Exact finite algebra, cubic-surface, code, module, and Lie-algebra results continuing the double-six / Levi-homology program.
Every noncommutative factor is split over its center. The Schur-index input is classical prior art; the block allocation is the exact continuation of the Pass4850 finite orbital algebra.
The 36 double-sixes have 1,200 overlap-six triangles. Exactly 120 have empty triple intersection; these are the independently reconstructed Steiner trihedral pairs. The 1,200×360 triangle-parity map has F₂-rank 325. Declaring exactly those 120 triangles odd determines one unique switching class modulo the 35-dimensional cut space.
The E₆ root-inner-product signing is an independent representative of that same class. The 1,080 even-Steiner triangles span the full dual [360,324,3]₂.
The quotient of the 64-dimensional Levi H₁ by the 54-dimensional oriented-K₃,₃ span is absolutely irreducible over F₃. The 36 norm-two points of the natural five-dimensional O₅(3) projective module reproduce the double-six graph. A common-generator calculation gives a unique rank-ten intertwiner
Via bivectors to skew endomorphisms, this is so₅(F₃) ≅ sp₄(F₃). The transported commutator bracket has center zero, full derived algebra, satisfies Jacobi, and is PGSp-invariant.
The complete non-cut switching coset has 2³⁵ words on 40 even weights from 120 through 198, beginning with 25,920 weight-120 Weyl-chamber signings. The ordinary cut-space Ising polynomial is still open. A certified hard witness and an exact LDL/spectral bound give
The exact covering radius is not promoted.
The intrinsic S₃⁴⁵ sheet action in C399 is genuine. A bijection from the three sheet cells at every recovered GQ point to its three incident quotient-line ports is the minimal local datum with trivial S₃ stabilizer. The global matching stabilizer is a diagonal copy of PGSp; adding the global chirality choice selects PSp.
Evidence boundary. These are exact finite algebra/code/incidence/module statements. No continuum gauge field, particle assignment, or automatic hardware realization is inferred. The ordinary cut-space enumerator and exact covering radius remain open.