A QUANTITATIVE CHOLLET INEQUALITY FOR RANK-FOUR MATRICES OF ORDER SEVEN
Scott Schubr, Luminary Oddities LLC
Manuscript version 1.1 - October 20, 2026

Contents
  Rank_Four_Chollet_Order_Seven.pdf       Complete paper
  manuscript/main.tex                   Complete LaTeX source
  certificate/singular5_rho3_5_certificate.json
                                        Exact rational certificate data
  code/verify_certificate.py            Standalone exact-integer checker
  code/build_pdf.py                     Portable local PDF build helper
  CITATION.bib                          Citation metadata
  MANIFEST.sha256                       File-integrity inventory

Results
For a complex Hermitian positive semidefinite matrix A of rank at most
four, put P=per(A) and Q=per(A o conjugate(A)). The paper proves
  order5 or order6: Q <= (3/5) P^2;
  order7:          Q <= (192/245) P^2.
The inequalities are strict when every diagonal entry is positive.
A zero diagonal gives P=Q=0. The corresponding two-factor estimates
require BOTH factors to have rank at most four.

The proof combines an exact singular-order-five certificate with a
rank-preserving border comparison. The inherited principal-minor SOS
feature cone is credited to Lavi and the border mechanism to
Li--Zhao--Luo. The paper proves the needed comparisons and does not assume
their full conjectural frameworks. No optimality of the constants is claimed.

Verify the essential certificate
Requires Python 3.9 or later and its standard library only. From this
package directory run:
  python code/verify_certificate.py
  python -O code/verify_certificate.py
An explicit certificate path may be passed as the sole positional argument.
The checker validates all 6,210 polynomial coefficients, all 240 symmetry
actions, all 38 exact positivity witnesses and the 119-feature strictness
block. A PASS is based on exact arithmetic, not a hash or numerical tolerance.
The code prints its result and does not change the input data.

The certificate and analytic argument are both essential. A file-integrity
hash alone does not prove an inequality. No optimizer, saved discovery
matrices, external certificate, numerical search, or private file is needed.

Rebuild the paper
Requires Tectonic with the needed LaTeX resources. From this directory run:
  python code/build_pdf.py --output-dir ../chollet-reader-build
Use --tectonic PATH_TO_TECTONIC if it is not on PATH; use --only-cached
for an offline build. An existing TeX cache may be selected with
--cache-dir PATH. The output directory must be new; logs are written
there, and the supplied sources are left unchanged.
Alternatively compile manuscript/main.tex with an ordinary compatible
LaTeX installation supporting the listed packages.

This package treats rank at most four at the stated orders. It does not
settle unrestricted-rank order seven, full rank-four order eight, or
general non-tight order nine. The source is a scholarly manuscript;
its inclusion here is not a statement of journal acceptance.
