% 317_Addendum_HopfFiber.tex
% Copyright Léon Fernando Vlegels -- CC BY 4.0
% OI-287-1. A316 closed the family-to-mode route in the real radial reduction and
% named the blocker: the three charged-lepton families are the three complex
% cube-root eigenspaces omega^k of the layer-cycle, but the real reduction folds
% omega and omega^2 together, since Re(omega^k) = (1, -1/2, -1/2) makes families
% two and three degenerate. A317 moves to a complex realization. On the Hopf
% fibration S^1 -> S^3 -> S^2 the layer-cycle psi -> psi + 2pi/3 acts on the
% fiber-momentum sector e^{i m psi} by omega^m, so the omega^k eigenspace is the
% set of m congruent to k modulo 3, with lowest representative m = k. The fiber
% momentum enters the radial operator as a genuine m-dependent fiber-Laplacian
% term, tested in two corpus-faithful forms: a flat additive m^2 (the fiber as an
% S^1 of fixed circumference) and an m^2/r^2 term (the fiber on the cone, scaling
% with the radius like the angular term). The canonical map sends family k to the
% ground radial state of the m = k sector. The finding has two parts. First, the
% A316 degeneracy is broken: m = 0, 1, 2 are three genuinely distinct operators and
% give three distinct diagonal entries, which the real reduction could not. Second,
% the canonical map overshoots: the additive form returns mu = 2.43 and the cone
% form mu = 2.29, both far above 1.5, with no peak, and neither improves on the
% real-reduction closest of distance 0.399 at mu = 1.222. The verdict is null on
% the magnitude piece: the genuine omega^k phase supplies a real structural result,
% the broken degeneracy, but not the magnitude, and the residual is elsewhere.
% Verifier: verify_P317.py.

\documentclass[11pt]{article}
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\newtheorem{proposition}{Proposition}[section]
\newtheorem{remark}[proposition]{Remark}

\title{Addendum 317: The Complex Hopf-Fibered Realization\\
\large The genuine cube-root phase breaks the degeneracy but does not supply the
magnitude}
\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 complex realization breaks the A316 family
degeneracy but does not reproduce the magnitude target with a peak, and does not
improve on the real-reduction closest) \quad|\quad
\textbf{Addresses:} OI-287-1 \quad|\quad
\textbf{Verifier:} \texttt{verify\_P317.py} --- all PASS

\begin{abstract}
A316 closed the family-to-mode route in the real radial reduction of the
charged-lepton selection rule and located the blocker in the complex structure of
the family definition. The three families are the three cube-root eigenspaces
$\omega^k$ of the layer-cycle, $\omega = e^{2\pi i/3}$, but the real radial
reduction folds $\omega$ and $\omega^2$ together: the real layer-cycle shift
$\mathrm{Re}(\omega^k) = (1, -\tfrac12, -\tfrac12)$ takes equal values on $k = 1$
and $k = 2$, so families two and three collapse to one radial state. This addendum
moves to a complex realization. On the Hopf fibration $S^1 \hookrightarrow S^3 \to
S^2$ the layer-cycle acts as a rotation $\psi \mapsto \psi + 2\pi/3$ of the fiber
coordinate, so on the fiber-momentum sector $e^{i m\psi}$ it multiplies by
$e^{i m\,2\pi/3} = \omega^m$, and the $\omega^k$ eigenspace is the set
$\{m \equiv k \pmod 3\}$ with lowest representative $m = k$. The fiber momentum
enters the radial operator through the fiber Laplacian, whose eigenvalue is $m^2$,
and $m = 0, 1, 2$ give three distinct values where $\mathrm{Re}(\omega^k)$ gave
only two. Two corpus-faithful forms of the fiber term are tested, an additive
$m^2$ and a cone-weighted $m^2/r^2$, with the canonical map sending family $k$ to
the ground radial state of the $m = k$ sector. The complex realization breaks the
A316 degeneracy: $m = 0, 1, 2$ are three genuinely distinct operators and produce
three distinct diagonal entries. But the canonical map overshoots, returning
$\mu = 2.43$ for the additive form and $\mu = 2.29$ for the cone form, both far
above the target $1.5$, with no peak, and neither improves on the real-reduction
closest at distance $0.399$ and $\mu = 1.222$. The verdict is null on the
magnitude piece: the genuine cube-root phase supplies the broken degeneracy, a
real structural result, but not the magnitude, which lies elsewhere.
\end{abstract}

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

\begin{proposition}[The complex realization breaks the degeneracy; the canonical
map overshoots the magnitude]
On the forced arena geometry of A311, the four-dimensional radial operator
$-d^2/dr^2 - (3/r)\,d/dr + V_{\text{eff}}(r)$, self-adjoint in $r^3\,dr$ with both
eigenstates and the degree-$k$ density overlap formed in $r^3\,dr$, gains the
fiber-momentum term of the Hopf fibration. The layer-cycle rotates the fiber
coordinate by $2\pi/3$, so the $\omega^k$ eigenspace is the fiber-momentum sector
$m \equiv k \pmod 3$ with lowest representative $m = k$, and the fiber Laplacian
contributes the eigenvalue $m^2$. Two corpus-faithful forms are tested: the
additive form $\hat O_m = -d^2/dr^2 - (3/r)\,d/dr + V_{\text{eff}}(r) + m^2$, the
fiber as a circle of fixed circumference, and the cone form
$\hat O_m = -d^2/dr^2 - (3/r)\,d/dr + m^2/r^2 + V_{\text{eff}}(r)$, the fiber
scaling with the radius like the angular term $l(l+2)/r^2$. The canonical map sends
family $k$ to the ground radial state of the $m = k$ sector, $k = 0, 1, 2$, and the
diagonal entry $n = k+1$ is the degree-$n$ density overlap of that state divided by
its eigenvalue. The three sectors $m = 0, 1, 2$ are three distinct operators: the
additive form gives ground eigenvalues $23.27, 24.27, 27.27$ and the cone form
$23.27, 29.31, 42.39$, all distinct, and at $m = 0$ both forms reduce to the pure
radial operator with ground eigenvalue $23.27$. Each form returns three distinct
diagonal entries, with no degenerate collapse, unlike the real reduction of A316.
The additive form returns the diagonal $(1, 2.427, 21.040)$ with $\mu = 2.427$ and
$\tau = 21.040$, at distance $13.07$ to the target; the cone form returns
$(1, 2.290, 20.726)$ with $\mu = 2.290$ and $\tau = 20.726$, at distance $12.75$.
Neither lands within twenty percent of $(1, 1.5, 8)$ and neither restores a valid
peak, the column-standardized argmax of column $k$ equal to $k$ for all three
columns while the sine-box control formed in the same measure fails the same test.
Neither improves on the real-reduction closest of A316, distance $0.399$ at
$\mu = 1.222$.
\end{proposition}

