Power Concavity of Low-Dimensional Unimodular Fiber Counts

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1. Edition 5 · 2026-11-02 · public
2. Edition 4 · 2026-10-27 · public
3. Edition 3 · 2026-10-24 · public
4. Edition 2 · 2026-10-24 · public
5. Edition 1 · 2026-10-22 · public

Subjects

Mathematics · Combinatorics and Graph Theory

Keywords

unimodular fibers, integer flows, power concavity, lattice-point counts, Kostant partition function, Brunn–Minkowski inequality

Manuscript date

2026-10-22

Reader reuse

No additional permission granted

Scope

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 results retain their integrability, integral-mean, feasibility and central-boundedness hypotheses.

Disclosures

AI-assisted tools supported research exploration, literature review, manuscript preparation and internal verification. The author retains responsibility for the work. Internal AI-assisted and computational checks are distinct from external human peer review.

Reproduction

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References and related resources

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