Sharp Quadratic Smoothing for Truncated Simplex Sums
Loading original first page…
Abstract
For every nonnegative integer radius, we prove joint log-concavity of the raw three-dimensional truncated simplex lattice sum at arbitrary real endpoints when each independent real power is at least two. The coordinatewise threshold two is sharp. Exhaustive cell coverage and exact positive-coefficient polynomial identities certify the quadratic endpoint over unbounded parameter orthants, rather than numerical samples or finite parameter cutoffs. In every dimension, the same endpoint holds with up to two unrestricted occupancies and arbitrarily many binary matroid companions with activities in [0,1]. Degree-elevated homogenization, classical complete-log-concavity and compensated composition supply that extension.
Read the full paper in the original PDF.
Edition files
- Paper.pdf149.0 KiB · pdf
SHA-256
e86d8aaafd9fe963b42b5e5d3e7cd0cac0eaba23aac2322a5a58739f5cfc5e6b - Publication_Package.zip173.3 KiB · package
SHA-256
388aaad3ef113b2aeab4dce15f2e4696637151be994eb50587df34c397a218d6
Available editions
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
For every nonnegative integer radius, we prove joint log-concavity of the raw three-dimensional truncated simplex lattice sum at arbitrary real endpoints when each independent real power is at least two. The coordinatewise threshold two is sharp. Exhaustive cell coverage and exact positive-coefficient polynomial identities certify the quadratic endpoint over unbounded parameter orthants, rather than numerical samples or finite parameter cutoffs. In every dimension, the same endpoint holds with up to two unrestricted occupancies and arbitrarily many binary matroid companions with activities in [0,1]. Degree-elevated homogenization, classical complete-log-concavity and compensated composition supply that extension. Integer-endpoint midpoint coverage in arbitrary dimension is prior Leake-Mohammadi Yekta via an explicit TU model; Speyer difference-bound overlap is stated only where actually established. Classical ALOV/BH/Prekopa and prior2D/fractional baseline are credited.
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 approved 10-member package contains the complete 14-page paper, editable LaTeX sources and dependencies, reader/build instructions, scholarly references and a public checksum manifest, with necessary exact verification programs and certificates. Follow the included README and retain the package together.