% 305_Addendum_JointKernel.tex
% Copyright Léon Fernando Vlegels -- CC BY 4.0
% OI-287-1. A ninth mechanism class is excluded: a non-separable joint
% kernel K[n,k] applied to the A299 base, suppressing entries with
% n != k, does not supply the missing localizing factor. None of the four
% declared joint kernels (Gaussian off-diagonal suppression, Kronecker
% band, self-structure overlap, hard projection) yields a valid diagonal
% peak, and every kernel that equals one on the matching pair leaves the
% normalized diagonal at the A299 value (1, 1.12, 7.50): off-diagonal
% suppression cannot lift a diagonal entry. The separable control does not
% beat the joint kernels. The localizing factor is not a kernel acting on
% the existing overlap matrix.
% Verifier: verify_P305.py.

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

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

\title{Addendum 305: The Joint Kernel\\
\large A ninth excluded class, and why off-diagonal suppression cannot
lift a diagonal entry}
\author{TOE Corpus}
\date{2026-06-13}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} Case D (mechanism class excluded; OI-287-1 lead
unchanged) \quad|\quad
\textbf{Addresses:} OI-287-1 \quad|\quad
\textbf{Verifier:} \texttt{verify\_P305.py} --- all PASS

\begin{abstract}
A300 and A303 together excluded the two separable hypotheses for the
factor missing from the A299 base
$B_{nk} = \langle\psi_n|c_k\pi^k x^k|\psi_n\rangle / \lambda_n$, whose
diagonal normalizes to $(1, 1.12, 7.50)$ against the target $(1, 1.5, 8)$.
A300 ruled out a factor depending only on the family index $n$; A303 ruled
out a factor depending only on the monomial degree $k$. Both pointed to the
same next hypothesis: the missing factor must couple $n$ and $k$ jointly,
acting within a column to raise the $n = k$ entry above its neighbors, an
off-diagonal-suppression effect. This addendum runs that test. Four
non-separable joint kernels $K_{nk}$ are declared in advance and applied as
$M_{nk} = B_{nk}K_{nk}$: a Gaussian off-diagonal suppression
$\exp(-(k-(n{+}1))^2/2)$ with fixed width, a Kronecker band
$(1/2)^{|k-(n{+}1)|}$ with fixed decay, the overlap's own structure as a
self-suppressing kernel, and a hard projection onto the diagonal. A declared
separable control $f(n)g(k)$ is run alongside. None of the joint kernels
yields a valid diagonal peak: the Gaussian moves only the third column's
argmax one row toward the diagonal and stalls; the hard projection collapses
every column's variance and so peaks trivially on the matched sine-box
control as well, failing the control clause. Every kernel that equals one on
the matched pair leaves the normalized diagonal exactly at $(1, 1.12, 7.50)$,
because off-diagonal suppression by construction does not touch the diagonal
entry it sits beside. The separable control lands at the same $L^2$ as the
joint kernels and does not beat them. A non-separable joint kernel acting on
the existing overlap matrix is therefore excluded as a ninth mechanism class.
The localizing factor is not a kernel that reweights the overlaps the
operator already produces; it must change the overlaps themselves.
\end{abstract}

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

\begin{proposition}[A joint kernel on the overlap cannot localize the diagonal]
Let $B_{nk} = \langle\psi_n|c_k\pi^k x^k|\psi_n\rangle / \lambda_n$ be the
A299 base, diagonal $(1, 1.12, 7.50)$, in which every column's largest entry
sits in row $0$. For a joint kernel $K_{nk}$ with $K_{nn} = 1$, the weighted
matrix $M_{nk} = B_{nk}K_{nk}$ has $M_{nn} = B_{nn}$, so its normalized
diagonal is identical to that of $B$. Evaluated on the declared kernels, each
of the Gaussian $\exp(-(k-(n{+}1))^2/2)$, the band $(1/2)^{|k-(n{+}1)|}$, and
the self-structure overlap returns the diagonal $(1, 1.12, 7.50)$ unchanged,
and none produces a valid diagonal peak under the column-standardized
criterion paired with the sine-box control. The hard projection $K = I$ peaks
on the diagonal but also peaks on the control, so it fails the control clause.
The closest candidate by $L^2$ sits at $0.63$, the A299 control distance, and
its $\mu$ entry remains at $1.12$, not moved toward $1.5$. The pre-registered
HIT criterion, a valid diagonal peak together with all three entries within
$20\%$, is met by no candidate. A non-separable joint kernel is excluded as a
ninth mechanism class. The separable control $f(n)g(k)$ lands at the same
$L^2$ and does not beat the joint kernels.
\end{proposition}

\begin{remark}[Why the kernel fails, and what it implies]
The failure has two parts, and both are structural rather than numerical. The
first part concerns the diagonal entries. A kernel that suppresses
off-diagonal entries by definition equals one on the matched pair, so it
multiplies $B_{nn}$ by one and leaves it fixed; the diagonal ratio is
invariant under any such kernel. The lift the target demands, $\mu$ from
$1.12$ toward $1.5$, raises a diagonal entry, and no off-diagonal suppression
can raise a diagonal entry. The second part concerns the peak. Suppressing the
off-diagonal does move a column's argmax toward the diagonal, and the Gaussian
shows this in the third column, where the argmax shifts from row $0$ to row
$1$; but the dominance of the first family in the raw overlaps is large enough
that finite suppression does not carry every column all the way to its diagonal
row, and the only kernel that forces it, the hard projection, destroys the
column variance that distinguishes the operator's eigenfunctions from a generic
basis and so peaks on the sine-box control too. Read together with A300 and
A303, the three exclusions close the space of post-hoc reweightings: the factor
is not a function of the family alone, not a function of the degree alone, and
not a kernel on the overlaps that depends on both. The honest narrowing is that
the localizing information is not present in the overlap matrix $B$ to be
recovered by any reweighting of it. It must enter earlier, in the overlaps
themselves, which means in the operator or the states, not in a correction
applied after the overlaps are formed.
\end{remark}

\paragraph{Status.} OI-287-1 remains open. Its form is unchanged by this
addendum: the missing factor lifts the $\mu$ entry by roughly $34\%$ and the
$\tau$ entry by roughly $7\%$, and is now known to be neither a function of the
family index alone (A300), nor of the monomial degree alone (A303), nor a
kernel applied to the A299 overlap matrix (this addendum). The next probe must
look inside the overlap, at the operator or the eigenstates, not at a
post-overlap correction. No published number changes.

\paragraph{Verifier.} \texttt{verify\_P305.py}: recomputes the eigenstructure
(eigenvalues ordered, node counts $(0,1,2)$), confirms the A299 base diagonal
reproduces $(1, 1.12, 7.50)$, confirms each declared joint kernel that equals
one on the matched pair leaves the diagonal at the A299 value and that none
yields a valid diagonal peak, and confirms the NULL conditions, that no
candidate lands within $20\%$, the closest sits at the control $L^2 = 0.63$
with $\mu$ unmoved, and the separable control does not beat the joint kernels.
All PASS.

\end{document}
