% 320_Addendum_CoupledBulkBoundary.tex
% Copyright Léon Fernando Vlegels -- CC BY 4.0
% OI-287-1. The radial reduction (A300-A318) undershoots the magnitude of the
% charged-lepton selection rule and the pure-boundary reduction (A319) overshoots;
% the target mu = 1.5 sits in a gap [1.34, 2.0] reached by neither. A320 builds the
% one arena not yet tested, the coupled bulk+boundary B^4 problem (P18 S4.3 O_{l,k}:
% the radial operator carrying BOTH the boundary-harmonic centrifugal term
% l(l+2)/r^2 AND the layer-cycle family term omega^k gamma together). The state is
% the radial bound state of the l=L sector times the boundary harmonic of degree L;
% the coupled spectrum is the radial eigenvalue plus the boundary Casimir L(L+2)
% plus the family phase Re(omega^k) gamma. The hypothesis was interpolation: the
% radial sector undershoots, the boundary sector overshoots, the coupling might land
% in the gap. The finding is null. The canonical coupled map (family n -> boundary
% degree L=n-1 coupled to its radial ground, the corpus O_{l,k}) gives mu = 7.77,
% tau = 152, far above the gap; the linked map gives mu = 3.90; both overshoot past
% 2.0, the canonical one harder than any prior arena. The boundary multiplicity
% factor (L+1)^2 = 1,4,9 compounds the overshoot rather than interpolating. The two
% controls bracket the gap (pure radial mu = 1.222, pure boundary mu = 3.38) but no
% coupled map lands inside it. The magnitude is in none of radial, boundary, or
% their coupling; the framing is the likely issue (companion probe A321).
% Verifier: verify_P320.py.

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

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

\title{Addendum 320: The Coupled Bulk-and-Boundary $B^4$ Problem\\
\large The arena that keeps both the radial bound state and the boundary harmonic
overshoots the magnitude and does not reach the gap either reduction missed}
\author{TOE Corpus}
\date{2026-06-14}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} Case C (null; the coupled bulk-and-boundary reduction keeps both
sectors, breaks the family degeneracy, but overshoots the magnitude past the gap
the single reductions also missed, and does not improve on the standing fit)
\quad|\quad
\textbf{Addresses:} OI-287-1 \quad|\quad
\textbf{Verifier:} \texttt{verify\_P320.py} --- all PASS

\begin{abstract}
Across seven arenas the magnitude entry of the charged-lepton selection rule splits
in two: the channels that do not distinguish the three families undershoot, the
pure-radial $\mu = 1.222$ and the $r^3$ measure $\mu = 1.337$, and the channels
that give three distinct families overshoot, the fiber $2.3$, the Dirac $3.14$, the
boundary-lens $3.86$, and the boundary-harmonic $6.28$. The target $\mu = 1.5$ sits
in a gap, $[1.34, 2.0]$, reached by neither the radial reduction of Addenda
300--318 nor the pure-boundary reduction of Addendum 319. This addendum builds the
one arena not yet tested, the coupled bulk-and-boundary $B^4$ problem. Paper 18
\S4.3 gives the explicit radial form of the master operator,
$\hat{O}_{l,k} = -\,d^2/dr^2 - (3/r)\,d/dr + l(l+2)/r^2 + V_{\rm eff}(r)
+ \omega^k \gamma$, which carries both the boundary-harmonic centrifugal term
$l(l+2)/r^2$ and the layer-cycle family term $\omega^k \gamma$ together. A state is
the radial bound state of the $l = L$ sector times the boundary harmonic of degree
$L$; the coupled eigenvalue is the radial eigenvalue plus the boundary Casimir
$L(L+2)$ plus the family phase $\mathrm{Re}(\omega^k)\gamma$. The hypothesis was
interpolation: the undershooting radial sector and the overshooting boundary sector
together might land in the gap. The finding is null. The canonical coupled map,
family $n$ to boundary degree $L = n-1$ coupled to its radial ground state, returns
$\mu = 7.77$ with $\tau = 152$, far above the gap and above every prior overshoot;
the linked map, family $n$ the $(n-1)$-th radial mode of the $l = n-1$ sector,
returns $\mu = 3.90$. Both overshoot past $2.0$, the canonical one hardest of all,
because the boundary multiplicity factor $(L+1)^2 = 1, 4, 9$ compounds the overshoot
rather than tempering it. The two controls bracket the gap, pure radial at
$\mu = 1.222$ and pure boundary at $\mu = 3.38$, yet no coupled map lands inside it.
The magnitude is in none of the radial, the boundary, or their coupling.
\end{abstract}

