Kite matrices over 𝔽₂: the exact reach of a doubly perfect construction

Abstract

The kite matrix Aₙ is the n × n zero–one matrix whose (i,j) entry is 1 exactly when i+j ≤ n+1. Gross and Goedicke, constructing doubly perfect functions on the discrete phase space ℤ₂²ⁿ — and through them two-unitary matrices and absolutely maximally entangled states — record as a remark, supported by computer experiments, that the block-diagonal quadratic form built from two copies of Aₙ appears to give such a function for every n ≢ 1 (mod 3); Goedicke's dissertation restates a sibling of that claim, for a single kite of order 2n, as the only conjecture it puts forward, supported by a search over n ∈ [500]. We prove both, in both directions and for every n. The mechanism is that Aₙ is the reversed partial-sum operator, so the kernel equation of Aₙ+I differences to a two-term recursion whose transfer matrix is the signed Fibonacci matrix M=smat(1, -1)10, and M³=-I over every commutative ring. The orbit therefore has period six, and matching it against the closing condition leaves exactly the residues n ≡ 1,4 (mod 6); over 𝔽₂ both are carried by the single factor Aₙ+I, and their union is n ≡ 1 (mod 3). We also settle the same matrices over ℤ, where the two residues split between the two factors of Aₙ²-I: det(Aₙ+I)=0 exactly when n ≡ 4 (mod 6) and det(Aₙ-I)=0 exactly when n ≡ 1 (mod 6), so that det(Aₙ²-I)=0 exactly when n ≡ 1 (mod 3) over ℤ as well. The modulus three is thus not a feature of characteristic two; what characteristic two does is merge the two factors. Every theorem below is machine-checked in Lean 4.

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