\begin{remark}[The verdict is null; what the cube-root phase did and did not
supply]
The construction is the complex realization A316 pointed to, built without fitting.
The family definition is the corpus one, the families as the $\omega^k$ eigenspaces
of the layer-cycle, and the realization carries the cube-root phase genuinely:
because the layer-cycle is a rotation of the Hopf fiber, the eigenspaces are the
fiber-momentum sectors $m \equiv k \pmod 3$, and the lowest representatives
$m = 0, 1, 2$ enter the radial operator through the fiber-Laplacian eigenvalue
$m^2$. The first result is structural and clean. The three sectors are three
distinct operators, $m^2 = 0, 1, 4$ in the additive form and $m^2/r^2$ in the cone
form, and they give three distinct diagonal entries with no collapse. This is the
quantity A316 could not produce: the real layer-cycle shift took two equal values
on the second and third families, and the real radial reduction had no further
structure to separate them; the complex fiber momentum separates them, since
$1 \neq 4$ where $\mathrm{Re}(\omega) = \mathrm{Re}(\omega^2)$. The degeneracy that
A316 named as the blocker is removed. The second result is the magnitude, and here
the canonical map does not close. Both forms send $\mu$ well above the target
rather than to it: the additive form to $2.43$ and the cone form to $2.29$, with
$\tau$ above $20$ where the target is $8$, and neither restores a peak. The genuine
cube-root phase, carried through the only corpus-faithful fiber forms, overshoots.
It does not move $\mu$ toward $1.5$ from the real-reduction value $1.222$; it
pushes it past, and the distance to the full target grows from $0.399$ to roughly
thirteen. A hit would have required the canonical map, family $k$ to the $m = k$
sector, to land the target; it does not, and no non-canonical reassignment of $m$
is invoked, so there is no fit to report and no derivation to claim. The reading is
that the complex Hopf phase supplies the structure A316 was missing, three distinct
families, but not the magnitude. The factor that lifts the second family to exactly
$1.5$ and sharpens the diagonal to a peak is not the fiber-momentum eigenvalue, in
either of its corpus-faithful forms. It lies elsewhere: in how the fiber sector
couples to the density overlap rather than in the additive or centrifugal energy it
contributes, or in a different reduction of the master operator than the radial one
this thread has used throughout. The magnitude piece of OI-287-1 does not close
here.
\end{remark}

\paragraph{Status.} OI-287-1 remains open. A317 confirms the structural prediction
of A316 and refutes the magnitude hope built on it. The complex Hopf-fibered
realization, with the families as the fiber-momentum sectors $m \equiv k \pmod 3$
that carry the cube-root phase $\omega^m$, breaks the A316 degeneracy: $m = 0, 1, 2$
are three distinct operators and give three distinct diagonal entries, where the
real layer-cycle shift gave only two. But the canonical map, family $k$ to the
ground state of the $m = k$ sector, overshoots the magnitude in both corpus-faithful
fiber forms, returning $\mu = 2.43$ for the additive $m^2$ term and $\mu = 2.29$ for
the cone $m^2/r^2$ term, with $\tau$ above $20$ and no peak in either. Neither
improves on the real-reduction closest of A316, which stands at distance $0.399$ and
$\mu = 1.222$. The genuine cube-root phase therefore supplies the broken degeneracy,
a real result, but not the magnitude. No published number changes. The remaining
quantity is the factor that lifts the second family to $1.5$ and restores the
diagonal peak, now shown to be neither a measure choice, nor an angular coupling
strength, nor a family-to-mode selection in the real reduction, nor the
fiber-momentum eigenvalue of the complex realization; the residual lies in the
coupling of the fiber sector to the overlap or in a reduction of the master operator
other than the radial one.

\paragraph{Verifier.} \texttt{verify\_P317.py}: recomputes both fiber forms from
scratch and confirms that $m = 0, 1, 2$ give three distinct operators in each form,
that both forms reduce to the pure radial operator at $m = 0$ with ground eigenvalue
$23.27$, that the additive form returns $(1, 2.427, 21.040)$ at distance $13.07$ and
the cone form $(1, 2.290, 20.726)$ at distance $12.75$, that neither lands within
twenty percent of the target and neither restores a valid peak, that the A316
degeneracy is broken with three distinct entries and no collapse in both forms, and
that the complex realization does not improve on the real-reduction closest at
distance $0.399$ and $\mu = 1.222$, so the null verdict's defining condition holds;
and no NaN appears. Eight checks, all PASS.

\end{document}
