GQ(2,4)
27 points, 45 three-point lines, point graph SRG(27,10,1,5).
GQ(4,2) closes the carrier tower.
Passes 3769–3786 reconstruct the dual generalized quadrangles, the full Veldkamp PG(5,2), all 200 ovoids, W33 on 40 plane ovoids, a new rank-24 axial algebra, and a controlled lattice symmetry-breaking certificate.
Exact geometry
GQ(4,2)
45 points, 27 five-point lines, point graph SRG(45,12,3,3).
T = 16
The point Terwilliger algebra has dimension 16, center dimension 5, and type M3 ⊕ M2 ⊕ C³.
Veldkamp closure
651 lines
PG(5,2) splits into 45, 216, 270, and 120 lines by singularity type.
120 = Norton
The all-nonsingular Veldkamp lines are exactly the prior Norton/Fischer triples.
27 · 36 · 40 · 45 · 120 · 135
All major carriers now live in one executable incidence tower.
Plane ovoid reconstruction of W33
200 = 40 + 160
The ovoids split into 40 plane ovoids and 160 tripods under the order-51,840 action.
SRG(40,12,2,4)
Join two plane ovoids when they share three quadrangle lines. The graph is explicitly isomorphic to W33.
72¹ 12²⁴ 0¹⁵
The plane-ovoid incidence Gram matrix is 8I + 2AW33 + J.
New rank-24 algebra
45 axes
The normalized columns of P=(A−12I)(A−3I) are idempotents.
1, 1/7, −1/3
Axis multiplicities are 1, 14, and 9.
M24
Multiplication operators span the full 24×24 matrix algebra; derivations vanish.
not 2A
The fusion law is not nontrivially Z₂-graded and is not promoted as a Monster Majorana algebra.
Lattice boundary
rank ≤ 16 mod 2
Every W(D4)-invariant alternating form is singular, ruling out a W(D4)-invariant even-unimodular child.
E8³ · C2
An exact determinant-one basis change gives an even-unimodular positive child whose surviving U4(2) stabilizer is C2.
Leech pending
The certified child has roots. No rootless canonical Leech basis is claimed.
Evidence boundary
Semantic certificate 8d3f383e362f30e58d8c482f48e2ac2b77414922366ba8053633859d6206313a. Explicit target-group words, an executed class fusion, a rootless Leech polarization, remote PDF evidence, and physical implementation remain fail-closed.