This page freezes the exact continuation of the Pass-4721–4726 support-12/involution/dual-code theorem. The headline result is an explicit PSp(4,3)-equivariant identification between the 270 four-line involution residues and the independently constructed Pass-4716 selected270 router.
The residue action has rank 12 with stabilizer subdegrees 1,12,16,48,16,6,24,96,12,12,24,3. The share-two-lines orbital is 12-regular with 1,620 edges, diameter 4, distance shell 1,12,67,160,30, and spectrum 12^1, 8^15, 2^84, (-1)^64, (-4)^60, (-6)^6, (1+√13)^20, (1-√13)^20.
A degree-3 orbital is 27 disjoint Petersen graphs. Equality of an order-96 selected270 point stabilizer with a unique residue stabilizer produces a well-defined equivariant bijection. Under it the 1,620 residue share-two-lines edges are exactly selected270's cold edges and the 405 Petersen edges are exactly its hot edges.
For a four-fixing outer square root h, K=C_PSp(h) has order 48 and N_PSp(K) is the order-96 residue stabilizer. Thus N(K)/K ≅ C2. Representatives satisfy t²=n²=1, h=tn, and nt=h⁻¹. The earlier two-sheet mismatch is therefore an explicit normalizer-quotient twist, not an asserted cohomology class.
| q | rank₂(A*) | kernel minimum | elliptic fixed lines |
|---|---|---|---|
| 2 | 14 | 15 | — |
| 3 | 10 | 4 | 4 |
| 5 | 26 | 6 exact | 8 |
| 7 | 50 | ≤8 only | 8 |
The q=5 MILP result is a direct counterexample to the naïve statement that involution fixed sets are always the complete adjacency-kernel minimum shell.
A faithful 24-coordinate permutation action would require |PSp(4,3)| to divide |M24|; its 3^4 factor cannot fit in M24's 3^3. The canonical complement of a 16-fixed-line involution gives punctured H10=[24,10,6], and complements of the 135 weight-16 H10 words all give [24,9,4]. Both contain words below extended-Golay distance 8. This does not exclude a genuinely non-coordinate Leech/Conway representation.
The 27 Petersen fibers partition the 270 checks. Each fiber contains ten mutually disjoint four-line checks covering all forty line coordinates. Every coordinate lies in 27 checks, proving a lower bound of 27 layers; the fiber schedule attains it and is PSp-equivariant.
The check rank is 30. For H10=[40,10,12], syndromes are unique through weight 5. At weight 6 exactly 18,480 syndrome classes are doubletons, arising from complementary 6+6 halves of the forty weight-12 H10 words; there are no triple collisions.
The residue-check dependency code is [270,240,3] with exactly 540 minimum triples. Each triple has pairwise two-line intersections and six-line union, and the 540 triangles partition all 1,620 cold edges once. The tempting equality with the older 540 support/root triangles is false: their order-48 stabilizers have different element-order censuses.
For S_R=∏_{i∈R}Z_i and H=-Σ_R S_R, the check rank is 30 and the computational ground space is H10 with degeneracy 1,024. The syndrome code is [270,30,27]. Exact MILP certification proves the complete weight-27 shell consists of the forty single-line syndromes, giving a finite-model gap of 54 and first-excited degeneracy 40,960.
One-line defects violate 27 terms; two skew lines violate 48, while two meeting lines violate 54. The model therefore encodes W33 incidence directly in its two-defect energy. This is a finite commuting-check theorem only.
analysis/w33_pass4737_4738_4741_4742_residue_router_cocycle_decoder.pyanalysis/w33_pass4739_w3q_involution_minimum_shell_probe.pyanalysis/w33_pass4740_4744_golay_intertwiner_minor_no_go.pyanalysis/w33_pass4743_residue_parity_hamiltonian.py