Supplementary Information
=========================

Manuscript: Fractional projective-gap inequalities for harmonic Gaussian chaoses
Authors: Khamit Kadyrbekov and Daniyal Kadirbekov

File
----
harmonic_cubic_phase_bernstein_exact.py

Purpose
-------
The script regenerates the exact rational Bernstein certificate used in
Theorem 6.1. It starts from the displayed canonical harmonic cubic, computes
all spherical moments in exact arithmetic, verifies the invariant reduction
and the factorization (17), and certifies the ten-leaf subdivision tree.
No floating-point inequality is used.

Requirements
------------
Python 3, SymPy, and NumPy.

Run
---
Run from a directory that does not contain a local module named struct.py:

OMP_NUM_THREADS=1 OPENBLAS_NUM_THREADS=1 MKL_NUM_THREADS=1 \
VECLIB_MAXIMUM_THREADS=1 NUMEXPR_NUM_THREADS=1 \
python3 harmonic_cubic_phase_bernstein_exact.py

Expected terminal summary
-------------------------
Endpoint initial minimum: 4209/1925
R0 initial minimum:
  2915214396979092454188/23165189182717957375
R2 initial minimum:
  90692005649292/445013694125
EXACT_CERTIFICATE: 10 leaves, maximum depth 5

The program aborts if an identity, positivity assertion, leaf path, or depth
check fails.

SHA-256 of the verifier
-----------------------
170c18d377c14c0a30c90b42033c07464b2930d53fc27e4d73d5d5007fb17e20

