% 316_Addendum_ModeMap.tex
% Copyright Léon Fernando Vlegels -- CC BY 4.0
% OI-287-1. A315 reduced the magnitude residual of the charged-lepton selection
% rule to a family-to-mode SELECTION question: with the inner-product measure
% forced (A311, the B^4 radial volume r^3 dr) and the residual shown not to be a
% one-parameter angular coupling (A315), what remains is which eigenstate of the
% forced 4D-radial operator each of the three families is mapped to. A316
% enumerates a short, pre-declared list of principled family-to-mode maps and asks,
% for each, whether it reproduces the target diagonal (1, 1.5, 8) with a valid
% column-standardized peak, and whether the map is corpus-FORCED or a fit. Two of
% the five maps are privileged BY THE CORPUS as the two principled family
% definitions: the node-count map (family n = mode n-1 of the l=0 operator, the
% A297-A301 reading) and the Z_3-character map (family n = the omega^k eigenspace
% of the layer-cycle operator T_cycle, P20). The finding is NULL. The node-count
% map gives the diagonal (1, 1.222, 8.286) -- the closest of all five maps, the
% A313 C_4Dself object -- but mu falls short of 1.5 and no peak is restored. The
% Z_3-character map, realized through the real Re(omega^k)gamma layer-cycle shift
% the radial reduction admits, is rank-degenerate: the k=1 and k=2 shifts are
% equal, so families two and three collapse to the same radial state and the map
% cannot produce three distinct entries. None of the non-canonical maps hits the
% target with a peak either. No declared map both lands within twenty percent of
% (1, 1.5, 8) and restores a valid peak. Verdict NULL: the family-to-mode rule that
% lands the target with a peak is not among the principled maps, and the corpus
% canonical maps do not close the magnitude piece. Verifier: verify_P316.py.

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

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

\title{Addendum 316: The Family-to-Mode Selection Rule\\
\large No principled mode map reproduces the magnitude target with a peak}
\author{TOE Corpus}
\date{2026-06-13}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} Case C (null; neither corpus-canonical family-to-mode map
reproduces the target diagonal with a valid peak, and no declared map does)
\quad|\quad
\textbf{Addresses:} OI-287-1 \quad|\quad
\textbf{Verifier:} \texttt{verify\_P316.py} --- all PASS

\begin{abstract}
A311 fixed the inner-product measure of the charged-lepton selection rule as the
geometrically forced radial volume $r^3\,dr$ of the $B^4$ arena, and A315 showed
the residual to the target ratio $\mu = 1.5$ is not a one-parameter angular
coupling strength: the magnitude motion lives in which eigenstate of the forced
four-dimensional radial operator each family occupies, the family-to-mode
selection rule. This addendum enumerates five principled family-to-mode maps,
declared before any number is computed, and tests for each whether it reproduces
the target diagonal $(1, 1.5, 8)$ within twenty percent with a valid
column-standardized peak. Two of the five are privileged by the corpus as the two
principled family definitions: the node-count map, taking family $n$ to the
$(n{-}1)$-th radial mode of the $l = 0$ operator, and the $\mathbb{Z}_3$-character
map, taking family $n$ to the $\omega^k$ eigenspace of the layer-cycle operator,
$k = n{-}1$. A hit requires one of these two canonical maps; a fit by a
non-canonical map is reported as a fit. The finding is null. The node-count map
returns the diagonal $(1, 1.222, 8.286)$, the closest of all five maps and
identical to the pure radial object of A311 and A313, but $\mu$ falls short of
$1.5$ and no peak is restored. The $\mathbb{Z}_3$-character map, realized through
the real layer-cycle shift $\mathrm{Re}(\omega^k)\gamma$ the radial reduction
admits, is rank-degenerate: the shifts $\gamma(1, -\tfrac12, -\tfrac12)$ make the
$k = 1$ and $k = 2$ operators identical, so families two and three collapse to one
radial state and the map cannot produce three distinct diagonal entries. The
non-canonical maps either overshoot, with the angular-sector map at $\mu = 2.009$
and the shifted-sector map at $\mu = 2.326$, or reduce to the node-count map, as
the self-consistent diagonal map does, selecting modes $0,1,2$ and giving no peak
even when it selects for one. No declared map both lands within twenty percent of
the target and restores a valid peak. The verdict is null: the family-to-mode rule
that lands the target with a peak is not among the principled maps, and neither
corpus-canonical map closes the magnitude piece.
\end{abstract}

