% 313_Addendum_FullArena.tex
% Copyright Léon Fernando Vlegels -- CC BY 4.0
% OI-287-1. A311 forced the inner-product measure to the radial volume r^3 dr of
% the B^4 arena and left two limits: the lift stops at mu = 1.34 rather than 1.5,
% and no measure restores the diagonal peak. A311's hypothesis was that the
% residual sits in the angular (l) and Hopf-fiber (S^1) sectors the radial
% reduction drops. A313 tests that hypothesis with the full-arena object of
% P18 S4.3, O_{l,k}, family n carrying angular momentum l = n and fiber index
% k = n. Five forced candidates run without fitting: the flat control, the A311
% r^3 measure, the A311 4D-radial operator, the angular full-arena candidate, and
% the Hopf-fiber full-arena candidate. The angular sector does carry strong
% magnitude motion, confirming the direction of A311's hypothesis, but the naive
% l = n assignment overshoots: mu moves from the pure-radial 1.34 to 2.01 for the
% angular candidate and 2.27 for the fiber candidate, with tau and the L2 fit
% degrading sharply. No candidate restores the diagonal peak. The closest fit
% remains the pure-radial 4D operator at L2 = 0.399. The verdict is
% strengthened-partial: the residual to 1.5 is partly in the dropped angular
% sector, the angular sector supplies most of the motion, but the simple l = n
% linkage over-applies it, so the magnitude piece of OI-287-1 is characterized,
% not closed. Verifier: verify_P313.py.

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

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

\title{Addendum 313: The Full Arena\\
\large Adding the angular and fiber sectors moves the magnitude past target,
not onto it}
\author{TOE Corpus}
\date{2026-06-13}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} Case B (strengthened partial; the angular sector carries the
residual motion but the forced $l=n$ linkage overshoots, OI-287-1 magnitude
characterized, not closed) \quad|\quad
\textbf{Addresses:} OI-287-1 \quad|\quad
\textbf{Verifier:} \texttt{verify\_P313.py} --- all PASS

\begin{abstract}
A311 forced the inner-product measure of the A299 selection rule to the radial
volume element $r^3\,dr$ of the $B^4$ arena and left two limits. The lift stops
at $\mu = 1.337$ rather than reaching $1.5$, and no measure restores the
column-standardized diagonal peak. A311 read the four-dimensional radial
operator's improved fit at $L^2 = 0.399$ as a sign that the residual is shared
with the angular and Hopf-fiber sectors that the pure radial reduction drops.
This addendum tests that reading. Paper 18 S2.2 gives the bulk radial equation
with the angular term $[-d^2/dr^2 - (3/r)\,d/dr + l(l+2)/r^2]R = ER$, and S4.3
the explicit assembled form $\hat O_{l,k} = -d^2/dr^2 - (3/r)\,d/dr + l(l+2)/r^2
+ V_{\text{eff}}(r) + \omega^k\gamma$, with $\omega = e^{2\pi i/3}$ the cube root
of unity and $\gamma = 3/4$ the layer-cycle coefficient. The radial node index
and the angular momentum enter the operator together through the unperturbed
spectrum $E_{n,l} = (2n + l + 2)^2$, and the three families are the three
layer-cycle eigenspaces of S5.3. The geometrically forced reading of family $n$
in the full arena is therefore $l = n$ in the angular sector and $k = n$ in the
fiber, each family the ground radial mode of its own $l$-sector operator. Five
forced candidates run without fitting: the flat control, the A311 $r^3$ measure,
the A311 four-dimensional radial operator, the angular full-arena candidate, and
the Hopf-fiber full-arena candidate with the lapse measure $r^3|\sin\pi r|$ and
the $\omega^k\gamma$ family shift. The angular sector carries strong magnitude
motion, confirming the direction of A311's reading, but the $l=n$ assignment
moves the magnitude past the target rather than onto it. The angular candidate
returns $\mu = 2.009$, the fiber candidate $\mu = 2.265$, both with $\tau$
inflating to $18.6$ and $21.2$ and the fit distance growing to $L^2 = 10.6$ and
$13.2$. The angular sector supplies most of the motion, a shift of $+0.672$ in
$\mu$ against the fiber's $+0.256$. No candidate restores a valid diagonal peak,
so the pre-registered HIT criterion is met by none, and the closest fit remains
the pure radial four-dimensional operator at $L^2 = 0.399$. The verdict is
strengthened-partial: the residual to $1.5$ is partly in the dropped angular
sector, the angular sector supplies most of the motion, and the simple forced
$l=n$ linkage over-applies it. The magnitude piece of OI-287-1 is characterized,
not closed.
\end{abstract}

