Exact finite graph, code, Weyl-group, orbital-algebra, and homology results continuing the 1080-cycle / 360-K₃,₃ incidence program.
The binary incidence kernel is K=[360,36,20]₂. Its 36 minimum words are exactly the classical 36 cubic-surface double-sixes, each represented on the 27-line carrier by a K₆,₆ minus perfect matching.
The same 36 objects are explicitly the 36 projective E₆ root pairs. The full order-51840 action on the double-sixes is conjugate to the Weyl W(E₆) action on projective roots. The extra generator σE₆ is the negative-inner-product edge signing.
The nontrivial coset over the 35-dimensional cut code has exact minimum weight 120. Its complete minimum shell has
Weyl-chamber signings. The characteristic-two extension by the one-dimensional quotient is nonsplit for both PSp and PGSp.
The low shell below weight 64 is A₂₀=36, A₃₈=360, A₄₀=270, A₅₄=1200, A₅₆=3240, A₅₈=2160, A₆₀=540. Its dual is
and its 1080 weight-three dual words are precisely the E₆-sign-even triangles, equivalently the binary Levi cycles. The full coordinate automorphism group of this 360-bit code has order 51840 and is the certified W(E₆) / PGSp action.
The PSp action on the 1080 binary minima has orbital algebra dimension 59 and center dimension 15, with complex type C⁷ × M₂(C)⁴ × M₃(C)⁴ and rational center Q⁹ × Q(√−3)³. The full PGSp action has dimension 49 and center dimension 13, with complex type C⁶ × M₂(C)⁴ × M₃(C)³ and fully split rational center Q¹³.
The K₃,₃ incidence Gram is only 3A₀+A₁+A₅ in the PGSp orbital basis and is not central.
The residual S₃⁴⁵ sheet action cannot be removed by any higher intrinsic code shell because it is already a true automorphism subgroup of the full classical code. The coordinate automorphism structure is
Breaking sheet symmetry therefore requires external placement, timing, labels, dynamics, or another structure not contained in the binary code.
The 1080 projective Levi eight-cycles span all 64 dimensions of H₁(Levi;F₃). The 360 projective K₃,₃ weight-six witnesses span only 54 dimensions. No global choice of projective ± orientations makes the unweighted 1080×360 incidence factor linearly to their homology map.
The correct PGSp-equivariant lift uses the 720-object orientation double cover with deck reversal acting by −1. Ten ternary homology dimensions remain outside the canonical K₃,₃ family.
Evidence boundary. These are exact finite root, graph, code, group, and homology statements. No physical E₆ field, particle assignment, experimental threshold, or hardware clock claim is inferred.