Index-two border identities for Hadamard matrices
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A Hadamard matrix has ±1 entries and mutually orthogonal rows. For a Goethals–Seidel core over a finite abelian group of even order n≥4, which four-row-and-column coset borders yield a Hadamard matrix when the subgroup has index two? This is the s=1 index-two border system in Jones’s note.
Why it matters and what counts as progress
Why it matters: The bounded formalization tests an alternative proof of an existing border theorem; it does not claim discovery of the underlying Hadamard matrices.
What counts as progress: Give an alternative integer derivation and complete formal proofs of the stated index-two identities; this verifies an existing theorem rather than establishing new matrix orders.
How the problem developed
J. D. Jones and Claude agents A public formalization disclosed an AI-derived integer proof route for the s=1, index-two clauses of Theorem D in Jones’s earlier Hadamard-border note. The note’s theorem and earlier proof predate this alternative derivation.
AI contribution
Claude formalizes an alternative Hadamard border argument
Claude agents devised an integer proof route for the s=1, index-two clauses of Theorem D in Jones’s Hadamard border note, replacing its rank and positive-semidefiniteness route. The formal development checks those Theorem D clauses.
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