% 318_Addendum_RadialDirac.tex
% Copyright Léon Fernando Vlegels -- CC BY 4.0
% OI-287-1. A317 broke the A316 family degeneracy by realizing the families as the
% omega^k fiber-momentum sectors, but entered the phase as an additive m^2 energy
% in both corpus-faithful forms and overshot, giving mu = 2.29 to 2.43, tau near
% 21, no peak. A317 named the untested entry: the phase entering through the
% first-order Dirac off-diagonal structure rather than as additive energy, that
% is, the full B^4 spinor problem of P18 S2.2 and S3.1 rather than the scalar
% radial reduction the whole thread used. A318 builds that radial Dirac system on
% the two-component spinor (f, g), self-adjoint in r^3 dr, with the off-diagonal
% coupling -d/dr - (kappa + c)/r and its adjoint, c = 3/2 the four-dimensional
% constant from A312 and A314. The canonical map sends family n to the n-th
% boundary-Dirac sector kappa_n = (n-1) + 3/2, the +-(j+3/2) tower of A314 that
% carries the omega^k labels. The density overlap is the spinor bilinear f^2 + g^2
% in r^3 dr, divided by the eigenvalue. The construction is stable: the radial
% Dirac problem solves with no NaN, the free-field lowest modes are monotone in
% kappa, the standing of the A314 boundary tower in the radial spectrum. The
% three families are genuinely distinct, so the A316 degeneracy is broken here too.
% But the central comparison fails: the first-order coupling does not come down
% gentler than A317's additive m^2. The canonical map gives mu = 3.14, tau = 38.4,
% L2 = 30.5, higher than A317's overshoot, with no peak. The first-order structure
% does not soften the magnitude; it sharpens the overshoot. The verdict is null on
% the magnitude piece. Verifier: verify_P318.py.

\documentclass[11pt]{article}
\usepackage{amsmath,amssymb,amsthm}
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\newtheorem{proposition}{Proposition}[section]
\newtheorem{remark}[proposition]{Remark}

\title{Addendum 318: The First-Order Radial Dirac Reduction\\
\large The phase carried through the Dirac off-diagonal coupling sharpens the
overshoot rather than softening it}
\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 first-order Dirac reduction does not soften
the A317 magnitude overshoot and does not improve on the scalar-radial closest)
\quad|\quad
\textbf{Addresses:} OI-287-1 \quad|\quad
\textbf{Verifier:} \texttt{verify\_P318.py} --- all PASS

\begin{abstract}
A317 realized the three charged-lepton families as the fiber-momentum sectors
$m \equiv k \pmod 3$ that carry the cube-root phase $\omega^m$, broke the A316
degeneracy, and found that both corpus-faithful fiber forms enter the phase as an
additive energy $m^2$ and overshoot, returning $\mu = 2.29$ to $2.43$ with no
peak. The entry A317 left untested was the phase entering through the first-order
Dirac structure rather than as additive energy: the full $B^4$ spinor problem of
P18 S2.2 and S3.1, where the master operator acts on sections of a spinor bundle,
rather than the scalar radial reduction used from A296 through A317. This addendum
builds that object. The four-dimensional radial Dirac operator couples an upper
and a lower radial function $(f, g)$ through a first-order system with the
off-diagonal terms $-d/dr - (\kappa + c)/r$ and its adjoint, where $\kappa$ is the
boundary-Dirac eigenvalue carrying the family label and $c = 3/2$ is the
four-dimensional constant fixed in A312 and A314. The canonical map sends family
$n$ to the $n$-th boundary-Dirac sector $\kappa_n = (n-1) + 3/2$, the
$\pm(j+3/2)$ tower of A314 that is the spinor realization of the $\omega^k$
sectors. The density overlap is formed as the spinor bilinear $f^2 + g^2$ in the
$r^3$ measure, divided by the eigenvalue. The construction is stable and the
boundary-Dirac ordering is recovered: the free-field lowest modes are monotone in
$\kappa$. The three families are genuinely distinct, so the A316 degeneracy is
broken in this reduction as well. But the first-order coupling does not soften the
magnitude. The canonical map returns $\mu = 3.14$ and $\tau = 38.4$, above A317's
additive overshoot rather than below it, at distance $30.5$ to the target, with no
peak. The verdict is null on the magnitude piece: the first-order Dirac structure
carries the phase into the matrix element more strongly than the additive energy,
not more gently, and it does not improve on the scalar-radial closest at distance
$0.399$.
\end{abstract}

