% 319_Addendum_BoundaryS3.tex
% Copyright Léon Fernando Vlegels -- CC BY 4.0
% OI-287-1. A318 and the cumulative arc through it established that the magnitude
% residual of the charged-lepton selection rule is not in the radial reduction by
% any channel: across radial channels the mu entry runs pure-radial 1.222
% (undershoot), r^3 measure 1.337, fiber m^2 energy 2.3 (overshoot), Dirac
% first-order 3.14 (worse overshoot), and the target 1.5 sits in a gap reached by
% none. A319 leaves the radial bound-state picture entirely and builds the boundary
% S^3 problem directly. The families are the SO(4) harmonics (degree L, eigenvalue
% L(L+2), multiplicity (L+1)^2, built from the so(4)=su(2)+su(2) Casimir as in A314)
% and the Z_3 = S^3/Z_3 lens-space characters (the omega^k eigenspaces of the
% 2pi/3 Hopf rotation, the complex structure native on the lens space). Two
% formulations are declared up front: B-harm, families = the lowest harmonic
% degrees L=0,1,2; and B-lens (canonical, per P20), families = the three Z_3
% character classes on L(3,1). The boundary degree-n coupling is the harmonic
% overlap: the moment c_n pi^n carried by the family-n boundary sector's harmonic
% multiplicity, normalized by its SO(4) Casimir spectral weight, the boundary
% analog of the radial <...c_n pi^n r^n...>/lambda_n. The finding is null on the
% magnitude. Both formulations give three distinct families, but both overshoot:
% B-lens (canonical) returns mu = 3.86 and B-harm mu = 6.28, both far above 1.5 and
% above the radial fiber overshoots. The boundary problem does not reach the
% [1.34, 2.0] gap, overshooting straight past it, and does not beat the radial-best
% L2 = 0.399. The magnitude is in neither the radial nor the pure-boundary
% reduction; the next arena is the full bulk+boundary coupled problem.
% Verifier: verify_P319.py.

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

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

\title{Addendum 319: The Boundary $S^3$ Problem Directly\\
\large The pure-boundary harmonic and lens-space reductions overshoot the
magnitude and do not reach the gap the radial 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 pure-boundary $S^3$ reduction breaks the family
degeneracy but overshoots the magnitude, does not reach the gap the radial
reduction could not, and does not improve on the radial-best fit) \quad|\quad
\textbf{Addresses:} OI-287-1 \quad|\quad
\textbf{Verifier:} \texttt{verify\_P319.py} --- all PASS

\begin{abstract}
The cumulative arc through Addendum 318 established that the magnitude residual of
the charged-lepton selection rule is not in the radial reduction by any channel.
Across radial channels the $\mu$ entry runs pure-radial $1.222$, $r^3$ measure
$1.337$, fiber $m^2$ energy $2.3$, and Dirac first-order $3.14$, while the target
$\mu = 1.5$ sits in a gap, $[1.34, 2.0]$, reached by none: the channels that carry
the cube-root phase overshoot it and the channels that do not undershoot. This
addendum leaves the radial bound-state picture and builds the boundary $S^3$
problem directly. The objects are the $SO(4)$ harmonics, degree $L$ with
Laplace-Beltrami eigenvalue $L(L+2)$ and multiplicity $(L+1)^2$, built from the
$\mathfrak{so}(4) = \mathfrak{su}(2) \oplus \mathfrak{su}(2)$ Casimir as in
Addendum 314, and the $\mathbb{Z}_3 = S^3 / \mathbb{Z}_3$ lens-space characters,
the $\omega^k$ eigenspaces of the $2\pi/3$ Hopf rotation. Two formulations are
declared in advance: $B$-harm, with families the lowest harmonic degrees
$L = 0, 1, 2$, and $B$-lens, the canonical reading per Paper 20, with families the
three $\mathbb{Z}_3$ character classes on $L(3,1)$. The boundary degree-$n$
coupling is the harmonic overlap, the moment $c_n \pi^n$ carried by the family-$n$
sector's harmonic multiplicity, normalized by its $SO(4)$ Casimir spectral weight,
the boundary analog of the radial
$\langle \psi_n | c_n \pi^n r^n | \psi_n \rangle / \lambda_n$. Both formulations
break the family degeneracy and give three distinct entries, but both overshoot the
magnitude: $B$-lens returns $\mu = 3.86$ and $B$-harm $\mu = 6.28$, both far above
the target and above the radial fiber overshoots. The boundary problem does not
reach the gap, overshooting straight past it, and does not improve on the
radial-best distance $0.399$. The verdict is null: the magnitude is in neither the
radial nor the pure-boundary reduction.
\end{abstract}

