Power Concavity of Low-Dimensional Unimodular Fiber Counts

Scope of the work

We prove a global sixth-root Jensen inequality for bounded unimodular fibers whose central affine dimension is at most three. Every integrable integer-valued demand with feasible fibers almost surely and integral mean satisfies the inequality when the central fiber is bounded and has affine dimension at most three, including countable support. A monotone concave extension of the sixth root of every type-A3 alcoved lattice count allows all twelve directed bounds to vary. Classical chamber formulas and Lorentzian volume theory combine with degree-six homogenization, wall gluing, shortest-path closure and a common integral rank-three reduction. Stronger planar powers, K4 extension sharpness and integral cube-root counterexamples are retained. Directed rounding and a triangle obstruction distinguish local methods from global Jensen. The general sixth-root exponent is not claimed optimal; unrestricted higher-dimensional unimodular Jensen remains open. Unrestricted higher central dimension, optimal general dimension-three exponent and optimal lattice-only K4 power remain open; integrability/integral mean/feasibility/central boundedness are hypotheses

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