GPT-5 Pro proves an improved convex-optimisation bound, which humans had already surpassed
Awaiting review
The takeaway
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.
Status
- Claim
Awaiting review
- Our reporting
- Medium confidence
- Verification
- Expert-checked
- Importance
- 2 of 5
- Last verified
- 19 October 2026
Your AI and this story
- GPT-6 AstraIn its training data
- Claude Opus 5.5In its training data
- Gemini 4 HydrogenIn its training data
- Grok 4.7In its training data
It happened before the cutoffs of all four assistants, so it can be in their training data.
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 |
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Sources
3 sources from 3 sites. Numbers match the chips in the text.
3 sources: 1 primary, 1 press, 1 reaction
Primary
- Sébastien Bubeck on Xx.com, official
Press
- What does GPT-5’s new math claim actually mean?allthings.how, press
Reactions
- whataifound.org: GPT-5 convex boundwhataifound.org, discussion
Changes
- Status confirmed → pending (the only check is by the OpenAI researcher who posed the question; no paper); verification made precise
- Filed