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

\begin{proposition}[The pure-boundary reduction overshoots the magnitude and does
not reach the gap]
On the boundary $S^3 = \partial B^4$ the natural objects are the $SO(4)$ scalar
harmonics, degree $L$ carrying the Laplace-Beltrami eigenvalue $L(L+2)$ with
multiplicity $(L+1)^2$, and the lens-space $L(3,1) = S^3 / \mathbb{Z}_3$ character
classes, the eigenspaces of the layer-cycle under which the Hopf fiber rotates by
$2\pi/3$. The harmonic data is built from the
$\mathfrak{so}(4) = \mathfrak{su}(2)_L \oplus \mathfrak{su}(2)_R$ Casimir on the
symmetric $(j,j)$ irreps, $4 j(j+1) = L(L+2)$ with $L = 2j$, the same construction
Addendum 314 used, giving $L(L+2) = 0, 3, 8$ and multiplicity $1, 4, 9$ for
$L = 0, 1, 2$. The boundary degree-$n$ density coupling is declared as the harmonic
overlap, the moment $c_n \pi^n$ carried by the family-$n$ sector's harmonic
multiplicity and normalized by that sector's spectral weight, the boundary analog
of the radial matrix element over its eigenvalue, with the constant harmonic's
zero Laplacian energy handled by the multiplicity floor $(L+1)^2 = 1$. Two
formulations are tested, both declared before any number is computed. In $B$-harm
the families are the lowest harmonic degrees $L = 0, 1, 2$, and the normalized
diagonal is $(1, 6.283, 88.83)$ with $\mu = 6.283$ and $\tau = 88.83$, at distance
$80.97$ to the target $(1, 1.5, 8)$. In $B$-lens, the canonical reading in which
the three families are the $\mathbb{Z}_3$ character classes of the lens space, the
normalized diagonal is $(1, 3.856, 64.60)$ with $\mu = 3.856$ and $\tau = 64.60$,
at distance $56.65$. Each formulation produces three distinct entries with no
degenerate collapse, since the three character classes are genuinely populated and
distinct, the complex $\omega^k$ structure native on the lens space. But each
overshoots the magnitude: both $\mu$ values lie above the target $1.5$ and above
the radial fiber overshoots of $2.3$ to $3.14$, and neither restores a valid
column-standardized peak against the flat control. Neither reaches the gap
$[1.34, 2.0]$ the radial reduction could not enter, overshooting straight past it,
and neither improves on the radial-best distance $0.399$ at $\mu = 1.222$.
\end{proposition}

\begin{remark}[The verdict is null; the boundary alone is not the arena]
The construction is the boundary problem the cumulative radial arc pointed to,
built without fitting. The boundary objects are the corpus ones, the $SO(4)$
harmonics and the lens-space characters, and the harmonic data is the genuine
$\mathfrak{so}(4)$ Casimir spectrum machine-built as in Addendum 314, not assumed.
The canonical family definition is the Paper 20 reading, the three families as the
$\mathbb{Z}_3$ character classes carrying the $\omega^k$ phase. The first result is
structural: both formulations break the degeneracy that defeated the real radial
reduction in Addendum 316, since the three character classes are distinct on the
lens space, where the real layer-cycle shift folded the second and third families
together. The second result is the magnitude, and here the boundary reduction does
not close. Both formulations overshoot, $B$-lens to $\mu = 3.86$ and $B$-harm to
$\mu = 6.28$, with $\tau$ far above the target $8$, and neither restores a peak.
The pattern that held across the radial channels holds again on the boundary, in
sharper form: the structure that carries the cube-root phase overshoots the
magnitude, now past even the radial overshoots. The target $\mu = 1.5$ sits in the
gap $[1.34, 2.0]$, and the boundary problem does not enter it. The reading is that
the magnitude is in neither the radial reduction nor the pure-boundary reduction.
The boundary carries the family distinction faithfully, as the radial fiber and
Dirac channels did, but the magnitude is not a property of the boundary spectral
weights any more than it was of the radial ones. What remains untested is the full
bulk-and-boundary coupled problem, in which the radial bulk states and the boundary
harmonics enter together rather than either reduction alone. The magnitude piece of
OI-287-1 does not close here.
\end{remark}

\paragraph{Status.} OI-287-1 remains open. A319 leaves the radial reduction and
tests the boundary $S^3$ problem directly, in two formulations declared in advance:
$B$-harm, families the lowest $SO(4)$ harmonic degrees $L = 0, 1, 2$, and $B$-lens,
the canonical reading with families the three $\mathbb{Z}_3$ character classes on
the lens space $L(3,1)$. Both break the family degeneracy and give three distinct
entries, the character classes genuinely distinct on the lens space. But both
overshoot the magnitude: $B$-lens returns $\mu = 3.86$ at distance $56.65$ and
$B$-harm $\mu = 6.28$ at distance $80.97$, both above the radial fiber overshoots,
and neither restores a peak. The boundary problem does not reach the gap
$[1.34, 2.0]$ the radial reduction could not enter, and does not improve on the
radial-best distance $0.399$ at $\mu = 1.222$. The pure-boundary reduction
therefore supplies the family distinction but not the magnitude, the same shape of
result the radial fiber and Dirac channels gave. No published number changes. The
magnitude is shown to be in neither the radial nor the pure-boundary reduction; the
remaining arena is the full bulk-and-boundary coupled problem, in which the radial
states and the boundary harmonics enter together.

\paragraph{Verifier.} \texttt{verify\_P319.py}: rebuilds the $SO(4)$ harmonic
spectrum from the $\mathfrak{so}(4)$ Casimir and confirms it is the genuine
boundary data, $L(L+2) = 0, 3, 8$ with multiplicity $(L+1)^2 = 1, 4, 9$ for
$L = 0, 1, 2$ as in Addendum 314; confirms that each formulation gives three
distinct families with all three $\mathbb{Z}_3$ classes populated; recomputes the
$B$-harm diagonal $(1, 6.28, 88.8)$ at distance $80.97$ and the $B$-lens diagonal
$(1, 3.86, 64.6)$ at distance $56.65$, with no valid peak in either; confirms that
the boundary problem does not reach the gap $[1.34, 2.0]$, both $\mu$ above $2.0$;
and confirms the null verdict's defining condition, that the canonical map does not
hit, the boundary does not reach the gap, and the boundary does not beat the
radial-best distance $0.399$, with no NaN. Eight checks, all PASS.

\end{document}
