Exact characteristic-zero ovoid algebra, W33 code/Leech transport, marked axial presentation, and double-K4 triality closure.
M₂(Q) ⊕ Q ⊕ M₂(Q) ⊕ Q ⊕ M₃(Q)
Exact ternary edge code with dX=3 and dZ=4.
Axes 3, 33, and 36 have Leech norm four and lift to nine D₄ frames.
The smallest tripod–Norton relation factors into forty complete bipartite components.
The 19-orbital coherent algebra has rational centre dimension five. Exact primitive central idempotents and minimal-left-ideal ranks prove that every component splits over Q. The 200-point permutation module decomposes as 1² + 15a² + 15b + 24³ + 81.
The canonical W33/plane-ovoid dictionary transports all 240 edges, 160 triangles, 40 lines, and 36 spreads. Centered plane carriers span rank 24. Exactly three axial numerator vectors are Leech minimal vectors; the surviving C₂ fixes one and swaps the other two.
The 45 axes modulo the 27 five-axis line sums have dimension 24. Together with the two pair-product laws, these relations present the marked rational algebra. Its marked automorphism group has order 51,840. All 148,995 four-axis subsets split into twenty exact orbits.
The two 40-block tripod fibrations produce nonisomorphic row and column SRG(40,12,2,4) carriers. No side swap exists. Complete cross-orbit systems connect 160 tripods, 120 Norton lines, 135 D₄ frames, and 135 ordinary GQ flags.
The forty-K₄,₃ incidence action and the directed-edge action both have size 480, but their character norms are 23 and 24, with cross inner product 18. Equal cardinality does not identify the carriers.
Semantic certificate: ede91cc799f9093209c589fb0b73c88fca4159bedcc8e1e7cd6680c5f0c778fd. Monster words, Monster class fusion, the full unmarked linear automorphism group, remote CI/PDF success, and physical claims remain pending.