This page continues the exact support-12 apartment-code frontier. It contains one correction to an old action-domain error and a chain of finite geometry, group, and coding theorems.
Exact.
The 1,620 corner-star thickenings from Pass 4703 form a graph under disjoint 12-line support. Every vertex has degree two and every component is K3, so the shell is exactly 540 disjoint triangles. Each triangle covers 36 of the 40 W33 lines and leaves a four-line pairwise-disjoint residue. There are 270 residues, two triangles per residue.
Correction to historical Pass 1830.
PSp(4,3) has 315 inner involutions in the W33 line action, split by fixed lines as 270 fixing four lines and 45 fixing sixteen. The 270 four-fixed-line sets are all distinct and exactly equal to the 270 support-12 residues.
The old Pass 1830 GAP script had defined its permutation group on W33 point positions and then applied that group to a set of line indices. Its reported orbit size 2,880 was therefore not the induced line action. With the correct line action the residue orbit has stabilizer 96 and size 25920/96=270.
Exact PSp(4,3)-G-set theorem.
For a residue R and its unique inner involution g, PGSp(4,3) contains eight outer square roots h with h²=g. Exactly two fix four W33 lines; they are the inverse pair h and h−1. The two support triangles above R are swapped by either such root.
The decisive certificate is subgroup equality: the PSp stabilizer of either support triangle is exactly the PSp centralizer of either four-fixing root, of order 48. Hence the 540 support triangles and the 540 four-fixing outer order-4 elements are isomorphic PSp(4,3)-sets. Both are two-fold covers of the same 270 residue/involution orbit.
The full PGSp triangle stabilizer and root centralizer both have order 96 but are distinct extensions with common inner intersection 48. Therefore no untwisted full-PGSp identification is claimed.
Exact matrix identity.
Let B be the 40×270 W33-line versus residue incidence matrix and A* the line-intersection graph. Every block has size four, every line lies in 27 blocks, every skew line-pair lies in exactly three blocks, and intersecting pairs lie in none. Therefore
BBT = 27I + 3(J − I − A*).
The real Gram spectrum is 1081 ⊕ 1824 ⊕ 3615, so B has real rank 40. Over F2, the residue-mask span has rank 30.
Exact binary-code identity.
Every residue mask r satisfies A*r=0 over F2. Since rank(A*)=10 and the 270 residues span dimension 30, their span is exactly
span(residues) = ker(A*) = im(A*)⊥ = [40,30,4].
An exhaustive search through all coordinate subsets of weights 1–4 finds no kernel words of weights 1, 2 or 3 and exactly 270 weight-4 words. Those 270 words are precisely the involution residues, so the group orbit is the complete minimum shell rather than a selected subset.
Exact dual reconstruction.
The established protected code is H10 = im(A*) = [40,10,12]. Symmetry of A* gives H10⊥=ker(A*). Therefore H10 is exactly the common orthogonal kernel of the 270 minimum residue checks.
The H10 weight enumerator is 1 + 40z12 + 135z16 + 672z20 + 135z24 + 40z28 + z40. Binary MacWilliams then yields the complete dual enumerator, beginning 1 + 270z4 + 6720z6 + 152685z8 + 1651392z10 + … + 270z36 + z40, totaling 230 words.
Haemers, Peeters and van Rijckevorsel, Binary Codes of Strongly Regular Graphs, Designs, Codes and Cryptography 17 (1999), 187–209, already computed the binary adjacency code of the relevant GQ(3,3) graph: dimension 10 with exactly the H10 weight enumerator above, and with the forty adjacency rows comprising all weight-12 words. H10 itself is therefore not claimed as newly discovered here.
The new content is the W33-specific identification of H10⊥ with the span of the 270 inner-involution fixed-line residues, the proof that those 270 words are the complete dual minimum shell, and the support-12 / outer-order-4 double cover feeding that shell.
The executable verifiers are analysis/w33_pass4721_4724_support12_involution_square_root_cover.py and analysis/w33_pass4725_4726_involution_residue_dual_code.py, with frozen JSON and focused regression tests. Fixed sets, square maps, stabilizers, centralizers, low-weight exhaustion, orthogonality, and MacWilliams are checked explicitly.
This is finite incidence geometry, permutation-group theory, and binary coding. No particle, field, lattice, or dynamical interpretation follows from these theorems.