bibkey: bedertbucickravitzmontgomerymuyesser2025cayley
authors: Benjamin Bedert, Matija Bucić, Noah Kravitz, Richard Montgomery, Alp Müyesser
year: 2025
title: On Graham's rearrangement conjecture over F₂ⁿ
doi: 10.48550/arXiv.2508.18254
claim: Section 4 localizes near-invariant vectors in proper stabilizers and derives subgroup concentration and Cayley cut bounds.
strata_touched:
  - D5/S3/Combinatorics/Sequenceability/CayleySubgroupConcentration
license: citation-only
triage: anchor

Cayley concentration through displacement geometry

The fixed source is arXiv:2508.18254v1, Section 4, especially Lemma 4.6, Claim 4.7, Theorem 1.5 and Corollary 4.1. The displacement of a unit vector under a finite isometric group action is subadditive and invariant under inversion. A second-moment bound puts most original labels in a small displacement ball. Separated translates force a plateau among successive balls, and the plateau’s left stabilizer is proper when the orbit sums to zero.

The formal localization uses the actual regular action and retains the original labels. The shell count is the natural number (floor(4/tau)+2)/3; the sufficient plateau bound is k <= 4/tau. The threshold argument divides by the positive squared radius, including zero correlation defect without dividing by that defect.

For nonsymmetric Cayley operators in nonabelian groups, an upper bound on eigenvalue real parts does not imply an upper bound on every real quadratic form. The numerical quadratic-form condition is the appropriate stopping condition for a subsequent concentration descent and directed cut argument. The general core permits identity loops and nonnormal subgroups.

This source supplies the structural cut input to later rainbow-path arguments. It does not certify their matching, coupling, connecting or absorption proofs. In particular directed proper colouring alone does not justify a two-edge rainbow connector. Path consumers require original nonidentity labels. The arbitrary-abelian proper-interval ordering problem and unrestricted AM remain open here.

Locator

• Fixed version: https://arxiv.org/abs/2508.18254v1
• Section 4: https://arxiv.org/html/2508.18254v1#S4
• DOI: https://doi.org/10.48550/arXiv.2508.18254