Diffusion budgets for contact-freezing lattice ellipsoids
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Abstract
We study bounded-covariance matrix martingales whose ellipsoids acquire and retain lattice contacts. On a common set of Haar lattices of probability tending to one, we bound the total nonnegative gain in a heat-occupation frame by the time-averaged square-root diffusion cost. A cost of o(log d) forces this gain to vanish at the horizon (16 log d + 8 log log d + O(1))/d². The bound is uniform over admissible policies on each selected lattice and also gives an averaged-budget theorem. Its proof combines deterministic pair envelopes, radial and angular Haar estimates, same-noise coupling and a potential-weighted restart. Applications include every fixed-strength normalized curvature-preserving tilt and rotating low-rank feedback. We also solve the instantaneous curvature-preserving optimization problem with the same diffusion cost, including noncommuting curvature and occupation operators. For fixed instantaneous operators, matching two-sided bounds give the sharp square-root small-budget law with an explicit linear error.
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Research details
- Subjects
- Mathematics · Probability and Statistics · Geometry and Optimal Transport · Optimization and Control
- Keywords
- Manuscript date
- 2026-10-27
- Reader reuse
- none
Scope
We study bounded-covariance matrix martingales whose ellipsoids acquire and retain lattice contacts.
Disclosures
OpenAI ChatGPT/Codex assisted with literature discovery, mathematical exploration and proof development, manuscript drafting, verification-code development and internal critical checking. The author assumes responsibility for all content. Scott Schubr owns Luminary Oddities, which operates the Research Archive hosting this preprint. This work received no specific grant from any funding agency, commercial or not-for-profit sector.
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Complete searchable manuscript and 10-member source package with editable sources and necessary reader-facing verification material. Build instructions and computational scope are stated in the package; proofs are given in the manuscript.
References and related resources
- B. Klartag. Lattice packing of spheres in high dimensions using a stochastically evolving ellipsoid. arXiv:2504.05042v2 (2026).
- B. Klartag. Poisson approximation of random lattices. arXiv:2605.10479v1 (2026), equation (7).
- D. Kelmer and S. Yu. The second moment of the Siegel transform in the space of symplectic lattices. arXiv:1802.09645v1 (2018), Introduction: classical Rogers formula.
- R. Bhatia, T. Jain and Y. Lim. On the Bures-Wasserstein distance between positive definite matrices. arXiv:1712.01504v1 (2017).
- N. Gargava, M. Gröbner and M. Simkin. A cyclotomic gluing construction for lattice sphere packings. arXiv:2610.12619v1 (2026).