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Research Article · Preprint Mathematics

Sharp Quadratic Smoothing for Truncated Simplex Sums

Scott Schubr

Luminary Oddities

  • Mathematics
  • Classical and Real Analysis
Primary subject
Mathematics
Paper type
Research Article
Manuscript stage
Preprint
Manuscript
Archived
Edition
2 · Permanent link
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Overview

Abstract

We prove joint log-concavity of the raw lattice-simplex sum ∑_(i+j+k≤ r)(A-i)_+^2(B-j)_+^2(C-k)_+^2 for every integer radius r≥0 and arbitrary positive real endpoints. The proof combines exhaustive cell coverage with exact positive-coefficient certificates over unbounded parameter orthants. Independent real exponents can all be at least two, and this coordinatewise threshold is sharp. In every dimension we also prove the corresponding theorem with up to two unrestricted occupancies and arbitrarily many binary matroid companions, including activities in [0,1]. This uses degree-elevated homogenization, the classical complete-log-concavity criterion and compensated composition.

Keywords Classical and Real Analysis · Combinatorics and Graph Theory

Scope of the work

Sharp joint log-concavity for three unrestricted truncated lattice-simplex coordinates at powers at least two, and an all-dimensional extension with at most two unrestricted coordinates and binary matroid companions.

Reproduction notes

14-page manuscript and 10 source-package members. Compile main.tex. Install sympy==1.14.0 and python-flint==0.9.0; run verify_baseline.py, verify_deep.py, verify_lorentzian.py, verify_two_rows.py and verify_cells.py all for the separate exact identities and unbounded-orthant cell certificates.

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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.

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Cite this edition

Scott Schubr. (2026). Sharp Quadratic Smoothing for Truncated Simplex Sums (Version 2). Luminary Research Archive. https://research.luminaryoddities.com/sharp-quadratic-smoothing-for-truncated-simplex-sums/v/2

BibTeX
@techreport{luminary_sharp_quadratic_smoothing_for_truncated_simplex_sums_v2,
  title = {Sharp Quadratic Smoothing for Truncated Simplex Sums},
  author = {Scott Schubr},
  year = {2026},
  institution = {Luminary Research Archive},
  type = {Research Article},
  note = {Version 2},
  url = {https://research.luminaryoddities.com/sharp-quadratic-smoothing-for-truncated-simplex-sums/v/2}
}

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