Technical Report · Preprint Mathematics
Self-concordance and allocation for spatial power sums
- Primary subject
- Mathematics
- Subject subfields
- Optimization and Control · Classical and Real Analysis
- Paper type
- Technical Report
- Manuscript stage
- Preprint
- Manuscript
- Archived
- Edition
- 1 · Permanent link
Overview
Abstract
This mathematical supplement studies the negative logarithm of one-sided product-power smoothing of arbitrary coordinatewise decreasing weights. A common real power p >= 16d gives a standard self-concordant barrier with parameter at most 2pd. For decreasing lattice inputs with at most r positive coordinates per state, p >= 32r suffices independently of ambient dimension and occupancy magnitudes. Residual Pareto tails, an exact third-cumulant identity and a sparse L3 bound supply the proof without input independence or log-concavity. Decreasing binary mixtures require power growing linearly with r for uniform convexity. Log-concave binary generating polynomials with deletion bounds admit dimension-free fixed-power statements and squared-logarithmic activity dependence. Classical damped Newton and projected-gradient methods give explicit allocation consequences. Constants are not sharp and no bit-complexity bound is asserted.
Keywords Optimization and Control · Classical and Real Analysis · mathematical research
Scope of the work
Common powers only. Decreasing continuous/lattice spatial smoothing: standard SC p>=16d; sparse active-coordinate support r: p>=32r. Ordinary-logQ-concave BINARY deletion1: absolute scale2560000 at every p>=4, parameter<=2Cpd, classical matroid premise. Deletion ratios<=M: p>=72(1+logM)^2, absolute scale360000, parameter<=3Cpd; rank-one squared-log order obstruction and literal p4/M625 curvature -192/25. Explicit-list Newton gap logm. Quartic matroid activities<=1: independence-oracle sampled projected gradient with polynomial expected arithmetic cost/high probability epsilon objective accuracy; all histories feasible. No bitbound, counting/value/Hessian oracle, generic input oracle, NA or hyperbolicity claim. Mathematical supplement to the companion manuscript Dimension Thresholds for Concavity of Normalized Mixed Moments; the companion is a separate manuscript. The approved supplement does not release the companion.
Reproduction notes
The exact publication package includes the 16-page manuscript, editable LaTeX sources, reader/build instructions, citation metadata, public checksums and any necessary mathematical or computational supplements. Keep its 11 members together and follow the included README for dependencies, build commands and any finite verification programs. Published finite checks support the stated arguments; they do not replace universal analytic proofs or establish external human refereeing.
The manuscript
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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.
A stable scholarly record
Cite this edition
Scott Schubr. (2026). Self-concordance and allocation for spatial power sums (Version 1). Luminary Research Archive. https://research.luminaryoddities.com/self-concordance-and-allocation-for-spatial-power-sums/v/1
BibTeX
@techreport{luminary_self_concordance_and_allocation_for_spatial_power_sums_v1,
title = {Self-concordance and allocation for spatial power sums},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Technical Report},
note = {Version 1},
url = {https://research.luminaryoddities.com/self-concordance-and-allocation-for-spatial-power-sums/v/1}
}Loading discussion…