Exact finite geometry · Passes 3751–3758

The six-bit parent behind the Fischer lattice and 36-port transform

Frozen semantic certificate: 6271dafcc58467d6e758cdbcc9a1b220fe21693b3ace3c727fb5b5499be60ce6.

[36,6,16]binary minus-type code

(64,36,20)quadratic difference set

1¹²2²⁴lattice Gram Smith form

55modular Terwilliger dimension

Rooted Construction-A lattice

The doubly-even, self-orthogonal code with weights 0¹ 16²⁷ 20³⁶ produces an even 36-dimensional lattice.

L(C)=2^−1/2{z∈Z³⁶:z mod 2∈C}, det G=2²⁴, L*/L≅(Z/2)²⁴.

The minimum norm is two and the complete root system is A1^36, containing 72 roots. This is an exact obstruction to identifying the lattice with a rootless unimodular lattice.

Quadratic Walsh parent

The 36 nonsingular vectors of the six-dimensional quadratic space form a (64,36,20) difference set. Every nonzero translation intersects it in exactly 20 points.

W₆₄W₆₄^T=64I, K=W_N−2I, H=K/6.

The W33 multiport is therefore a shifted principal minor of the full six-bit bilinear Walsh transform.

Exact reflection network

The 21-dimensional negative eigenspace of H has an explicit pairwise-orthogonal primitive integer basis. The corresponding rational Householder reflections commute and multiply exactly to H.

H=R₁R₂⋯R₂₁.

Integral triple torsion

SNF(D)=1³⁰2⁵6, coker(D)≅(Z/2)⁵×Z/6, |coker(D)|=192.

The two-primary component recovers the six-dimensional orthogonal module. The three-primary component is one invariant conservation quotient. The value 192 is not treated as a tomotope isomorphism.

Cover rigidity and holonomy frame

The 120-vertex cover has degree 27, diameter three, full finite symmetry U4(2):2, and stabilizer subdegrees 1,2,27,36,54. Two different distance-two intersection profiles prove that it is not distance-regular.

Q₂₀²=108Q₂₀, rank(E₂₀)=20.

The projected 120-vector frame has normalized angles −1/2, −1/6, 0, 1/3; each three-point fiber becomes an equilateral triangle summing to zero.

Monster boundary

The exact abstract standard-pair and cycle fingerprints are search filters, not serialized Monster words. A self-reporting GAP workflow publishes an actual CTblLib result or a structured failure artifact. No Monster multiplicity is claimed before that file is observed.