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

\begin{proposition}[Neither corpus-canonical mode map hits the target with a peak]
On the forced arena geometry of A311, the four-dimensional radial operator
$\hat O_l = -d^2/dr^2 - (3/r)\,d/dr + l(l+2)/r^2 + V_{\text{eff}}(r)$,
self-adjoint in $r^3\,dr$ with both eigenstates and the degree-$k$ density overlap
formed in $r^3\,dr$, consider five family-to-mode maps. The node-count map
(S-radial) takes family $n$ to the $(n{-}1)$-th radial mode of $\hat O_0$; the
angular-sector map (S-lsector) takes family $n$ to the ground mode of
$\hat O_{n-1}$; the shifted-sector map (S-lground) takes family $n$ to the ground
mode of $\hat O_n$; the $\mathbb{Z}_3$-character map (S-Z3) takes family $n$ to the
ground mode of $\hat O_0 + \gamma\cos(2\pi(n{-}1)/3)$, with $\gamma = 3/4$ the
layer-cycle coefficient, the real footprint of the $\omega^k$ phase on the radial
operator; the self-consistent map (S-diag) takes family $n$ to the low mode of
$\hat O_0$ whose degree-$n$ overlap is maximal. Each diagonal entry $n$ is the
degree-$n$ density overlap of the selected state divided by its eigenvalue, and a
valid peak requires the column-standardized argmax of column $k$ to equal $k$ for
all three columns while the sine-box control formed in the same measure fails the
same test. Computed on the standard grid, the node-count map returns the diagonal
$(1, 1.222, 8.286)$ with $\mu = 1.222$ and $\tau = 8.286$, node counts $0,1,2$,
and distance $0.399$ to the target, the closest of the five and identical to the
A313 pure radial object; it restores no valid peak and $\mu$ is short of $1.5$. The
$\mathbb{Z}_3$-character map returns $\mu = 2.656$ and is rank-degenerate: the
shifts $\gamma(1, -\tfrac12, -\tfrac12)$ make the $k = 1$ and $k = 2$ operators
identical, so families two and three share an eigenvalue and an entire overlap row,
and the map yields only two distinct radial states. The angular-sector map gives
$\mu = 2.009$, the shifted-sector map $\mu = 2.326$, and the self-consistent map
reduces to the node-count map at $\mu = 1.222$ with no peak. No declared map both
lands within twenty percent of $(1, 1.5, 8)$ and restores a valid peak; in
particular neither of the two corpus-canonical maps does.
\end{proposition}

\begin{remark}[The verdict is null; what remains]
The two maps privileged by the corpus are the two principled definitions of a
fermion family. The node-count reading, that the three families are the three
lowest radial modes distinguished by node count, is the one carried through the
selection-rule thread; the $\mathbb{Z}_3$-character reading, that the families are
the three eigenspaces of the layer-cycle operator, is the one stated in the
family-origin proposition of the embedding paper. Both were declared before
computing, so a hit by either would have been a genuine derivation. Neither hits.
The node-count map is the closest object in the entire arc but stops at the same
$\mu = 1.222$ the pure radial operator gives, with the diagonal peak still
unrecovered. The character map, in the only form the real radial reduction admits,
is degenerate by construction: the real part of the cube-root phase takes two equal
values, and the radial operator has no further structure to separate them, so the
map collapses two of the three families. This is itself an honest reading of the
gap. The $\mathbb{Z}_3$ phase is a complex label; the radial reduction is real, and
the part of the family structure that distinguishes the second and third families
is exactly the part the reduction cannot carry. Among the non-canonical maps, the
angular and shifted-sector maps overshoot the target and the self-consistent map,
which selects modes to maximize the diagonal, recovers only the node-count map and
still produces no peak, which says the peak is not available to any selection among
the low radial modes of a single operator. The closest fit on the full target is
unchanged from A311 and A313. The verdict is null. The family-to-mode rule that
both lands $(1, 1.5, 8)$ and restores the peak is not among the principled maps
tested here, and the corpus-canonical maps, the node-count and the
$\mathbb{Z}_3$-character definitions, do not close the magnitude piece. What
remains is a family-to-mode rule that carries the complex $\mathbb{Z}_3$ phase, or
a structure beyond the real radial reduction, capable of distinguishing all three
families and producing a diagonal peak.
\end{remark}

\paragraph{Status.} OI-287-1 remains open. A316 closes the family-to-mode route as
posed: among five pre-declared principled maps, neither corpus-canonical map, the
node-count map nor the $\mathbb{Z}_3$-character map, reproduces the target diagonal
$(1, 1.5, 8)$ within twenty percent with a valid peak, and no non-canonical map
does either. The node-count map is the closest, returning $(1, 1.222, 8.286)$ at
distance $0.399$, the same pure radial object as A311 and A313, short of $\mu = 1.5$
with no peak. The $\mathbb{Z}_3$-character map is rank-degenerate in the real radial
reduction, collapsing two of the three families, because the real part of the
cube-root phase takes two equal values. The closest fit is still the pure radial
four-dimensional operator, and the diagonal peak remains unrecovered. No published
number changes. The remaining quantity is a family-to-mode rule that carries the
complex $\mathbb{Z}_3$ phase or reaches beyond the real radial reduction, together
with the unrecovered peak.

\paragraph{Verifier.} \texttt{verify\_P316.py}: recomputes all five maps from
scratch and confirms the node-count map reproduces the pure radial diagonal
$(1, 1.222, 8.286)$ at distance $0.399$, the closest map, with $\mu$ short of $1.5$
and no valid peak; the angular-sector map overshoots at $\mu = 2.009$; the
shifted-sector and self-consistent maps do not hit, the latter reducing to the
node-count map; the $\mathbb{Z}_3$-character map is rank-degenerate with families
two and three collapsed; no canonical map and no non-canonical map hits, so the
null verdict's defining condition holds, that no declared map both lands within
twenty percent and restores a valid peak; and no NaN appears. Eight checks, all
PASS.

\end{document}
