Pass5284–5291: post-distance shell geometry
Authoritative q=5 status: Pass5262 proves the apartment code is [73125,625,625]₂, and the 936 chamber stars are exactly its minimum words.
What this packet adds
- Complete K₀ block-minimum word shell: 2,340 words, 15 on each of the 156 W-point footprints; every one is a difference of two same-point chamber stars and has apartment weight 1,000.
- PG(3,2) fiber: the 15 minimum words over a W point are the nonzero vectors of
E₆/<111111> ≅ F₂⁴. - Controller-sheet diagonal: under the q=5
(a,b,c,d)→(a,c) chart, the 4D fiber is a constant section over all 25 sites of the 5×5 P-block sheet. - Replicated simplex: locally the fiber is a 5-fold
[15,4,8] simplex, giving [75,4,40] after zero-coordinate removal; globally it is the 125-fold simplex [1875,4,1000]. - Hoffman shortening: the 312 complement coordinates split into four exact cover-relative types
(6,5,2)^144, (6,6,1)^96, (6,3,4)^48, (6,1,6)^24. The shortened minimum remains open in {28,32,36,40}. - q=9 anchor: the point/P-component incidence matrix has rank 369 over F₂ and an explicit weight-16 dual dependence. No q=9 distance-81 theorem is claimed.
- All-odd rank reduction: every W-line is orthogonal mod 2 to every P-carrier, proving
rank₂(F) ≤ q(q²+1)/2. The q=3,5,7,9 anchors all attain equality.
Evidence firewall
The q=5 distance-625 theorem is finite and closed. The all-q apartment-distance law, the all-odd footprint-rank equality, the q=9 footprint distance, and the exact Hoffman-shortened minimum remain separate open problems.