CHOLLET'S INEQUALITY FOR POSITIVE SEMIDEFINITE MATRICES OF RANK AT MOST FOUR
Scott Schubr, October 22, 2026

Contents
  Rank_Four_Chollet_All_Orders.pdf       Complete specialist note
  manuscript/main.tex                  Complete LaTeX source
  certificate/singular5_rho3_5_certificate.json
                                       Unchanged prior singular-five certificate
  code/verify_certificate.py           Unchanged exact-integer base verifier
  code/verify_comparisons.py           Finite scalar comparison checks

Mathematical dependencies
The new contribution is the controlled-loss border transfer and its full
rank-at-most-four order-eight/order-nine consequences. The singular-five
theorem and certificate were already established in Schubr's order-seven
paper. The occupation identities are credited to Li, Zhao and Luo. The
all-orders corollary cites Rodtes for order four (with identity padding for
smaller orders) and Ma, Theorem 1.2, for every order at least ten.
Both matrices must have rank at most four. Arbitrary-rank Chollet and optimal
matrix constants are outside the note. Ma's tight-nine result is not used.

Verification
From this directory, using Python 3 with its standard library:
  python -B code/verify_certificate.py
  python -B code/verify_comparisons.py
The first command reconstructs the entire 6,210-coefficient polynomial
identity and checks 38 positive-definiteness witnesses. The certificate
format and PSD interpretation are described in the paper and its cited
predecessor. The second command checks the finite arithmetic of the transfer.
Neither command re-proves Rodtes or replays Ma's large-order computations.
No saved PASS output is required or trusted by either command.

The base certificate SHA-256 is:
  e41c28e21094316d5813cac779e95b23e64be66b4ddd8f98587b148faa8aa074
The unchanged base verifier SHA-256 is:
  0893bf023977d895a4d3c19fbebfb2c8a0948b444fe6367ba432a86d0fc17423
These identify the inherited files; mathematical validity comes from their
exact checks and the analytic proof, not their hashes.

Build
Compile manuscript/main.tex with a current LaTeX distribution supporting
amsmath, amssymb, amsthm, booktabs, newtxtext, newtxmath, microtype, geometry
and hyperref; run the compiler until references settle. Tectonic is one
option: tectonic manuscript/main.tex --outdir <chosen-output-directory>
The supplied PDF is for readers who do not wish to install TeX.

The cited large-order theorem is not re-proved in this concise note. A
separate independent reconstruction of its computational evidence is not
part of this reader package. Numerical searches are not proof inputs.
