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Factored adjacency-matrix identity

Proved
Conway99Formal.SrgCore.adjacency_polynomial

by harry · Oct 4, 2026 · Mathlib 0df444a (Lean v4.33.1)

adjacency-matrixcompiled-reuse-review-requiredconway99-formal-project-20261003matrix-polynomialpossible-catalog-reuse-adapterstrongly-regular-graphs

If G has strongly regular parameters (99,14,1,2), then over any commutative ring its adjacency matrix A satisfies

(A−3I)(A+4I)=2J(A-3I)(A+4I)=2J

Role: This is the factored form of the Conway adjacency identity.

Preamble
import Mathlib

namespace Conway99Formal.SrgCore
end Conway99Formal.SrgCore

set_option autoImplicit false

/-! Graph-owned parameter and adjacency identities for a hypothetical SRG(99,14,1,2).
Sources: `Conway99/Conway99/Core.lean` §§1–3, 8.1;
`Conway99/Conway99/Claims/C01srgcorealgebra.lean` §§0, 3, 6;
`Conway99/results/R005_star_complement_square_discriminant.md`.
-/

open Conway99Formal.SrgCore

open SimpleGraph Matrix Finset

variable {V : Type*} [Fintype V] [DecidableEq V]
variable (G : SimpleGraph V) [DecidableRel G.Adj]
Formal statement
theorem Conway99Formal.SrgCore.adjacency_polynomial (h : G.IsSRGWith 99 14 1 2)
    (α : Type*) [CommRing α] [DecidableEq α] :
    (G.adjMatrix α - 3 • (1 : Matrix V V α)) *
      (G.adjMatrix α + 4 • (1 : Matrix V V α)) =
        2 • (of 1 : Matrix V V α) := by sorry
Source
Exact original Lean source: formalization/2026-10-03/srg-core/Core.lean#L351-L363; source commit a45708acebe3f397faccb1b646be906f24f23ee5; source SHA-256 64ce9b86d07bbdd11a61266b80c3043c34c08d7f939471fff2c44dc34ff37904. Mechanically extracted declaration: blob/a45708acebe3f397faccb1b646be906f24f23ee5/formalization/2026-10-03/srg-core/Core.lean#L351-L363.

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