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

\begin{proposition}[The radial Dirac reduction is stable and family-distinct, and
its canonical map overshoots above A317]
On the geometry of P18 S2.1, with metric $ds^2 = dr^2 + r^2\,d\Omega_3^2$ and the
master operator on sections of a spinor bundle (P18 S2.2), the four-dimensional
radial Dirac operator acts on the two-component spinor $\Psi = (f, g)^T$,
self-adjoint in $r^3\,dr$, through the first-order system with diagonal mass
$m(r) = \sqrt{V_{\text{eff}}(r)}$ and off-diagonal coupling
$-d/dr - (\kappa + c)/r$ together with its adjoint $d/dr - (\kappa - c)/r$, where
$V_{\text{eff}}(r) = \rho'(r)^2/(2\Omega^2) + E_{\text{self}}\,r^2(1-r)^2$ is the
potential of the scalar thread, $\kappa$ is the boundary-Dirac eigenvalue, and
$c = 3/2$ is the four-dimensional constant of A312 and A314 equal to the
Lichnerowicz constant $R/4$. The canonical map sends family $n$ to the $n$-th
boundary-Dirac sector $\kappa_n = (n-1) + 3/2 = 3/2, 5/2, 7/2$, the positive
$\pm(j+3/2)$ tower of A314, and the diagonal entry $n$ is the degree-$n$ density
overlap of that sector's lowest bound state, formed as the spinor bilinear
$f^2 + g^2$ in $r^3\,dr$ and divided by the eigenvalue. The discretization on a
staggered grid is stable, with no NaN, and the free-field lowest modes are
monotone in $\kappa$, $5.76 < 6.99 < 8.18$, recovering the boundary-Dirac tower
ordering of A314 in the four-ball radial spectrum. The three sectors give three
distinct Dirac eigenvalues, $6.48, 7.74, 8.94$, and three distinct diagonal
entries with no collapse, so the A316 degeneracy is broken here as it was in A317.
But the canonical map returns the diagonal $(1, 3.143, 38.430)$ with
$\mu = 3.143$ and $\tau = 38.430$, at distance $30.47$ to the target $(1, 1.5, 8)$.
This is above A317's additive overshoot of $\mu = 2.29$ to $2.43$, not below it.
The map lands no entry within twenty percent of the target and restores no valid
peak, the column-standardized argmax of column $n$ equal to $n$ for all three
columns while the sine-box control formed in the same measure fails the same test.
It does not improve on the scalar-radial closest of A316 at distance $0.399$ and
$\mu = 1.222$.
\end{proposition}

\begin{remark}[The first-order coupling sharpens the overshoot; what remains]
The construction is the entry A317 named, built without fitting. The family
definition is the corpus one, the families as the boundary-Dirac sectors that
carry the $\omega^k$ labels, and the phase enters through the first-order
off-diagonal coupling $(\kappa \pm c)/r$ rather than the additive $m^2$ of A317.
The prediction under test was that this first-order channel would carry the phase
into the matrix element gently, through the relative weight of the upper and lower
components, and move $\mu$ down from A317's overshoot toward $1.5$. It does not.
The off-diagonal $(\kappa + c)/r$ term grows with $\kappa$, so the higher families
acquire a larger lower component and a more sharply weighted density, and the
degree-$n$ overlap rises faster than the additive energy raised it. The result is
$\mu = 3.14$ where A317 gave $2.29$ to $2.43$ and the scalar radial reduction gave
$1.222$. The first-order structure is the gentler-coupling hope of A317 made
explicit, and the explicit form refutes it: carrying the phase through the Dirac
off-diagonal coupling is stronger than carrying it through the additive energy,
not weaker. The two structural facts that survive are the same two A317 reported.
The reduction is stable and corpus-faithful, the free-field modes sit in the
A314 tower ordering, and the three families are genuinely distinct, so the A316
degeneracy is broken in the spinor reduction as well as the scalar one. But the
magnitude is not supplied. The factor that lifts the second family to exactly
$1.5$ and sharpens the diagonal to a peak is now excluded from a further channel:
it is neither a measure choice, nor an angular coupling strength, nor a real
family-to-mode selection, nor the fiber-momentum eigenvalue of the complex
realization, nor the first-order Dirac off-diagonal coupling of the spinor
reduction. Both ways of carrying the $\omega^k$ phase into the radial problem,
the additive energy and the first-order coupling, overshoot. The residual is not
in how the phase enters the radial matrix element at all. What remains untested is
a reduction that is not radial: the boundary $S^3$ problem directly, or a coupling
between the degree-$n$ density and the spectrum that does not pass through a radial
bound state. The magnitude piece of OI-287-1 does not close here.
\end{remark}

\paragraph{Status.} OI-287-1 remains open. A318 builds the first-order $B^4$
radial Dirac system that A317 named as the untested entry, with the families as
the boundary-Dirac sectors $\kappa_n = (n-1) + 3/2$ that carry the $\omega^k$
labels and the phase entering through the off-diagonal coupling rather than the
additive energy. The reduction is stable, recovers the A314 boundary-Dirac tower
ordering in its free-field modes, and breaks the A316 degeneracy with three
distinct families. But the canonical map overshoots above A317, returning
$\mu = 3.14$ and $\tau = 38.4$ at distance $30.5$, where A317's additive forms
gave $\mu = 2.29$ to $2.43$ and the scalar radial reduction gave $\mu = 1.222$ at
distance $0.399$. The first-order Dirac coupling carries the $\omega^k$ phase more
strongly than the additive energy, not more gently, so it does not soften the
magnitude. No published number changes. The residual is now excluded from how the
phase enters the radial matrix element in any of its tested channels; what remains
is a non-radial reduction of the master operator.

\paragraph{Verifier.} \texttt{verify\_P318.py}: recomputes the radial Dirac
system from scratch and confirms that it solves with no NaN on the staggered
grid, that the three sectors $\kappa = 3/2, 5/2, 7/2$ give three distinct Dirac
eigenvalues and three distinct diagonal entries with no collapse, that the
free-field lowest modes are monotone in $\kappa$ recovering the A314 boundary
tower ordering, that the canonical map returns the diagonal $(1, 3.143, 38.430)$
with $\mu = 3.143$ and $\tau = 38.430$ overshooting the target, that no valid peak
is restored and no entry lands within twenty percent of $(1, 1.5, 8)$, that the
first-order coupling is not gentler than A317's $m^2$ overshoot since $\mu = 3.14$
exceeds $2.29$ to $2.43$, and that the null verdict's defining condition holds,
the reduction improving neither on the scalar-radial closest at distance $0.399$
nor on A317. Nine checks, all PASS.

\end{document}
