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OpenAI Says Its Models Disproved a Core Complexity Conjecture. Now Experts Have to Check the Proof

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  • OpenAI Says Its Models Disproved a Core Complexity Conjecture. Now Experts Have to Check the Proof
A mathematician writing a proof in chalk on a blackboard covered with equations.

OpenAI announced on October 26 that its internal models have produced three new scientific results: a proof of Sogge's local smoothing conjecture in harmonic analysis, a disproof of the Exponential Time Hypothesis in computer science, and a rigorous existence proof for quantum chromodynamics and related quantum field theories in physics. None of the three has been peer reviewed. OpenAI is unusually candid about that, calling the manuscripts raw research data and handing them to outside experts to examine, explain, and, if they hold up, turn into actual scientific understanding.

The computer science claim is the one with the widest blast radius. The Exponential Time Hypothesis, proposed by Russell Impagliazzo, Ramamohan Paturi, and Francis Zane, holds that 3-SAT cannot be solved in subexponential time, and hundreds of papers have used it to derive conditional hardness results across graph theory, databases, computational biology, and geometry. OpenAI says its system found a natural-language disproof on October 16 and formalized it in the Lean proof assistant the next day. Both the writeup and the Lean proof are public on GitHub. Fittingly, the examination will be led by Impagliazzo and Paturi themselves, alongside MIT's Ryan Williams, and OpenAI has agreed to fund a conference on the implications. If the result stands, it does not make hard problems fast overnight; OpenAI itself notes that turning it into practical algorithms will take significant further research.

In mathematics, the claimed proof of Christopher Sogge's 1991 local smoothing conjecture covers every spatial dimension, which would also resolve the Kakeya conjecture and Stein's restriction conjecture in all dimensions; human mathematicians had settled local smoothing in two dimensions in 2019 and Kakeya in three dimensions in 2025. The AI output arrived on October 18 and was formalized in Lean a day later. The physics claim is the most closely held: existence proofs for four-dimensional quantum field theories including QCD, built on Balaban's approach to constructive quantum field theory, with pure Yang-Mills existence, part of a Clay Millennium Prize problem, as a subproblem. A group led by Nobel laureate David Gross will shepherd that work, and unlike the other two, its materials are being withheld until that group writes them up.

The release model may matter as much as the results. OpenAI says it seeks no academic credit, set no nondisclosure agreements, and made participation voluntary and unpaid, with three different strategies chosen by the experts: open release for the mathematics, public release plus an external group for computer science, and expert-led delay for physics. That is a more honest template than a victory-lap announcement, and it acknowledges the asymmetry baked in: the lab still chooses who sees extraordinary claims first.

For people building with AI, the lesson is not that conjectures are falling like dominoes. It is that generation has outrun verification, again. A Lean formalization is a real check on the argument as formalized, but experts still have to confirm the formal statement matches the conjecture mathematicians actually care about, which is why none of this counts as settled until the examiners report. The same trust gap showed up in miniature on October 27, when TechCrunch reported Bank of America emailing customers not to share their banking passwords with AI assistants. Whether the claim is a disproved hypothesis or an agent asking for your login, the productive question is the same: what, exactly, has been verified, and by whom?

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Claude Duet Reaches Out First: Anthropic's Proactive Agent Joins the Always-On Race

Anthropic CEO Dario Amodei speaking on stage at a technology conference.

Anthropic unveiled Claude Duet on Wednesday, a standalone agent app that does not wait to be asked. Announced on October 21 and opening as a waitlisted research preview for United States subscribers on the Pro and Max plans, Duet monitors the apps you have connected, sends a briefing each morning, checks back every few hours, and alerts you when something needs your attention. It is built around Anthropic's Opus 5.5 model, runs on iOS, Android and claude.ai, and works in one ongoing thread that reuses the memory, skills and connectors from your Claude account.

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OpenAI Moves Into Google Cloud: The Frontier Is Now Sold Everywhere

Server racks with colorful status lights inside a Google data center.

OpenAI announced on Wednesday, October 21, that its frontier models and its Codex agentic workflows are coming to Google Cloud's Model Garden, with general availability planned in the coming weeks and customers able to request access now. Google Cloud customers will be able to buy the models through Google Cloud Marketplace and draw the purchases down against existing Google Cloud spend commitments, and OpenAI said the models will soon also be available inside Google Antigravity and the Gemini Agent. Reuters reported that the offering includes OpenAI's latest models such as GPT-6 Astra.

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Anthropic Needs More Compute Than Its Own Cloud Partners Will Sell It

Anthropic CEO Dario Amodei speaking on stage at a technology conference.

Anthropic's problem, in its own draft IPO prospectus, is not demand. It is supply. The Wall Street Journal reported on October 20 that it had viewed the draft S-1, and TechCrunch summarized the filing the same day: Anthropic has sought more cloud capacity than Amazon Web Services, Google and Microsoft were willing to provide, in part because those partners increasingly see the Claude maker as a competitor rather than just a customer.

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