Passes 3694–3700 reconstruct the result from finite geometry and freeze every promoted claim in an executable certificate.
Centered line–spread incidence. Coherence 1/5. Welch bound saturated.
Minimal orthogonal guard rank for an exact 36-mode Naimark/unitary completion.
Primitive idempotents with spectrum 1¹, (−1/2)⁵, (1/6)⁹.
Three triples per W33 line; ten triples through every spread.
C = B − J/4, CᵀC = 18E₁₅, E₁₅ = I/2 + A/6 − J/12.
The normalized Gram entries are +1/5 when two spreads share four lines and −1/5 when they share one line. The Seidel matrix obeys S²−2S−35I=0, with spectrum 7¹⁵ and (−5)²¹.
x⋆y = E₁₅(x∘y), aᵢ = 6E₁₅eᵢ, aᵢ⋆aᵢ=aᵢ.
Adjacent spreads satisfy aᵢ⋆aⱼ=(aᵢ+aⱼ)/6. Every nonadjacent pair selects a unique third axis:
aᵢ⋆aⱼ=−(aᵢ+aⱼ)/6+aₖ/3.
Those selections are exactly the 120 triples. Each triple shares one W33 line, and the nine spreads through any line split into three disjoint triples.
The 36-ray magic orthogonality graph is 11-regular with spectrum 11¹, 2²⁰, (−1)³, (−4)¹². The spread graph is 15-regular with spectrum 15¹, 3¹⁵, (−3)²⁰. They are not isomorphic.