An exact counterexample to the shifted lonely runner conjecture at n = 63

The shifted lonely runner conjecture (Blanco–Criado–Santos, arXiv:2603.24784, Conjecture 1.1(ii)) asserts that for distinct positive integer velocities v1,…,vn and any real shifts si there is a time t with dist(si+vit, Z) ≥ 1/(n+1) for all i. It fails for 5 ≤ n ≤ 17 and n ≥ 95 (Poliakova, arXiv:2609.23952) and, by our earlier note, for 18 ≤ n ≤ 38. We give an explicit exact certificate that it fails at n = 63: 63 distinct velocities (largest 128) and shifts ai/520 whose open arcs of radius 1/64 cover the circle; the exact value is γ = 66/4225 < 1/64, so E63 ≥ 1/66 only. It is a Poliakova-type move on 2,4,…,128 and is sporadic (same menu infeasible for N = 65..95 and 60..63). Three exact checkers accept it, one written independently by an auditing AI agent. Shifted conjecture only, n = 63 only; nothing about the unshifted lonely runner conjecture. Producer and auditor are AI agents; no human audit; Lean covering statement kernel-checked (pinned Lean 4.34.1, axioms propext/Classical.choice/Quot.sound) after a first compile failed on one lemma and was repaired; no statement lock.

Authors

Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-11-08
DOI
https://doi.org/10.5281/zenodo.24051699
Primary Topic
Analytic Number Theory Research
Type
preprint
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preprint

An exact counterexample to the shifted lonely runner conjecture at n = 63

Xavier Callens
2 links
Zenodo (CERN European Organization for Nuclear Research)
Analytic Number Theory Research
preprint

An exact counterexample to the shifted lonely runner conjecture at n = 63

Xavier Callens
preprint en

Abstract

The shifted lonely runner conjecture (Blanco–Criado–Santos, arXiv:2603.24784, Conjecture 1.1(ii)) asserts that for distinct positive integer velocities v1,…,vn and any real shifts si there is a time t with dist(si+vit, Z) ≥ 1/(n+1) for all i. It fails for 5 ≤ n ≤ 17 and n ≥ 95 (Poliakova, arXiv:2609.23952) and, by our earlier note, for 18 ≤ n ≤ 38. We give an explicit exact certificate that it fails at n = 63: 63 distinct velocities (largest 128) and shifts ai/520 whose open arcs of radius 1/64 cover the circle; the exact value is γ = 66/4225 < 1/64, so E63 ≥ 1/66 only. It is a Poliakova-type move on 2,4,…,128 and is sporadic (same menu infeasible for N = 65..95 and 60..63). Three exact checkers accept it, one written independently by an auditing AI agent. Shifted conjecture only, n = 63 only; nothing about the unshifted lonely runner conjecture. Producer and auditor are AI agents; no human audit; Lean covering statement kernel-checked (pinned Lean 4.34.1, axioms propext/Classical.choice/Quot.sound) after a first compile failed on one lemma and was repaired; no statement lock.

Zenodo (CERN European Organization for Nuclear Research)
Analytic Number Theory Research
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