PASSES 3500–3505 · EXACT GRAPH-THEORY SUPPLEMENT

Seven graphs, two different 57s, and one exact W33 ladder.

The known triangle-free strongly regular graphs form real descendant and spectral structures. The missing degree-57 Moore graph is a separate 3250-vertex problem, and its edge-rooted construction is a coupled permutation system—not a source of 56! automorphisms.

The seven known examples

Graph	Parameters	Restricted spectrum
C5	(5,2,0,1)	(−1±√5)/2
Petersen	(10,3,0,1)	1, −2
Clebsch	(16,5,0,2)	1, −3
Hoffman–Singleton	(50,7,0,1)	2, −3
Gewirtz	(56,10,0,2)	2, −4
M22	(77,16,0,4)	2, −6
Higman–Sims	(100,22,0,6)	2, −8

The μ=4, r=2 ladder

(77,16,0,4) → (57,14,1,4) → (40,12,2,4) → (26,10,3,4) → (15,8,4,4)

M22 → hole → W33

Fixing μ=4 and the positive restricted eigenvalue r=2 forces a five-rung parameter family. The 57-vertex rung was proved nonexistent in 1983.

v(λ)=(λ−7)(3λ−22)/2

The degree, negative eigenvalue, and both multiplicities are polynomial functions of λ. W33 is the central λ=2 rung.

57 vertices ≠ degree 57

The impossible SRG(57,14,1,4) is not the hypothetical Moore graph SRG(3250,57,0,1).

W33 and Gewirtz share a quadratic kernel

(A−2I)(A+4I)=0 on augmentation

Both graphs have nonprincipal eigenvalues 2 and −4. Their complement operators have nonprincipal eigenvalues ±3, so the centered complement squares to 9I. This is a shared functional calculus, not a canonical graph or module identification.

The corrected Moore-57 edge chart

2 + 56 + 56 + 56²

Fixing an edge partitions the hypothetical graph into its endpoints, two punctured neighborhoods, and a 3136-vertex residual core.

56 × 56

Each residual vertex is the unique common neighbor of one branch vertex on each side, forcing a row-column grid.

55 residual neighbors

Every pair of distinct rows—and every pair of distinct columns—is joined by a perfect matching.

short products derange

The matching permutations and their three-row products must be fixed-point-free to prevent triangles.

Automorphism correction. 92,625 is the edge count. The 56! coordinate relabellings are construction gauge, not automatically automorphisms of a fixed completed graph. Regular bipartite graphs also need not be complete bipartite.

The four-way 57 firewall

Object	What “57” means	Status
Missing M57	degree 57; 3250 vertices	Open
μ=4 ladder hole	57 vertices; degree 14	Nonexistent
Perkel graph	57 vertices; degree 6	Exists
Regular 57-cell	57 polytope cells	Exists

A June 2026 preprint claims any missing Moore graph has no involutory automorphisms. If correct, its automorphism group is odd, so the even-order PSL(2,19) symmetry of the 57-cell/Perkel object cannot embed. This is a conditional symmetry firewall, not a nonexistence proof.

Machine certificate

PASS_6_FRONTS
e308364dd480970803061b90ab27bf86afa2ae4946399b656292f02682c17fd9

The verifier and frozen JSON certificate are in analysis/bt3500_3505_triangle_free_srg_m57_bridge.py and data/PART_BT3500_BT3505_TRIANGLE_FREE_SRG_M57_BRIDGE_results.json.