Six-Cycle Insertion, an Exact Log-Concavity Threshold, and Hook-Unimodality in Higher Lie Characters

Let M_λ(x) denote the hook-multiplicity polynomial of the higher Lie character attached to a conjugacy class of cycle type λ. Adin–Hegedüs–Roichman conjectured that the coefficient sequence of M_λ(x) is unimodal for every partition λ, and separately conjectured log-concavity for rectangular types (r^s) with even r ≠ 6. Using their product formula, we determine the six-cycle family explicitly for every s ≥ 1. We prove that M_(6^s)(x) is unimodal for all s, that (1+x)M_(6^s)(x) is strictly log-concave on its nonzero support for all s, and that the unsmoothed polynomial M_(6^s)(x) itself is strictly log-concave on its nonzero support if and only if s ≥ 7. The threshold is controlled by the explicit quartic determinant Δ_(s,2) = (s^4 − 8s^3 + 13s^2 − 6s − 12)/4. We then use discrete strong unimodality to prove a six-cycle insertion theorem. Consequently, every nonempty partition whose parts lie in {1,2,3,4,5,6} has a unimodal hook-multiplicity sequence, with no bound on multiplicities or total size. The accompanying reproducibility package provides exact-arithmetic reconstruction from the Adin–Hegedüs–Roichman product identity, symbolic verification of the polynomial certificates, independent coefficient extraction, and direct permutation/descent-set cross-checks. This work establishes an infinite subfamily of Adin–Hegedüs–Roichman Conjecture 8.1; it does not claim the conjecture for arbitrary partitions. Historical priority for the exact s = 7 threshold remains subject to specialist literature review.