Row-dependent bounds for strong permanent contraction on four-by-eight supports
Scott Schubr, Luminary Oddities LLC

This package contains the paper, editable LaTeX source and a standalone exact
Python certificate verifier. The theorem concerns all complex 4-by-8 operator
contractions X, with A(X) having diagonal blocks I4 and I8. It proves
per(|A(X)|^2) <= per(A(X)), equivalently Q(B) <= d(B) P(B) for Hermitian
positive semidefinite matrices supported in K4,8. Absolute squares are entrywise.

Run with standard-library Python 3.10 or newer:
    python -B verify_k48_certificate.py
    python -O -B verify_k48_certificate.py

The verifier reads no other files, writes no output files and uses no network,
optimizer or floating-point tolerance. It reconstructs the exact fixed-mixture
obstruction, 720 small-row corner/pairing evaluations and all 65,536 cubic tensor
Bernstein coefficients across 256 balanced-region boxes. It also checks exact
interpolation reconstruction away from the grid. A successful run exits zero;
any failed acceptance condition raises an explicit exception, including under -O.

The paper proves the analytic matrix reductions, domain coverage, coordinate
degree bounds and Bernstein partition-of-unity argument. Finite code alone is
not a replacement for those arguments. The heavy-entry input is the explicitly
cited Lemma 4 of the author's published through-six-columns paper. The retained
third-minor bound is credited to the published four-by-seven continuation.

Build the manuscript with a LaTeX installation supporting amsmath, amssymb,
amsthm, booktabs, geometry and hyperref. For example:
    tectonic manuscript/main.tex
The engine may retrieve standard TeX dependencies on its first run. No private
inputs or external figures are required. The supplied PDF is the reader copy.

MANIFEST.json records SHA-256 hashes for the five primary payload files.
SHA256SUMS additionally covers MANIFEST.json; neither file claims to hash itself.

This result does not prove K4,9, K5,5, all four-row bipartite supports, an
arbitrary-second-factor inequality, or unrestricted Chollet. The computer-assisted
proof and its scope are stated in the paper. No optimality or equality
classification is claimed.