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

\begin{proposition}[The coupled reduction overshoots the magnitude and does not
reach the gap]
On the 4-ball $B^4 = (\text{radial } r \in [0,1]) \times (\text{boundary } S^3)$ the
master operator in the sector with fixed boundary angular momentum $l$ and family
index $k$ is, by Paper 18 \S4.3,
\[
  \hat{O}_{l,k} = -\frac{d^2}{dr^2} - \frac{3}{r}\frac{d}{dr}
  + \frac{l(l+2)}{r^2} + V_{\rm eff}(r) + \omega^k \gamma,
  \qquad \gamma = \tfrac34, \quad \omega = e^{2\pi i/3},
\]
with $V_{\rm eff}$ the canonical effective potential held verbatim from the radial
thread. This is the coupled object: the same radial operator the radial reduction
used, now carrying both the boundary-harmonic centrifugal term $l(l+2)/r^2$, the
$S^3$ sector entering the radial equation, and the layer-cycle family term
$\omega^k \gamma$, the $\mathbb{Z}_3$ sector. A state is $\psi_{n,L}(r)\, Y_L(S^3)$,
the radial bound state of the $l = L$ sector times the boundary harmonic of degree
$L$; the radial reduction set $l = 0$ and dropped $\omega^k \gamma$, and the
pure-boundary reduction kept $L(L+2)$ and dropped the radial bound state. The
coupled eigenvalue is $\lambda_{n,L} = \varepsilon^{\rm rad}_{n,L} + L(L+2)
+ \mathrm{Re}(\omega^k)\gamma$, the radial eigenvalue plus the boundary Casimir plus
the family phase, and the full $B^4$ overlap is the bulk density moment in the $r^3$
radial measure times the boundary harmonic multiplicity $(L+1)^2$, divided by
$\lambda_{n,L}$. Two coupled maps are declared in advance. In the canonical map,
family $n$ is the boundary degree $L_n = n-1$ coupled to the radial ground state of
the $l = L_n$ sector with family index $k = n-1$, the corpus $\hat{O}_{l,k}$
structure; the normalized diagonal is $(1, 7.77, 152.3)$ with $\mu = 7.77$ and
$\tau = 152.3$, at distance $144.5$ to the target $(1, 1.5, 8)$. In the linked map,
the radial node index and the boundary degree are linked so that family $n$ is the
$(n-1)$-th radial mode of the $l = n-1$ sector; the normalized diagonal is
$(1, 3.90, 50.5)$ with $\mu = 3.90$, at distance $42.6$. Each map produces three
distinct entries, since the coupled sectors are genuinely distinct, but each
overshoots, both $\mu$ values lying above the target and above the gap $[1.34, 2.0]$,
and neither restores a valid column-standardized peak against the box control. The
two controls reproduce the two limits the coupling was to interpolate between: the
pure-radial control, $L = 0$ for all families, returns $\mu = 1.222$, recovering the
radial undershoot, and the pure-boundary control, the radial ground fixed with the
boundary Casimir and multiplicity carrying the distinction, returns $\mu = 3.38$,
recovering the boundary overshoot. The controls bracket the gap, yet no coupled map
lands inside it; neither improves on the standing-best distance $0.399$ at
$\mu = 1.222$.
\end{proposition}

