% 315_Addendum_AngularScan.tex
% Copyright Léon Fernando Vlegels -- CC BY 4.0
% OI-287-1. A313 bracketed the magnitude piece between two forced objects on the
% r^3 dr arena measure: the pure 4D-radial operator (C_4Dself, mu=1.222, modes
% 0,1,2 of one operator) and the full l=n angular identity (C_ang, mu=2.009,
% ground mode of each l-sector). The stated next step was to scale the centrifugal
% term by a scalar coupling g in [0,1] and locate g* where mu=1.5, then test
% whether g* is a corpus constant (forced) or a tuned value (a fit). A315 runs
% that probe. The finding is that the two A313 endpoints differ in mode-selection,
% not in centrifugal strength: holding the construction fixed and scaling only g,
% the n-th-mode family runs mu DOWN from 1.222 to 0.975 and the ground-mode family
% runs mu DOWN from 2.532 to 2.009. Neither single-knob family crosses 1.5; the
% target lies in the gap between the two disjoint mu-ranges. There is therefore no
% g* to test against any constant, and the question "is g* corpus-forced?" is
% vacuous. The magnitude residual is not a one-parameter coupling-strength
% question; the A313 1.222->2.009 motion was mode-reindexing. Verdict NULL with a
% structural redirection. Honest outcome, no overclaiming: there is no fitted g*
% and no forced g*, because no scalar g brackets the target. Verifier:
% verify_P315.py.

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

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

\title{Addendum 315: The Angular Coupling Scan\\
\large No single-parameter centrifugal coupling brackets the magnitude target}
\author{TOE Corpus}
\date{2026-06-13}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} Case C (null with structural redirection; the magnitude
residual is not a one-parameter coupling strength, so there is no $g^\ast$ to test
against a corpus constant) \quad|\quad
\textbf{Addresses:} OI-287-1 \quad|\quad
\textbf{Verifier:} \texttt{verify\_P315.py} --- all PASS

\begin{abstract}
A313 bracketed the magnitude piece of OI-287-1 between two forced objects on the
forced $r^3\,dr$ arena measure of A311: the pure four-dimensional radial operator,
whose three families are modes $0,1,2$ of one operator, returning $\mu = 1.222$;
and the full angular identity $l = n$, whose three families are the ground radial
mode of each $l$-sector operator, returning $\mu = 2.009$. A313 read the magnitude
residual to the target $\mu = 1.5$ as lying between these and proposed that a
scalar angular coupling $g \in [0,1]$, scaling the centrifugal term, would
interpolate $\mu$ from $1.222$ to $2.009$ and cross $1.5$ at some $g^\ast$ whose
status as a corpus constant would decide whether the coupling is forced or fitted.
This addendum runs that probe, with the candidate corpus constants for $g^\ast$
declared before any value is computed. The finding is that the question is
ill-posed as stated, and the value of the result is in saying so plainly. The two
A313 endpoints differ in their mode-selection rule, not in their centrifugal
strength. Holding the construction fixed and scaling only $g$ gives two
mode-consistent families: the $n$-th-mode family, which reproduces the
four-dimensional radial endpoint at $g = 0$ and runs $\mu$ downward from $1.222$
to $0.975$; and the ground-mode family, which reproduces the $l = n$ endpoint at
$g = 1$ and runs $\mu$ downward from $2.532$ to $2.009$. Both are monotone in $g$,
and neither crosses $1.5$. The target lies in the gap between the disjoint ranges
$[0.975, 1.222]$ and $[2.009, 2.532]$, unreachable by either single knob. There
is therefore no $g^\ast$ to compare against the ten pre-declared corpus constants;
the fit-versus-forced test is vacuous because there is neither a fitted nor a
forced coupling. The $1.222 \to 2.009$ motion A313 attributed to angular strength
was mode-reindexing, the change from modes $0,1,2$ of one operator to the ground
mode of each $l$-sector, not a tunable coupling. The verdict is null with a
structural redirection: the magnitude residual is not a one-parameter
coupling-strength quantity, and the search must address the mode-selection rule
rather than a centrifugal coefficient.
\end{abstract}

