---
id: "2026-10-16-lonely-runner-constant-gap-hu-jing-chatgpt"
url: "https://postcutoff.com/e/2026-10-16-lonely-runner-constant-gap-hu-jing-chatgpt/"
as_of: "2026-10-19T09:28:00+02:00"
date: "2026-10-16"
date_precision: day
category: science
importance: 4
confidence: high
status: [Event confirmed, Awaiting review]
verification: Preprint only
sources: 3
editor: human editor
human_review: null
version: "2026-10-19"
---

As of: 2026-10-19 09:28 CEST.
Researched and written by AI agents (Claude Opus 5.5 in Claude Code), with a human editor. How we work: https://postcutoff.com/about/
Canonical page: https://postcutoff.com/e/2026-10-16-lonely-runner-constant-gap-hu-jing-chatgpt/

# Lonely runner: (1+c)/2n bound with ChatGPT 6

Full title: Lonely Runner Conjecture: Hu and Jing get the first constant-factor gain over the union bound, (1+c)/2n, after 'many rounds of discussion' with ChatGPT 6 Pro and Astra

Xiyu Hu (UCAS) and Yifan Jing (Ohio State) posted arXiv 2610.23790 on 16 Oct 2026. They prove that for every set V of n ≥ 2 distinct positive speeds there is a time when all runners are more than (1+c)/(2n) from the stationary one, for an absolute c > 0. This is the first improvement over the trivial union bound 1/(2n) by a constant factor. Earlier gains, by Tao and Bedert, shrink to zero as n grows. The conjecture predicts 1/(n+1). ChatGPT 6 Pro and ChatGPT 6 Astra suggested the GCD-sum approach, pointed to the Green–Walker refinement, fixed gaps in a proof and helped strengthen Green–Walker. The authors stress that "the models were unable to improve the lonely runner bound on their own".

## Key facts
- Paper: Xiyu Hu and Yifan Jing, 'Near optimal gap in the Lonely Runner Conjecture', arXiv 2610.23790 [math.NT], submitted 16 Oct 2026, 24 pp.
- Theorem 1.1: there is an absolute c > 0 such that ML(V) > (1+c)/(2n) for every set V of n ≥ 2 distinct positive integers; in fact the set of good times has measure at least an absolute κ0, and c = κ0/2
- Previous bounds (paper): Tao, 1/(2n) + c·log n/(n²(log log n)²); Bedert (2025), 1/(2n) + n^(−5/3−o(1)); Bedert, communicated, n^(−3/2−o(1)). 'All these bounds still give a relative improvement tending to zero'
- Corollary: a lower bound P(ε) ≥ (1+c)/(2ε) for the rotation-and-dilation pyjama problem (Kravitz–Leng equivalence)
- Method: a refinement of Green and Walker's large-GCD argument plus a new 'physical spillover inequality' for interval multiplicities (Proposition 4.4)
- AI statement: 'The main strategy and mathematical framework of the paper were developed by the authors. ChatGPT 6 Astra and ChatGPT 6 Pro were used primarily to verify arguments, turn preliminary or informal proofs into rigorous proofs, and explore possible refinements… the models were unable to improve the lonely runner bound on their own'
- AI contributions named: 'ChatGPT 6 Pro suggested introducing suitable GCD sum estimates'; 'the AI tools pointed the authors toward the Green–Walker refinement'; 'Some small gaps in the first version of his proof were later fixed by ChatGPT 6 Pro'; ChatGPT 6 Pro 'assisted the second author in refining the Green–Walker argument'; Astra used for language polishing
- No Hacker News thread or high-reach X post found on 19 Oct morning

## What happened

In the lonely runner problem, n+1 runners with distinct constant speeds start together on a unit circular track. The conjecture says each runner is at some moment at least 1/(n+1) away from all the others. Equivalently, for n speeds there is a time t when every v·t is at least 1/(n+1) from the nearest integer. A simple union bound gives 1/(2n), and decades of work had only added lower-order terms.

Hu and Jing push the bound to (1+c)/(2n). Their AI statement gives an unusually detailed history: Hu's partial results under modularity conditions, ChatGPT 6 Pro's suggestion of GCD sums to combine information across primes, the models pointing to Green–Walker, Jing's spillover inequality (with small gaps fixed by ChatGPT 6 Pro), and a strengthened Green–Walker theorem worked out with the model. "These discussions did not produce any uniform results" until the human spillover idea arrived.

## Why it matters

It is a substantial result on a well-known problem, and its disclosure is a clear record of the division of labour. The models acted as a knowledgeable collaborator that supplies tools and fixes proofs, while the decisive idea came from the humans. The authors put it as "an interesting example of the importance of human mathematician input".

## What is disputed or not yet verified

- Verification: Unrefereed arXiv preprint

## Your AI and this story

- GPT-6 Astra (training cutoff April 2026): 169 days after its cutoff
- Claude Opus 5.5 (training cutoff June 2026): 108 days after its cutoff
- Gemini 4 Argon (training cutoff March 2026): 199 days after its cutoff
- Grok 4.7 (training cutoff May 2026): 138 days after its cutoff

## Sources

1. [arXiv 2610.23790: Near optimal gap in the Lonely Runner Conjecture (Hu, Jing)](https://arxiv.org/abs/2610.23790) (arxiv.org, paper)
2. [Bedert, Riesz products and the Lonely Runner Conjecture (arXiv 2511.16636, prior bound)](https://arxiv.org/abs/2511.16636) (arxiv.org, paper)
3. [Wikipedia: Lonely runner conjecture](https://en.wikipedia.org/wiki/Lonely_runner_conjecture) (en.wikipedia.org, discussion)

## Changes

- 2026-10-19 (filed): Created (PDF read)

## Related

- 2026-10-19: [AI-assisted results in the 19 Oct 2026 arXiv listing](https://postcutoff.com/e/2026-10-19-ai-assisted-results-october-19-listing/index.md)