Which gamma laws are freely infinitely divisible, and strict enlargement of the Hankel hierarchy of free cumulants

Abstract

For a probability measure with all moments finite, free infinite divisibility forces every shifted free-cumulant Hankel matrix H_N=(κᵢ₊ⱼ₊₂)_(i,j=0)^(N) to be positive semidefinite. A single vector on which one of those matrices is negative therefore certifies that the measure is not freely infinitely divisible, and the point of this paper is that one such vector can be held fixed while a parameter of the law moves. For the classical gamma law boldsymbol(γ)ₚ the moments mₙ=(p)ₙ are integer polynomials in the shape p, hence so are the free cumulants, so the quadratic form Q(p)=u^(T)H_N(boldsymbol(γ)ₚ)u of a fixed integer vector u is one polynomial of degree at most N+1, whose sign on a closed rational interval is decided by a Bernstein certificate. One vector at order 20 gives H₂₀(boldsymbol(γ)ₚ)notsucceq0 for every real p ∈ [2/3,4/3]; since Hasebe's set I covers (1/2,2/3) ∪ (4/3,3/2), this settles part (3) of his 2014 conjecture — boldsymbol(γ)ₚ is not freely infinitely divisible for any p ∈ (1/2,3/2) — and, with the positive half of his Corollary 1.3(1), completes the classification: boldsymbol(γ)ₚ is freely infinitely divisible exactly for p ∈ (0,1/2] ∪ [3/2,∞). A second vector, at order 13 on [(49)/(50),6/5], gives the same conclusion with no appeal to compiled evaluation, and already covers infinitely many of the intervals I omits, where the exponential law was the only shape previously known. The rest of the paper studies the hierarchy {H_Nsucceq0}_(N ≥ 0) itself, which is where the witness vectors come from. Yu exhibited, for the beta laws boldsymbol(β)_(p,q), exact rational parameters at which the (N+1) × (N+1) test is the first to fire, for 2 ≤ N ≤ 11, and asked whether the hierarchy keeps enlarging at higher orders. It does, at infinitely many orders, by a completeness statement — for a measure whose moment sequence is determinate, positive semidefiniteness of every H_N is not merely necessary for free infinite divisibility but sufficient — together with the fact that no test fires at p=1/2. That argument assembles published theorems and is the one part of this paper that is not machine-checked; it is labelled as such throughout, as is the observation that the second half of Yu's open sentence — that every non-freely-infinitely-divisible beta law is eventually separated by a finite Hankel matrix — is the contrapositive of the criterion Yu's paper itself cites. The computational content: the witness ladder continued from N=11 to N=20 with exact rational parameters and bracketed thresholds; the gamma ceiling ν(p), the first order at which the hierarchy of boldsymbol(γ)ₚ fails, computed at 36 rational shapes and shown to take every value from 13 to 25; a Schur reduction of the whole hierarchy to the sign of one scalar Φ_N, non-increasing in N wherever the trailing Hankel matrices are positive definite, so that the exclusion never un-fires there; and the first negative Hankel determinant of the free cumulants of the exponential law, of size 14 × 14, sharper than the 16 × 16 one on record and certified together with it. Everything computational is verified in Lean 4 by exact integer arithmetic — 159 theorems, no floating point anywhere — and all of it but the order-20 certificate is evaluated by the proof kernel itself.

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