\begin{remark}[The verdict is null; the coupling does not interpolate]
The construction is the corpus coupled object, Paper 18 \S4.3's own
$\hat{O}_{l,k}$, built without fitting: the boundary Casimir $L(L+2) = 0, 3, 8$ and
multiplicity $(L+1)^2 = 1, 4, 9$ are the genuine $\mathfrak{so}(4)$ data of Addendum
314, the radial bound states are the same ones the radial thread used, and the
family phase $\omega^k \gamma$ carries $\gamma = 3/4$ from Paper 18 \S4.2. The
canonical map is declared up front as the corpus $\hat{O}_{l,k}$ pairing, family $n$
to boundary degree $L = n-1$ coupled to its radial ground; a tuned pairing would be
a fit, and none is invoked. The hypothesis was interpolation, that the
undershooting radial sector and the overshooting boundary sector together would land
in the gap the single reductions missed. The coupling does the opposite. The
canonical map overshoots to $\mu = 7.77$, hardest of any arena tested, because the
boundary multiplicity factor $(L+1)^2$ that distinguishes the families is itself the
quantity that drives the overshoot: it enters the overlap multiplicatively as
$1, 4, 9$ across the three families, and the radial coupling does not damp it. The
linked map overshoots more gently, to $\mu = 3.90$, but still past $2.0$. The two
controls confirm the limits, pure radial at $1.222$ below the gap and pure boundary
at $3.38$ above it, and confirm that the coupled problem inherits the boundary
overshoot rather than tempering it with the radial undershoot. The seven-arena
pattern holds one level further out: the structure that distinguishes the families
overshoots the magnitude, now in the arena built precisely to interpolate. The
reading is that the magnitude is in none of the radial reduction, the pure-boundary
reduction, or their coupling. What this leaves is the framing. Three reductions and
their union all place the magnitude outside reach, with the channels that
distinguish the families overshooting and the channels that do not undershooting,
and nothing landing between. That is the signature of a quantity computed as the
wrong functional rather than in the wrong arena. Whether the target $(2, 3, 16)$ is
an overlap-diagonal quantity at all is the question the companion probe takes up.
The magnitude piece of OI-287-1 does not close here.
\end{remark}

\paragraph{Status.} OI-287-1 remains open. A320 builds the coupled bulk-and-boundary
$B^4$ problem, the arena that keeps both the radial bound state and the boundary
harmonic, in the corpus form $\hat{O}_{l,k}$ of Paper 18 \S4.3, with two coupled
maps declared in advance: the canonical map, family $n$ to boundary degree $L = n-1$
coupled to its radial ground, and the linked map, family $n$ the $(n-1)$-th radial
mode of the $l = n-1$ sector. Both break the family degeneracy and give three
distinct entries, but both overshoot the magnitude: the canonical map returns
$\mu = 7.77$ at distance $144.5$ and the linked map $\mu = 3.90$ at distance $42.6$,
both above the gap $[1.34, 2.0]$, the canonical one above every prior overshoot, and
neither restores a peak. The two controls bracket the gap, pure radial at
$\mu = 1.222$ and pure boundary at $\mu = 3.38$, yet no coupled map lands inside it,
and neither improves on the standing-best distance $0.399$. The coupling does not
interpolate; the boundary multiplicity factor compounds the overshoot. The magnitude
is shown to be in none of the radial, the boundary, or their coupling. No published
number changes. The remaining direction is not a further arena but the framing
itself, whether $(2, 3, 16)$ is an overlap-diagonal quantity, which the companion
probe addresses.

\paragraph{Verifier.} \texttt{verify\_P320.py}: rebuilds the coupled operator
$\hat{O}_{l,k}$ from scratch and confirms the two controls recover the two limits,
the pure-radial control at $\mu = 1.222$ and the pure-boundary control at
$\mu \ge 2.0$; confirms the full coupled spectrum
$\lambda_{n,L} = \varepsilon^{\rm rad} + L(L+2) + \mathrm{Re}(\omega^k)\gamma$ is
positive with the boundary Casimir $L(L+2) = 0, 3, 8$ entering the spectrum;
recomputes the canonical diagonal $(1, 7.77, 152.3)$ at distance $144.5$ and the
linked diagonal $(1, 3.90, 50.5)$ at distance $42.6$, with no valid peak in either;
confirms the coupled problem does not reach the gap $[1.34, 2.0]$, both $\mu$ above
$2.0$; confirms the controls bracket the gap while no coupled map lands inside it;
and confirms the null verdict's defining condition, that the canonical map does not
hit, the coupled problem does not reach the gap, and the coupled problem does not
beat the standing-best distance $0.399$, with no NaN. Nine checks, all PASS.

\end{document}
