Passes 4873–4874 · exact finite certificates

One number hid two groups. Four relations expose the Steiner cover.

The marked-double-six residue and the exceptional outer automorphism of S₆ both produce order 1440, but not the same group. Separately, the 120 Steiner triangles carry a complete imprimitive four-class association scheme whose quotient is Q(4,3), the line-intersection graph of W(3,3), not its point graph.

1440 ≠ identification

different centers, involution counts, and element orders

120 = 40 × 3

forty intrinsic three-element Steiner fibers

1, 2, 27, 36, 54

exact scheme valencies

15 ≠ 11 over F₃

Q43 line carrier separated from W33 points

Pass 4873: the two order-1440 extensions are nonisomorphic

Carrier	Group	Center	Involutions	Order-8 elements
marked double-six residue	S₆ × C₂	2	151	0
duad–syntheme outer action	Aut(S₆) = S₆:Out(S₆)	1	111	360

The first C₂ is the central operation that fixes duads and complements triads. The second is the exceptional outer-automorphism direction, certified by sending a transposition to a triple transposition. Their common order is a count coincidence, not a group identification.

Pass 4874: the complete Steiner association scheme

The four nonidentity relations on the 120 Steiner triangles have the exact meanings:

R₁: 120 pairs, valency 2 = forty K₃ fibers R₂: 1620 pairs, valency 27 = a perfect matching across each nonedge fiber pair R₃: 2160 pairs, valency 36 = all nine pairs across each Q(4,3) edge R₄: 3240 pairs, valency 54 = the complementary six pairs across each nonedge

The exact first eigenmatrix is

[ 1, 2, 27, 36, 54] [ 1, 2, -3, 6, -6] [ 1, 2, 3, -12, 6] [ 1, -1, 9, 0, -9] [ 1, -1, -3, 0, 3]

with primitive multiplicities 1,24,15,20,60. The first three sectors are the 40-dimensional fiber-constant Q(4,3) line-side Bose–Mesner module: dividing the lifted adjacency eigenvalues 36,6,−12 by the fiber size three recovers 12,2,−4. The transverse 80-space splits as 20+60 and is annihilated by the Q(4,3) adjacency lift. The forty maximal K₄ pencils recover the dual W33 point carrier; rank_F3(A+I) is 15 on the line side and 11 on the point side.

Reproduce

python3 analysis/w33_pass4873_two_order1440_extensions.py
python3 analysis/w33_pass4874_steiner_w33_association_scheme.py

Evidence boundary. The association relations, including the 3+6 refinement over every Q(4,3) nonedge (a pair of disjoint W33 lines), are canonical. Naming the three elements inside each fiber is additional gauge or coordinate data. The group theorem distinguishes two finite actions but selects neither as a physical symmetry. The complete [360,36,20]₂ weight enumerator is closed, but its covering radius remains open at the certified bound 124 ≤ ρ(K) ≤ 179.