Linear Algorithm and Density Asymptotics for Huang's Quadratic Form

Let G = N a, b be the gap set of the numerical semigroup generated by coprime a < b, and N = |G|. Yifeng Huang (2026) defined the quadratic form Q (n) = K (j-i) nᵢ nⱼ on RG, where K (d) = 1₃ ₀ - 1₃ ₀ - 1₃ ₁ + 1₃ ₀+₁, and showed that it recovers the dinv statistic on rational Dyck paths. The naive evaluation of Q requires O (N²) operations. This paper extends these findings in two main directions: Linear-Time Evaluation (O (N) Algorithm): We prove that the interval structure of K allows one to reduce Q to a linear combination of sliding-window sums, yielding an O (N) algorithm (Theorem 1. 1). Generalization to n Generators: We generalize the framework to semigroups p₁,. . . , pₙ with n generators. The generalized kernel K^ (n), defined by inclusion-exclusion with 2ⁿ terms, has at most w (n) ₙ active windows due to parity cancellation. These windows are computable incrementally in time O (n ₙ) (Theorem 1. 2). Empirically, w (n) ₙ while w (n) 2ⁿ: exponential combinatorics reduces to a sparse interval structure. For a sequence of pairwise distinct integers pᵢ 2 satisfying pₙ = o (n), we prove that the density of active windows becomes asymptotically negligible relative to the semigroup density (Theorem 1. 5): w (n) 2ⁿ - 1 = o (₈=₁^n (1 - 1pᵢ) ), n