit at
$L^2 = 0.399$, confirms the analytic fact that $\rho$ has leading term
$16\pi^3 r^3$ carrying $0.933$ of $\rho(1)$ so $r^3$ and $\rho$ share the leading
behaviour, confirms the comparison density measure reproduces A307's $\mu = 1.34$
and that $W = r^3$ matches it within $0.01$, and confirms the
strengthened-partial conditions, that a forced arena measure lifts $\mu$ into the
band while no candidate yields a valid diagonal peak. Eight checks, all PASS.

\end{document}
