bibkey: iliopoulos2017commutative
authors: "Fotis Iliopoulos"
year: 2017
title: "Commutative Algorithms Approximate the LLL-distribution"
doi: 10.48550/arXiv.1704.02796
url: https://arxiv.org/abs/1704.02796
claim: "Commutative resampling algorithms satisfying the stated cluster-expansion or Shearer condition have bounded expected flaw counts and query probabilities."
strata_touched:
  - D5/S3/Arith/Congruence/TwoOddPrimeUncoveredDensity
license: citation-only
triage: anchor

Commutative resampling and its required hypotheses

The inspected version is v6, dated 8 June 2019; first submission 10 April 2017. Checked 16 September 2026. Section 2.2 defines causality and charges, Definition 2.1 requires a probability-preserving injective swap, and Theorem 3.2(1) bounds expected flaw addresses. Remark 3.1 supplies the corresponding Shearer ratios.

The Erdős #7 dossier checks these hypotheses for canonical partial assignments using the actual conflict graph. Its independent rare-query derivation gives the terminal bound nu(E) <= Omega(E) Z_(V minus N(E))/Z_V for one fixed original output law. That ratio is a proved specialization, not the literal statement of Theorem 3.2(2). The source does not establish Shearer positivity for arbitrary distinct odd congruence families or the dossier’s quantitative head target.

Citation and source-boundary note only; no source text or code is vendored.