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Power Concavity of Low-Dimensional Unimodular Fiber Counts

Scott Schubr

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Abstract

We prove a global power-Jensen inequality for unimodular integer fibers whose central affine dimension is at most three. For an integrable integer-valued demand A with feasible fibers almost surely and integral mean b, a bounded nonempty central fiber of dimension at most three satisfies E[N(A)^(1/6)]≤N(b)^(1/6), and hence logarithmic Jensen. The expectations are finite and countable support is permitted. The proof constructs a monotone concave extension of the sixth root of type-A₃ lattice counts with all twelve directed bounds varying. Classical chamber formulas and Lorentzian volume theory combine with a degree-six certificate, wall gluing, shortest-path closure and a common integral rank-three reduction. We also prove a sharp corrected square-root inequality in dimension two and determine its optimal leading large-count exponent deficit. For the complete four-vertex DAG, the classical continuous count extension has optimal concavity power 4/13; exact integral examples give a separate cube-root obstruction. A product-preserving directed-rounding theorem for opposite-edge deformations and a full-dimensional triangle obstruction describe the scope of local balancing methods. Exact finite certificates accompany the complete analytic and geometric proofs.

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Available editions

  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

Research details

Subjects
Mathematics · Combinatorics and Graph Theory
Keywords
unimodular fibers, lattice-point enumeration, power concavity, Lorentzian polynomials, integer flows
Manuscript date
2026-10-22
Reader reuse
No additional permission granted

Scope

Exact unimodular-fiber sixth-root and logarithmic Jensen for integrable integer demands with almost-sure feasible fibers, integral mean and bounded nonempty central fiber of affine dimension at most three. Arbitrary column/equation counts and countable support are allowed. Sharper planar corrected concavity and leading large-count deficit, continuous four-vertex-DAG power 4/13 with separate integral-demand bounds, opposite-edge directed rounding with fixed cross intervals, local triangle obstruction and block products are retained with their stated domains.

Disclosures

AI-assisted tools were used in developing and internally checking the arguments and preparing the manuscript.

Reproduction

The 20-page manuscript contains complete analytic/geometric proofs. Four unchanged standard-library Python verifiers accompany the source: exhaustive A₃ chamber/coefficient and rank-three normal-form checks, exact K₄ witnesses, and stated finite rounding diagnostics. README.txt distinguishes their scopes and documents normal/optimized execution. The portable build.py uses a fixed PDF timestamp; all public member hashes are listed in SHA256SUMS.txt.

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