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Power Concavity of Low-Dimensional Unimodular Fiber Counts

Scott Schubr

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Abstract

We prove a corrected fourth-root 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 satisfies E[(N(A)−1/12)^(1/4)]≤(N(b)-1/12)^(1/4). This implies a 2/11-power inequality and logarithmic Jensen, with finite expectations and countable support. The proof constructs a monotone continuous extension of type-A₃ lattice counts with all twelve directed bounds varying. A quartic Lorentzian certificate gives the corrected fourth root; shortest-path closure and a common integral rank-three reduction transfer it to arbitrary unimodular systems. For this specified continuous count extension, 2/11 is the largest concavity power and 1/12 is the least fourth-root correction. Endpoint-count thresholds give stronger powers tending to 1/4. We also prove sharp corrected square-root Jensen in dimension two and its optimal leading large-count exponent deficit. For the complete acyclic four-vertex digraph, the classical continuous count extension has optimal power 4/13, while exact integral examples obstruct cube-root Jensen. Directed rounding and a full-dimensional local obstruction explain the role of a global continuous extension.

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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, integer flows, power concavity, lattice-point counts, Kostant partition function, Brunn–Minkowski inequality
Manuscript date
2026-10-22
Reader reuse
none

Scope

Corrected fourth-root Jensen for unimodular integer fibers of central affine dimension at most three, including countable integrable demands. Sharp correction and extension-specific powers, endpoint refinements, planar results and separate continuous-extension versus integral-demand obstructions retain their stated domains.

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

The 21-page manuscript and complete editable source are supplied in a 10-member public package. README.txt gives build instructions, dependencies and reproduction scope. Build reproduction is qualified by the compiler, support files and timestamp settings specified in the supplied instructions. The mathematical conclusions have analytic proofs; accompanying finite checks, where present, retain the limited scopes stated in the package.

References and related resources

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