Exact values of the De Klerk–Laurent hypercube constants: C₄ = 1/24, C₆ = 1/48, C_(4,6) = 1/224, and vertex-dual lower bounds

Abstract

For the hypercube [0,1]ⁿ with generators gᵢ = xᵢ - xᵢ², let C_(n,2d) be the smallest constant such that x₁… xₙ + C_(n,2d) lies in the degree-2d truncated quadratic module M_(n,2d)(g), and let Cₙ = C_(n,n). De Klerk and Laurent conjectured Cₙ = 1/(n(n+2)) for even n on the strength of a computer verification for n = 2, 4, 6 that they do not detail; Polak (arXiv:2605.31169) recently disproved the value at n = 8 and tabulated numerical values of C_(n,2d), marking 1/224 at (n,2d) = (4,6) and 1/360 at (6,8) as "suggested value from numerics". We prove C₄ = 1/24, C₆ = 1/48 and C_(4,6) = 1/224 exactly, and C_(6,8) ≥ 1/360, C_(6,10) ≥ 1/3968. The lower bounds are signed weightings of the 2ⁿ cube vertices: since every generator vanishes at every vertex, such a weighting is a dual certificate for the truncated quadratic module as soon as its 0/1 moment matrix of order d is positive semidefinite; the resulting relaxation is the Boolean-quotient relaxation that Polak discusses, and its exact rational optimum agrees with the numerical value of C_(n,2d) at every cell with 2d ≤ n+2 treated here and is strictly smaller at (6,10). The upper bounds at (4,6) and (6,6) are exact rational Gram certificates of orders 35 and 84, found on the optimal face by complementary slackness against the exact dual and checked as integer polynomial identities by a certificate checker whose soundness theorem yields membership in the truncated module with bounded-degree multipliers. Four constants are thus known exactly (C₂, C₄, C_(4,6), C₆); the bounds at (6,8) and (6,10) remain one-sided. Combined with a lemma of Polak, the exact values show that his Boolean-quotient constant Cᵇᵒᵒˡ_(n,n) equals Cₙ for n = 2, 4, 6. Every theorem is verified in Lean 4 with Mathlib.

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