Strict Log-Concavity Across All Scales for Exact-Height Plane Binary Trees

Let B₀(x) = x, B_h+1(x) = x(1 + B_h(x))² and N_h = B_h − B_h−1: N_k,h counts rooted plane binary trees with k vertices and exact height h. We prove that for every sufficiently large h the coefficient column is strictly log-concave on its whole interior, N_k,h² > N_k−1,hN_k+1,h for h+2 ≤ k ≤ 2^h+1−2. The proof passes through eight asymptotic regimes — left edge, subcritical saddle, critical h² law, quasi-pole corridor, h³ crossover, supercritical Fatou–Böttcher regime, bulk and right edge — sewn together on explicit windows.

Status. Working preprint. The proof was developed with substantial assistance from AI systems; it has been checked internally (exact computation, symbolic verification of fragile identities) but has not yet been read by a specialist in analytic combinatorics. The theorem is eventual: no explicit h₀ is given. Exact computation confirms the inequality for every h ≤ 16 (two independent implementations: h ≤ 12 and h ≤ 16).

The polynomials B_h enumerate the subsets closed under doubling of the complete binary trees in the 2-adic lifting of doubling modulo n (papers III–IV, doi:10.5281/zenodo.23088601).

Licences. Text: CC BY 4.0. Code: AGPL v3; commercial licence on request.