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

\begin{proposition}[The angular sector carries the residual motion and overshoots]
Let $B^W_{nk} = \langle\phi_n|c_k\pi^k r^k|\phi_n\rangle_W /
\langle\phi_n|\phi_n\rangle_W / \lambda_n$ be the A299-style diagonal overlap in
the measure $W\,dr$, with $\phi_n$ the family-$n$ mode. In the pure radial
reduction $\phi_n$ is the $n$-th mode of one operator; in the full arena $\phi_n$
is the ground radial mode of the $l=n$ sector operator
$\hat O_{l=n} = -d^2/dr^2 - (3/r)\,d/dr + n(n+2)/r^2 + V_{\text{eff}}(r) +
\omega^n\gamma$ of P18 S4.3, self-adjoint in $r^3\,dr$ for the angular candidate
and in the Hopf-lapse measure $r^3|\sin\pi r|\,dr$ for the fiber candidate. 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
$\mu = 1.337$, and the four-dimensional radial operator in its $r^3$ measure
returns $(1, 1.222, 8.286)$ at $L^2 = 0.399$, the closest of all, reproducing
A311. The full-arena angular candidate, family $n$ in the $l=n$ sector, returns
$(1, 2.009, 18.635)$ with $\mu = 2.009$ at $L^2 = 10.65$. The full-arena fiber
candidate returns $(1, 2.265, 21.188)$ with $\mu = 2.265$ at $L^2 = 13.21$. The
angular sector supplies a shift of $+0.672$ in $\mu$ relative to the pure radial
$r^3$ value; the fiber adds a further $+0.256$. Both full-arena values exceed the
target $1.5$. 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 residual is in the angular sector, the linkage over-applies it]
A311 located the lift in the forced arena volume and read the residual gap to
$1.5$ as carried by the angular and fiber sectors the radial reduction drops.
A313 confirms the direction and refines the reading. The angular sector does
carry magnitude: moving family $n$ into its own $l=n$ sector shifts $\mu$ by
$+0.672$, the largest single move in the arc, and the fiber adds a further
$+0.256$ through the lapse measure and the $\omega^k\gamma$ family term. The
residual to $1.5$ is therefore not in the measure normalization alone; it is in
the sector structure A311 named. The limit is that the forced $l=n$ linkage moves
the magnitude past the target rather than onto it. The centrifugal term
$l(l+2)/r^2$ pushes the higher-$l$ ground modes outward toward the boundary,
where the cubic head of $\rho$ is largest, so the degree-three overlap grows
faster than the degree-two one and both grow past the target. The naive identity
of the family index with the angular momentum is too strong a coupling; the
overlap responds to it more than the target asks. The honest statement is that
the magnitude motion lives where A311 said it would, in the dropped sectors, and
that the correct strength of the angular coupling is weaker than the forced $l=n$
identity. The closest fit remains the pure radial four-dimensional operator at
$L^2 = 0.399$, which neither overshoots nor restores the peak. The diagonal peak
is unrecovered by every candidate. The magnitude piece of OI-287-1 is thereby
characterized rather than closed: the sector is identified, the strength is not.
\end{remark}

\paragraph{Status.} OI-287-1 remains open. The A311 reading that the residual
sits in the angular and fiber sectors is confirmed in direction: the angular
sector supplies most of the magnitude motion, a shift of $+0.672$ in $\mu$, and
the fiber adds $+0.256$. The forced $l=n$ identity of P18 S4.3 over-applies the
coupling, carrying $\mu$ to $2.009$ for the angular candidate and $2.265$ for the
fiber, past the target $1.5$, with $\tau$ and the fit distance degrading. No
candidate restores the diagonal peak, and the closest fit is still the pure
radial four-dimensional operator of A311 at $L^2 = 0.399$. The remaining quantity
is the correct strength of the angular coupling, weaker than the forced $l=n$
identity, together with the unrecovered peak. No published number changes.

\paragraph{Verifier.} \texttt{verify\_P313.py}: recomputes the flat
eigenstructure (eigenvalues ordered $(16.51, 51.24, 102.02)$, node counts
$(0,1,2)$), confirms the flat control reproduces $(1, 1.12, 7.50)$, confirms the
forced $r^3$ measure gives $\mu = 1.337$ and the four-dimensional radial operator
gives $\mu = 1.222$ at $L^2 = 0.399$ as the closest candidate (A311), confirms
the full-arena angular candidate moves $\mu$ to $2.009$ past the target, the
fiber candidate to $2.265$ further still, and that the angular sector supplies
the larger share of the motion ($+0.672$ against $+0.256$), and confirms the
strengthened-partial conditions, that a full-arena candidate exceeds $\mu = 1.40$
while no candidate yields a valid diagonal peak and the magnitude overshoots
rather than closes. Eight checks, all PASS.

\end{document}
