---
id: "2025-08-20-gpt-5-pro-convex-optimization-proof"
url: "https://postcutoff.com/e/2025-08-20-gpt-5-pro-convex-optimization-proof/"
as_of: "2026-11-13T13:49:00+01:00"
date: "2025-08-20"
date_precision: day
category: science
importance: 2
confidence: medium
status: [Awaiting review]
verification: Expert-checked
sources: 3
editor: human editor
human_review: null
version: "2026-10-19"
---

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/2025-08-20-gpt-5-pro-convex-optimization-proof/

# GPT-5 Pro proves an improved convex-optimisation bound, which humans had already surpassed

OpenAI's Sébastien Bubeck reported that GPT-5 Pro, in about 17 minutes, proved that gradient descent on L-smooth convex functions yields a convex sequence of function values for step sizes up to 1.5/L. The paper's v1 had proved it for 1/L. However, the authors' own v2 had already proved the tight 1.75/L bound.

## Key facts
- Problem: for which step sizes η is the optimisation curve of gradient descent convex? v1 proved η ≤ 1/L and gave a counterexample above 1.75/L
- GPT-5 Pro proved η ≤ 1.5/L by a different argument; Bubeck checked it
- The human authors' updated version had already closed the gap at 1.75/L
- Bubeck: 'Claim: gpt-5-pro can prove new interesting mathematics.'

## What happened

Bubeck gave GPT-5 Pro the open question from v1 of a paper; the model produced a valid proof of an intermediate bound.

## Why it matters

It was one of the first widely discussed cases of an LLM producing correct new research-level mathematics. The fact that humans had already done better also foreshadowed later disputes over novelty.

## What is disputed or not yet verified

- Verification: Informal proof checked by Sébastien Bubeck (OpenAI), who posed the question; not written up as a paper or refereed; the human authors' revised paper already had the tight 1.75/L bound

## Your AI and this story

- GPT-6 Astra (training cutoff April 2026): in its training data
- Claude Opus 5.5 (training cutoff June 2026): in its training data
- Gemini 4 Hydrogen (training cutoff March 2026): in its training data
- Grok 4.7 (training cutoff May 2026): in its training data

## Sources

1. [Sébastien Bubeck on X](https://x.com/SebastienBubeck/status/1958198661139009862) (x.com, official)
2. [What does GPT-5's new math claim actually mean?](https://allthings.how/what-does-gpt-5s-new-math-claim-actually-mean/) (allthings.how, press)
3. [whataifound.org: GPT-5 convex bound](https://whataifound.org/finding/2025-08-gpt5-convex-bound) (whataifound.org, discussion)

## Changes

- 2026-10-19: Status confirmed → pending (the only check is by the OpenAI researcher who posed the question; no paper); verification made precise
- 2026-09-29 (filed): Created

## Related

- 2025-10-17: [OpenAI researchers claim GPT-5 'solved' 10 Erdős problems](https://postcutoff.com/e/2025-10-17-gpt-5-erdos-problems-controversy/index.md)
- 2025-08-07: [OpenAI launches GPT-5](https://postcutoff.com/e/2025-08-07-gpt-5/index.md)
- People: [Sébastien Bubeck](https://postcutoff.com/person/sebastien-bubeck/)
- Archived: 1 post, listed in https://postcutoff.com/e/2025-08-20-gpt-5-pro-convex-optimization-proof/index.json