HEXAGONbeta

Quartic exponential matching in the unit square

Primarily AI-generated textHuman understanding: all partsmath.PR — Probability

Contributed by Will Howe, Grant Huffman, Jensen Kohlmeyer ↗, Liam Kruer

Version 1 / Nov 11, 2026 / CC BY 4.0

Abstract

Take two independent samples of nn uniform points in the unit square. We prove that, with probability at least one half, the points can be paired so that the average of the exponential of the fourth power of the normalized pair distance is at most two. The distance normalization is Llog⁡n/nL\sqrt{\log n/n}, with LL an absolute constant. One pairing therefore satisfies both coordinate inequalities in the symmetric (4,4)(4,4) case of Talagrand's matching problem. The argument first controls a vector field transporting smoothed point densities, and then converts that control to a pairing. The main estimate comes from a Gaussian construction whose successive partitions depend only on noise already revealed; a correction using the classical bottleneck matching preserves the required density balance.

Provenance statement

The mathematical draft and exposition were produced primarily with OpenAI Codex under the direction of the listed contributors. This is the revised exposition prepared following Michel Talagrand's request for a clearer presentation. Lean was used for supporting lemmas; the complete matching theorem is not yet formally verified. No independent referee acceptance is claimed.

Tools used

OpenAI
Codex
Lean
Lean

References

  1. M. Costabel and A. McIntosh, On Bogovskiĭ\ and regularized Poincaré integral operators for de Rham complexes on Lipschitz domains , Math. Z. 265 (2010), no. 2, 297--320. doi:10.1007/s00209-009-0517-8 (https://doi.org/10.1007/s00209-009-0517-8).DOI
  2. R. Lachièze-Rey and S. Muirhead, Asymptotics for the critical level and a strong invariance principle for high intensity shot noise fields , arXiv:2111.09180v2 (2022), Lemma 4.5. https://arxiv.org/abs/2111.09180.arXiv
  3. M. Talagrand, Upper and Lower Bounds for Stochastic Processes: Decomposition Theorems , second ed., Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 60, Springer, 2021. doi:10.1007/978-3-030-82595-9 (https://doi.org/10.1007/978-3-030-82595-9).DOI
  4. M. Talagrand, Matchings , problem note, available online (https://michel.talagrand.net/prizes/matchings.pdf).link

Version history

  1. v1 · 13 pagesSubmitted by Jensen KohlmeyerInitial depositCurrentNov 10, 2026