W33-Theory · machine-verifiable finite continuation · Passes 4865–4872

From Steiner parity to an intrinsic ternary Lie algebra — without crossing the evidence firewall

This packet closes the ordinary cut-space Ising polynomial, exposes a canonical radical and composition filtration inside ternary Levi homology, makes the 10-dimensional sp₄(F₃) bracket intrinsic to incidence, and proves a characteristic obstruction that prevents a tempting linear Steiner-to-adjoint identification.

64 → 29 → 25 ⊥ 10

The canonical edge pairing on Levi H₁(F₃) has rank 35 and radical 29, entirely inside the 54D oriented-K₃,₃ module.

14 < {19,24} < 29

The radical has an exact invariant diamond; the 54D module has irreducible factors 14, 5, 10, 25 and its 25D quotient is irreducible.

f=(36,360,1200,1080,216)

The double-six clique complex has 216 maximal K₅'s and 120 maximal K₃'s. Those maximal triangles are exactly the Steiner triples.

β=(1,0,120,109,0)

The same Betti vector occurs over F₂ and F₃, but the 1080 non-Steiner triangle boundary span jumps from rank 324 to 325.

156 × 2²⁰

A marked K₆ chart reduces the 2³⁵ cut search to 156 unlabeled six-vertex graphs times all triad states, with exact S₆ orbit weights.

K=[360,36,20]₂

The complete 2³⁶-word weight enumerator is now closed. MacWilliams independently returns A₃⊥=1080.

4865 · Ternary Levi radical filtration

Let H₁ be the 64-dimensional ternary cycle space of the GQ(4,2) Levi graph, and K₅₄ the 54-dimensional submodule generated by oriented local K₃,₃ witnesses. The ordinary edge dot product does not stay nondegenerate in characteristic three:

rank Gram(H₁)=35, dim rad(H₁)=29, rad(H₁)⊂K₅₄.

The radical has the PSp/PGSp-invariant diamond 14 < {19,24} < 29. Exact splitting systems prove both 0→29→54→25→0 and 0→54→64→10→0 are nonsplit. Quotienting the radical gives a nondegenerate 35-space with canonical orthogonal decomposition 25 ⊥ 10.

4866 · Steiner triples are exactly the maximal triangles

In the double-six graph SRG(36,20,10,12), the 120 Steiner triangles can be recognized without E₆ roots or cubic-surface coordinates: they are exactly the maximal K₃'s, equivalently exactly the triangles with zero common neighbors. Every non-Steiner triangle has four common neighbors and belongs to two K₅'s.

The characteristic split is the key negative result:

rank ∂(1080 non-Steiner triangles): 324 over F₂, 325 over F₃.

The one-dimensional parity defect is therefore genuinely binary. Over F₃, H₂ becomes the 120-dimensional permutation module on the Steiner triangles, and exact common-generator calculations give zero Hom spaces in both directions between this H₂ and the adjoint Q₁₀.

Obstruction: there is no linear PSp-equivariant Steiner-H₂ ↔ adjoint-Q₁₀ bridge. A real bridge must be nonlinear, characteristic-changing, or carry additional structure.

4867 · Complete cut and code enumerators

Marking one double-six supplies six opposite columns. The other 35 double-sixes become 15 duads plus 20 triads. The duad state is just a labeled graph on six vertices; quotienting by S₆ leaves 156 unlabeled graph representatives.

occupied cut weights

minimum nonzero cut weight

216

maximum cut weight

120

maximum cuts

Adding the already-certified nontrivial switching coset closes the entire binary code enumerator. The first dual MacWilliams coefficients are 1,0,0,1080,10530,127656,2329680,37193040.

Still open: the covering radius was not solved by the enumerator. The exact prior bound remains 124 ≤ ρ(K) ≤ 179.

4869 · A marked double-six is a literal six-bit symplectic chart

The 15 nonneighbors are the duads C(6,2); the 20 neighbors are the triads C(6,3). Writing these as weight-2 and weight-3 vectors in F₂⁶, the entire 35-vertex residue adjacency is

x ~ y ⇔ B(x,y)=x·y + wt(x)wt(y)=1 (mod 2), matrix(B)=I₆+J₆.

The form is alternating and nondegenerate, and the full residue automorphism group is exactly S₆×C₂ of order 1440. This upgrades the repo's earlier K₆-duad carrier from a convenient model to an incidence-derived chart once a double-six is marked.

4871 · The sp₄(F₃) bracket is intrinsic to incidence

Starting from only the exact 10×10 action matrices reconstructed from Levi incidence, the equivariant alternating-product space is one-dimensional for both PSp and PGSp:

dim Hom(Λ²Q₁₀,Q₁₀)=1.

The unique nonzero map has rank 10, satisfies Jacobi on all basis triples, has zero center, and is perfect. The O₅(3)/exterior-square realization remains valuable—but it is now a cross-certificate, not the definition of the bracket.

4872 · Native port compiler

The minimal local six-state sheet-to-port matching is exactly AGL(1,3):

i ↦ (−1)ᵇ i + r mod 3, r∈F₃, b∈F₂.

So each of the 45 selectors is one trit plus one bit. The complete table has information 45 log₂6 ≈ 116.3233 bits: an optimally packed fixed binary table needs 117 bits, or 118 including the global chirality bit. The symbolic irreversible-reset lower bound is 45 k_BT ln6, but reversible routing itself does not automatically pay an erasure cost.

Executable evidence lives under analysis/w33_pass4865_4866_..., w33_pass4867_..., w33_pass4869_..., w33_pass4871_..., and w33_pass4872_..., with frozen data/PART_W33_PASS... certificates. The shared manuscript frontier imports the same theorem packet into W33 Paper, Photonic Holonet, and the machine blueprint.

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