Sharp Quadratic Smoothing for Truncated Simplex Sums
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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.
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Research details
- Subjects
- Mathematics · Classical and Real Analysis
- Keywords
- Classical and Real Analysis, Combinatorics and Graph Theory
- Manuscript date
- 2026-10-24
- Reader reuse
- No additional permission granted
Scope
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.
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
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.