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

\begin{proposition}[The two A313 endpoints lie in disjoint $\mu$-ranges]
Let $\hat O_{l=n}(g) = -d^2/dr^2 - (3/r)\,d/dr + g\,n(n+2)/r^2 + V_{\text{eff}}(r)$,
self-adjoint in $r^3\,dr$, with the centrifugal term scaled by the scalar
$g \in [0,1]$ and the operator otherwise verbatim from A313. Two mode-consistent
single-parameter families hold the construction fixed while turning $g$:
family~A takes family $n$ to be the $n$-th radial mode of $\hat O_{l=n}(g)$, so
that at $g = 0$ the three sector operators coincide and the modes are $0,1,2$ of
the pure radial operator, reproducing A313's four-dimensional radial endpoint with
$\mu = 1.222$; family~B takes family $n$ to be the ground radial mode of
$\hat O_{l=n}(g)$, so that at $g = 1$ it is the full $l = n$ identity, reproducing
A313's angular endpoint with $\mu = 2.009$. Computed on the standard grid,
family~A runs $\mu$ monotonically from $1.222$ at $g = 0$ to $0.975$ at $g = 1$,
spanning $[0.975, 1.222]$; family~B runs $\mu$ monotonically from $2.532$ at
$g = 0$, where its three families collapse to a common degenerate ground mode, to
$2.009$ at $g = 1$, spanning $[2.009, 2.532]$. The two ranges are disjoint and the
target $\mu = 1.5$ lies in the gap between them. Neither family crosses $1.5$ for
any $g \in [0,1]$, so no crossing $g^\ast$ exists. With no $g^\ast$ defined, the
ten pre-declared corpus constants have nothing to be compared against, and the
fit-versus-forced criterion is vacuous: there is neither a fitted coupling nor a
forced one.
\end{proposition}

\begin{remark}[The residual is in the mode-selection rule, not a coupling strength]
A313 located strong magnitude motion in the angular sector and read the
$1.222 \to 2.009$ change as the effect of a centrifugal coupling carried past the
target. A315 isolates the source of that change. When the centrifugal strength is
the only quantity varied, with the mode-selection rule held fixed, the magnitude
moves in the opposite direction and stays inside one of two disjoint ranges that
exclude the target. The whole of the $1.222 \to 2.009$ motion came from the change
in which radial mode each family occupies, from modes $0,1,2$ of a single operator
to the ground mode of three distinct $l$-sector operators, not from the strength of
the centrifugal term. The honest consequence is that the magnitude residual to
$1.5$ cannot be expressed as a single angular-coupling coefficient, and the
question of whether such a coefficient is a corpus constant does not arise. There
is no number to overclaim. The closest fit on the full target is unchanged from
A313 and A311, the four-dimensional radial operator, and within the families
scanned here family~A at small $g$ reduces the fit distance slightly while leaving
$\mu$ below the target and restoring no diagonal peak. The redirection this records
is concrete: the next quantity to fix is the mode-selection rule that maps family
$n$ to a radial state, the rule that carries the $1.222 \to 2.009$ motion, rather
than a centrifugal coefficient interpolating between the two A313 readings.
\end{remark}

\paragraph{Status.} OI-287-1 remains open. A315 closes one proposed route: the
magnitude residual to $\mu = 1.5$ is not a one-parameter angular-coupling
strength. The two A313 endpoints differ in mode-selection, not centrifugal
strength, and the two mode-consistent single-knob families span disjoint
$\mu$-ranges $[0.975, 1.222]$ and $[2.009, 2.532]$ that straddle the target
without reaching it, so no crossing coupling $g^\ast$ exists and the
fit-versus-forced test on $g^\ast$ is vacuous. The magnitude motion lives in the
mode-selection rule, not in a centrifugal coefficient. The closest fit is still
the pure radial four-dimensional operator of A311, and the diagonal peak remains
unrecovered. No published number changes. The remaining quantity is the
mode-selection rule for the family-to-state map, together with the unrecovered
peak.

\paragraph{Verifier.} \texttt{verify\_P315.py}: recomputes both families from
scratch and confirms family~A reproduces the four-dimensional radial endpoint at
$g = 0$ ($\mu = 1.222$), family~B reproduces the $l = n$ endpoint at $g = 1$
($\mu = 2.009$), each family is monotone in $g$, family~A stays below $1.5$ with
maximum $1.222$ and family~B stays above $1.5$ with minimum $2.009$, the two
$\mu$-ranges are disjoint so the $1.222 \to 2.009$ motion is mode-reindexing, no
single-knob family crosses $\mu = 1.5$ on $[0,1]$, the verdict is null so there is
no $g^\ast$ to test against the ten pre-declared corpus constants, and no NaN
appears. Eight checks, all PASS.

\end{document}
