W33-Theory · Pass 4870 · corrected by Passes 4953–4956

The Steiner triangles refine the W33 line action — and the first adjoint bridge is quadratic

The quadratic-Hom theorem of Pass4870 survives, but one graph identification does not. The forty Steiner fibers are not the standard W(3,3) point graph. Exact re-audit shows that their quotient is the intersection graph of the forty W33 lines, equivalently the point graph of the odd-q dual generalized quadrangle Q(4,3).

Correction: the former statement “the forty-fiber quotient is explicitly isomorphic to the standard symplectic W(3,3) point graph” is withdrawn. The owner producer now proves the quotient is the W33 line-intersection graph Q(4,3), recovers W33 points as forty maximal K4 pencils, and separates the carriers by rank_F3(A+I)=15 versus 11. Standard W33 point triads have census 1²⁸⁸⁰4³⁶⁰; the Steiner/Q43 quotient has 0¹⁰⁸⁰2²¹⁶⁰.

Prior-art firewall: the 120 Steiner trihedral pairs forming forty triads is classical cubic-surface geometry. The repo-derived content is the exact four-orbital refinement, the corrected Q(4,3) line-action quotient, and the characteristic-three Hom calculation.

120 = 40 × 3

The 120-pair relation gives forty three-element Steiner fibers.

2160 = 240 × 9

Each quotient adjacency edge lifts to a complete K3,3 between two fibers.

dim = 2

Hom_PSp(Sym² H2,Q10) is two-dimensional; all other pair-orbit contributions vanish.

Four intrinsic pair relations

pair count	degree	triangle intersection	cross edges	role
120	2	0	0	forty K3 fibers
1620	27	1	6	nonadjacent-fiber refinement
2160	36	0	6	Q(4,3) adjacency lift
3240	54	0	4	nonadjacent-fiber refinement

The corrected degree-40 quotient

Collapse the forty K3 fibers. The 2160-pair relation gives an SRG(40,12,2,4), but those parameters do not identify W33: the repo had already documented multiple nonisomorphic graphs with those parameters. Pass4954 compares against both degree-40 generalized-quadrangle actions and proves:

Steiner quotient = W33 line-intersection graph = Q(4,3) point graph ≠ W33 point graph

Pass4955 then resolves the duality from the 120 maximum-cut shell: the maximum cuts collapse 3:1 onto the forty W33 points, the Steiner triangles collapse 3:1 onto the forty W33 lines, and their non-splitting relation is literally the 40×40 W33 point-line incidence matrix.

The quadratic theorem survives

Pass4866 established Hom_PSp(H2,Q10)=0 and the reverse Hom also vanishes. Over F3, quadratic maps correspond to symmetric bilinear maps. Frobenius reciprocity on the five transitive diagonal/pair modules gives:

orbit	stabilizer order	dim Q10^stab
diagonal	216	0
pair 120	216	0
pair 1620	16	0
pair 2160	12	2
pair 3240	8	0

dim Hom_PSp(Sym² H2,Q10) = 2

Both dimensions occur exclusively on the 2160-pair Q(4,3) adjacency lift. Pass4941 subsequently combines any basis q1,q2 of this plane by the intrinsic adjoint bracket, F(x)=[q1(x),q2(x)], producing a projectively canonical outer-even quartic with full Q10 image.

Evidence boundary: the quadratic and quartic statements are finite characteristic-three module theorems. They do not choose a physical coupling or continuum normalization. The legacy Pass4870 producer contains historical logic; Passes4954–4956 are the authoritative graph-duality corrections.

Corrected frozen certificate: data/PART_W33_PASS4870_STEINER_W33_QUADRATIC_BRIDGE.json
Correction producers: analysis/w33_pass4954_steiner_quotient_is_q43_dual.py, analysis/w33_pass4955_maxcut_points_steiner_lines_incidence.py, analysis/w33_pass4956_point_line_24d_intertwiner.py

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