complex-compression-note.tex

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\title{Reusable compression for the complex finite network}
\author{Douglas Colkitt\thanks{Prepared with assistance from OpenAI Codex.
This is not independent review or endorsement by OpenAI.}}
\date{October 7, 2026}
\begin{document}
\maketitle
\begin{abstract}
We give two reusable methods for compressing weighted side computations
while retaining the binary phase-frame contract in OpenAI's integer
multiplication manuscript. Coordinate enclosures support disjoint-triple
rectangles; an explicit orthonormal family supports shared intersection-two
sums. Signed transparent computation restores arbitrary scratch, and an
even-ground-size matching reuses complete first/third-stage auxiliary banks.
At $h=26$ the construction supports complex saving $5\cdot10^{-9}$,
exceeding the retained bit saving. An independent combined upstream patch
integrates this construction and the retained compact-control arguments,
supporting the conditional bound $\kappa=2^{-31}$. The previously published
$2^{-34}$ artifacts are unchanged.
\end{abstract}
The retained manuscript is pinned at
\begin{center}\texttt{adc7f1241b42e322a6451854ab7e4b4c146bf78a}.\end{center}
Its algorithmic interfaces remain assumptions. This local research note
concerns the same fixed finite-alphabet, fixed-tape model and makes no
practical runtime or unrestricted optimality claim.
\input{complex-compression.tex}
\section{Artifacts}
\begin{itemize}
\item \path{scripts/complex_compression.py}: construction and exact accounting.
\item \path{scripts/binary_phase_frames.py}: reusable binary-space checker.
\item \path{patches/complex-compression-31.patch}: complete independent patch
against the pinned original, generated by
\path{scripts/make_complex_compression_patch.py}.
\item \path{certificates/complex-compression.json}: construction-source hash,
full disjoint-pair enumeration, finite frame controls, matching, network counts
and conditional assembly margins.
\item \path{tests/test_complex_compression.py}: exact rational scalar maps
with independent dirty variables, both signed directions, shared scratch
composition, phase identities, and rejection of alternating or degenerate
residuals.
\item \path{tests/test_complex_compression_patch.py}: source dependencies,
separate arities, exact parameter substitution and reference checks.
\end{itemize}
Finite examples do not replace the general proofs.
\end{document}