Research Article · Preprint Mathematics
Sharp Quadratic Smoothing for Truncated Simplex Sums
- Primary subject
- Mathematics
- Subject subfields
- Classical and Real Analysis
- Paper type
- Research Article
- Manuscript stage
- Preprint
- Manuscript
- Archived
- Edition
- 1 · Permanent link
Overview
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. The unrestricted quadratic simplex problem in dimensions four and higher remains open.
Keywords Classical and Real Analysis · Combinatorics and Graph Theory
Scope of the work
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. The unrestricted quadratic simplex problem in dimensions four and higher remains open. 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. No new integer TU principle, full higher-dimensional kernel solution, three-free matroid theorem, optimal degree elevation, worldwide priority or external human refereeing is claimed.
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Cite this edition
Scott Schubr. (2026). Sharp Quadratic Smoothing for Truncated Simplex Sums (Version 1). Luminary Research Archive. https://research.luminaryoddities.com/sharp-quadratic-smoothing-for-truncated-simplex-sums/v/1
BibTeX
@techreport{luminary_sharp_quadratic_smoothing_for_truncated_simplex_sums_v1,
title = {Sharp Quadratic Smoothing for Truncated Simplex Sums},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 1},
url = {https://research.luminaryoddities.com/sharp-quadratic-smoothing-for-truncated-simplex-sums/v/1}
}Loading discussion…