---
id: "2026-08-30-bounded-prime-gaps-186"
url: "https://postcutoff.com/e/2026-08-30-bounded-prime-gaps-186/"
as_of: "2026-11-13T13:49:00+01:00"
date: "2026-08-30"
date_precision: day
category: science
importance: 4
confidence: medium
status: [Awaiting review]
verification: "Lean proof, not yet reviewed"
sources: 4
editor: human editor
human_review: null
version: "2026-10-04"
---

As of: 2026-11-13 13:49 CET.
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/2026-08-30-bounded-prime-gaps-186/

# GPT-6 Astra lowers the bounded prime gaps record from 246 to 186

An OpenAI preprint (30 Aug 2026) claims lim inf (p_{n+1} − p_n) ≤ 186, improving Polymath8b's bound of 246, which had stood since 2014. It uses 'triply densely divisible' conditions feeding a multidimensional Selberg sieve and was announced with a Lean formalisation. Julia Stadlmann independently reached 240 at about the same time.

## Key facts
- Previous record: 246 (Polymath8b, 2014), building on Zhang (2013) and Maynard (2013)
- New claimed bound: 186
- Lean formalisation announced (Weijie Su); independent human verification not complete
- Human counterpart: Julia Stadlmann (UIUC), arXiv 2608.31126 (submitted 31 Aug 2026), proves 240 alone, 'with the assistance of traditional numerical computation, but not modern AI tools' (Tao); key idea: Motohashi–Pintz–Zhang estimates for only 'partly smooth' moduli

## What happened

OpenAI's model found a refinement of the Maynard–Tao sieve set-up that substantially improves the gap bound.

## Why it matters

Bounded prime gaps were one of the celebrated stories of 2013–14. An AI improving the collaborative record is a striking, if still pending, result.

## What is disputed or not yet verified

- Verification: Formal proof in Lean (announced); not yet independently peer-reviewed

## Your AI and this story

- GPT-6 Astra (training cutoff April 2026): 122 days after its cutoff
- Claude Opus 5.5 (training cutoff June 2026): 61 days after its cutoff
- Gemini 4 Hydrogen (training cutoff March 2026): 152 days after its cutoff
- Grok 4.7 (training cutoff May 2026): 91 days after its cutoff

## Sources

1. [OpenAI: short gaps between primes (PDF)](https://cdn.openai.com/pdf/51126fac-1b68-4128-9666-c908bcc16033/short_gaps.pdf) (cdn.openai.com, paper)
2. [Julia Stadlmann: Bounded gaps between primes (arXiv 2608.31126; human-only, bound 240)](https://arxiv.org/abs/2608.31126) (arxiv.org, paper)
3. [Terence Tao on Mathstodon: Stadlmann shaves 246 to 240 without modern AI tools](https://mathstodon.xyz/@tao/117197525544971208) (mathstodon.xyz, discussion)
4. [Weijie Su on X (Lean formalisation)](https://x.com/weijie444/status/2095600108956262911) (x.com, discussion)

## Changes

- 2026-09-29: Corrected arXiv 2608.31126 label (it is Stadlmann's human paper, not OpenAI's); added Tao's Mathstodon post on it
- 2026-09-29 (filed): Created

## Related

- 2026-09-03: [OpenAI releases GPT-6 Astra, its first GPT-6 model](https://postcutoff.com/e/2026-09-03-gpt-6-astra/index.md)
- 2026-08-01: [OpenAI's unreleased 'Astra' model claims ten advances in maths and theoretical CS, with Lean proofs](https://postcutoff.com/e/2026-08-01-openai-astra-ten-advances/index.md)
- People: [Terence Tao](https://postcutoff.com/person/terence-tao/)
- Archived: 2 posts, listed in https://postcutoff.com/e/2026-08-30-bounded-prime-gaps-186/index.json