GPT-5.6 Closed a 30-Year Optimization Gap. The Prompt Took a Year.

The headline version of this story is that an AI model solved a thirty-year-old math problem in a single sitting. That version is true, and it is also the least interesting thing about what happened.

In a session lasting roughly 148 minutes, GPT-5.6 Sol Pro produced a proof establishing a lower bound of Ω(d² / log(d+1)) on the number of function evaluations required to minimize a convex Lipschitz function when the solver has access only to exact function values — no gradients. That bound essentially matches, up to polylogarithmic factors, the O(d² log² d) upper bound that Protasov's value-only method established in 1996. A gap that had sat open in the complexity theory of convex optimization for three decades closed.

The proof was formalized in Lean 4 against Mathlib. It compiles without a single sorry — the marker Lean uses for a step asserted but not proven. Two independent groups have since confirmed the key lemma using computer algebra, one via CVXPY and one with a custom SDP solver in Julia. No counterexamples have surfaced. The result reached the top of Hacker News on July 17 with 513 points and 328 comments.

And the mathematician who ran the session, Phillip Kerger, is the person most actively deflating it.

"I wouldn't really say that this result is using or creating some fundamentally new techniques in convex geometry or optimization theory," Kerger said.

He is in a position to know. He had spent roughly a year on the problem before the model was involved.

What the prompt actually contained

The session that produced the proof began with a document about ten pages long. It was modeled on the prompting approach OpenAI had used for its own CDC proof work, and its contents were not a question. They were a decomposition: specialized mathematical context, a direction of attack, and a structure that broke the open problem into small lemmas that could each be stated formally and checked independently.

The workflow followed from that structure. Each lemma was converted into a formal statement in Lean 4 before any informal argument was attempted. When a lemma failed to compile, that lemma — and only that lemma — was iterated on. The model was not asked to produce a proof and then trusted. It was asked to produce candidate steps inside a scaffold where every step was mechanically adjudicated.

This is the part that generalizes, and it is not a story about model capability in isolation. It is a story about a loop with three participants: a domain expert who knows which decomposition is likely to work, a model that supplies mathematical steps faster than a human can write them, and a proof assistant that has no opinions and cannot be persuaded.

Remove the expert and there is no ten-page prompt. Remove the proof assistant and there is no way to distinguish this from the substantial genre of confident, fluent, wrong mathematical output that language models have been generating for years.

Formalization is why this one is different

The past two years have produced a great many "AI solved an open problem" claims, and a large fraction of them dissolved on inspection — the problem turned out to be already solved, or the proof contained a gap that survived casual reading, or the result was true but the model had reproduced a known argument it had memorized.

Lean changes the epistemics of that entirely. A Lean proof that compiles without sorry is not a persuasive argument; it is a checked one. The verification does not depend on a reviewer's attention, patience, or willingness to be impressed. Anyone can run it. That property is what separates this result from the noise, and it is why the independent lemma confirmations in CVXPY and Julia — belt-and-suspenders on an already-machine-checked claim — read as thoroughness rather than doubt.

It is also the reason the appropriate frame here is verification infrastructure rather than model benchmarks. The bottleneck on AI contributions to mathematics was never purely whether a model could generate a valid argument. It was whether anyone could cheaply tell the valid ones from the plausible ones at the volume models produce them. Formalization solves that, and does so in a way that scales.

The paper is on arXiv and has not completed formal peer review — though in this specific case, the Lean artifact carries more evidentiary weight than a referee report typically would.

Where this sits

The result arrives in a year that has already reset expectations. In May, OpenAI announced that an internal model had disproved the Erdős planar unit distance conjecture, first posed in 1946, with the model producing the core ideas and draft proof that mathematicians then verified and published. Google DeepMind's AlphaProof Nexus solved nine further Erdős problems in parallel, working through Lean throughout. Fields Medalist Tim Gowers has endorsed both sets of results as genuine contributions.

Kerger's convex optimization bound is smaller than those in reach and larger in what it demonstrates about method. It is not a lab announcement with a curated narrative attached. It is a working mathematician, a commercial model, a public prompt strategy, and a machine-checkable artifact — a workflow any sufficiently expert researcher can now run.

Which relocates the scarce input. It is no longer access to a frontier model; GPT-5.6 is a subscription. It is no longer verification; Lean is free. What remains scarce is the year of work that precedes the prompt — knowing the problem well enough to know which lemmas are worth asking for, and which decomposition has a chance of closing.

Kerger's insistence that the mathematics contains nothing fundamentally new is not modesty. It is the finding. The techniques were available. The gap stayed open for thirty years anyway, because assembling them took more sustained effort than the result seemed to justify. What changed is the cost of the assembly, and cost is what determines which open problems are worth a year of someone's attention.

That is a smaller claim than "AI does mathematics now." It is also the one that will move more results.