OMEGA PAPER IX — FOLLOW-UP REPRODUCTION
Date: 2026-11-01

Fixed problem:
  M = 13I + A on the pinned 120-vertex 600-cell graph;
  B = {0,...,73}^120; Delta = det(M);
  F_max = max_c #{u in B : [u]=c in Z^120/MZ^120}.

Current certified result: 307 <= ceil(log2(F_max)) <= 352 bits.
The exact maximum fiber and optimal width are not computed.

Requirements: Python 3.10+ and a C++17 compiler named g++.

Reproduce the retained baseline and follow-up certificates from this folder:

  python3 verify_orbit_cover.py --output verification_normal.json
  python3 -O verify_orbit_cover.py --output verification_optimized.json
  python3 verify_neighborhood_bound.py --output neighborhood_normal.json
  python3 -O verify_neighborhood_bound.py --output neighborhood_optimized.json
  python3 verify_neighborhood_order6.py
  python3 verify_neighborhood_order7.py
  python3 verify_cover17_order5.py
  python3 -O verify_cover17_order5.py
  python3 verify_cover17_order6.py
  python3 verify_dense_15_16_a7_internal.py
  python3 verify_dense15_a7_one_external.py
  python3 verify_cover19_screen.py
  python3 verify_matching_shearer.py
  python3 -O verify_matching_shearer.py
  python3 verify_cover20_order5.py
  python3 -O verify_cover20_order5.py
  OMP_NUM_THREADS=9 python3 verify_cover20_order6.py
  python3 verify_cover20_nested_bound.py
  python3 -O verify_cover20_nested_bound.py

The first four commands regenerate reports consumed by later commands. Exact
determinants use Bareiss elimination; threshold decisions use integer powers.
The 17-coordinate coefficients use all 120 projected columns of M.

Important limits:
  * The order-seven coefficient for the 13-cover proves that this particular
    projected-volume inequality cannot yield a 351-bit upper bound.
  * The tested 15-, 16-, 17-, and restricted 19-coordinate covers did not
    improve the upper endpoint. The 19-coordinate scan covers all six-vertex
    additions to one fixed 13-neighborhood, not all 19-subsets.
  * The local matching/Shearer relaxation certified only 353 bits at size 19
    and 354 bits at size 13; it is weaker than the retained projected-volume
    certificate.
* The exact 20-coordinate order-six projection calculation proves 352 bits
  but not 351. Adding the nested degree-seven coefficient sharpens that
  20-coordinate bound, but the result is still weaker than the retained
  13-coordinate upper bound. The exact order-six run takes about six minutes
  here using nine OpenMP threads.
  * No finite run here counts F_max, computes C_10, or proves a global limit
    over all possible covers or methods.
