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Power Concavity of Low-Dimensional Unimodular Fiber Counts

Scott Schubr

  • Mathematics
  • Combinatorics and Graph Theory
Primary subject
Mathematics
Paper type
Research Article
Manuscript stage
Preprint
Manuscript
Archived
Edition
4 · Permanent link
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Overview

Abstract

We prove a global power-Jensen inequality for bounded unimodular fibers whose central affine dimension is at most three. For every integrable integer-valued demand A with feasible fibers almost surely and integral mean b, if P(b) is bounded and has affine dimension at most three, then E N(A)^(1/6)≤ N(b)^(1/6). In particular logarithmic Jensen holds, including countably supported distributions. The proof constructs a monotone concave extension of the sixth root of every type-A_3 alcoved lattice count, with all twelve directed bounds varying. It combines the classical polytrope chamber formulas and Lorentzian volume theory with a degree-six homogenization, wall gluing and shortest-path closure. A common integral rank-three reduction transfers the result to general unimodular fibers. In dimension two we retain the stronger sharp corrected square-root inequality, the optimal leading large-count exponent deficit, and a fixed network obstruction. For the complete four-vertex DAG, 4/13 is the optimal power of its classical continuous extension, while cube-root concavity fails even on integral demands. A product-preserving directed rounding theorem for opposite edges and a full-dimensional triangle obstruction distinguish local rounding methods from global Jensen.

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

Scope of the work

Integrable integer demands in a fixed unimodular system satisfy sixth-root and logarithmic Jensen when the integral-mean central fiber is bounded, nonempty and of affine dimension at most three, with the sharper planar and specified K4/rounding refinements.

Reproduction notes

The 20-page manuscript and editable sources are included in this 9-member package. README.txt gives LaTeX build commands and four optional Python 3.9+ standard-library checks. verify_A3_structure.py reconstructs finite chamber/algebraic identities, verify_rank_three.py exhausts 1024 direction subsets, and verify_counterexample.py/verify_rounding.py check exact lattice witnesses and finite rounding controls. The manuscript supplies analytic concavity, wall gluing, common-lattice transfer and countable-support arguments.

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Author 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.

References and related publications

  1. J. Leake and M. Mohammadi Yekta, Log-concavity and Approximate Counting for Totally Unimodular Polytopes, arXiv:2609.39917v1 (2026). arxiv.org/abs/2609.39917.
  2. A. Khare, J. P. Matherne and A. St. Dizier, Log-concavity of characters of parabolic Verma modules, and of restricted Kostant partition functions , arXiv:2504.01623v2 (2025). arxiv.org/abs/2504.01623.
  3. J. A. De Loera and B. Sturmfels, Algebraic unimodular counting , Math. Program. 96 (2003), 183–203. arXiv:math/0104286.
  4. T. Lam and A. Postnikov, Alcoved polytopes I , Discrete Comput. Geom. 38 (2007), 453–478. arXiv:math/0501246.
  5. T. Lam and A. Postnikov, Alcoved polytopes II , arXiv:1202.4015 (2012). arxiv.org/abs/1202.4015.
  6. D. Alonso-Gutiérrez, M. A. Hernández Cifre and J. Yepes Nicolás, Further inequalities for the (generalized) Wills functional , Commun. Contemp. Math. 23 (2021), 2050011. arxiv.org/abs/1912.07993.
  7. D. Iglesias, J. Yepes Nicolás and A. Zvavitch, Brunn–Minkowski type inequalities for the lattice point enumerator. Author manuscript (2019) .
  8. A. Barvinok, Brunn–Minkowski inequalities for contingency tables and integer flows , Adv. Math. 211 (2007), 105–122. doi:10.1016/j.aim.2006.07.012.
  9. J. Leake and A. H. Morales, Capacity bounds on integral flows and the Kostant partition function, arXiv:2406.07838v3 (2024). arxiv.org/abs/2406.07838.
  10. J. Tarski, The partition function for certain simple Lie algebras , Technical Report No. 7, AFOSR TN58-657 (AD162188), Department of Mathematics, University of California, Berkeley (August 1958). HathiTrust scan.

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Cite this edition

Scott Schubr. (2026). Power Concavity of Low-Dimensional Unimodular Fiber Counts (Version 4). Luminary Research Archive. https://research.luminaryoddities.com/power-concavity-of-low-dimensional-unimodular-fiber-counts/v/4

BibTeX
@techreport{luminary_power_concavity_of_low_dimensional_unimodular_fiber_counts_v4,
  title = {Power Concavity of Low-Dimensional Unimodular Fiber Counts},
  author = {Scott Schubr},
  year = {2026},
  institution = {Luminary Research Archive},
  type = {Research Article},
  note = {Version 4},
  url = {https://research.luminaryoddities.com/power-concavity-of-low-dimensional-unimodular-fiber-counts/v/4}
}

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