A thread posted to r/math on July 15, 2026 reports that GPT-5.6 closed a 30-year gap in the complexity theory of convex optimization. The user describes a three-step process: asking the model to break the open problem into small lemmas, converting each into a formal statement in Lean 4 before attempting the informal argument, and iterating only on the lemma that failed to compile.
The result, according to the thread, extends the optimal convergence bound for first-order optimization methods to the nonsmooth convex case with bounded subgradients. That is exactly the zone where the problem had remained unresolved for three decades. The complete proof compiles with lake build without warnings and without the sorryAx axiom, the signal that Lean has accepted an unproven step.
What makes this case interesting is not that a large model produced a plausible argument. That happens all the time on r/math and usually ends up refuted in the comments. What matters is that the argument survived the most hostile step possible: translation into a language with no room for rhetorical ambiguity. Lean does not accept “it can be shown that” or “it is easy to see that”. It demands the complete logical step or rejects the compilation.
The lineage of the problem
Convex optimization is the ground where much of machine learning model training lives. Minimizing a convex loss function is what an optimizer like SGD or Adam does at every step. The underlying question that has chased this field for decades is easy to state and hard to answer: what is the best possible convergence rate for a method that can only query the gradient or subgradient of the function, without second-order information?
In 1983, Yurii Nesterov published his accelerated gradient method, achieving a convergence rate of O(1/k²) in the smooth case, well above the O(1/k) of classical gradient descent. Nemirovski and Yudin had proven a lower bound that same year matching the order of magnitude, but not the exact constant. That constant gap remained open for 31 years, until Donghwan Kim and Jeffrey Fessler published the Optimized Gradient Method (OGM) in 2014, closing the exact gap for the smooth case.
What remained unresolved was extending that result to the nonsmooth case, where the function does not have a gradient at every point and you have to work with bounded subgradients. That is the typical scenario for loss functions with L1 regularization, SVMs, or theoretical analysis of networks with nondifferentiable activations like ReLU. That, according to the r/math thread, is the 30-year gap the GPT-5.6-assisted proof claims to have closed.
How the prompt worked
The prompt ran to roughly 1,200 tokens and required two clarification rounds before the final proof appeared, according to a separate analysis on PromptZone. It combined three elements: a precise statement of the open problem, a request for a proof in the style of Boyd and Vandenberghe, and an instruction to flag any unproven steps. Early reproductions on the thread show that removing any of these components caused the model to revert to known partial results rather than the new claim.
The user applied a variant of the same prompt structure OpenAI used weeks earlier for its proof of the CDC problem (Complexity of Descent Conjecture). That is a reproducible workflow using tools anyone can install today: elan for the Lean version installer, and the project repository cloned from GitHub.
The verification question
Lean accepting the proof confirms internal consistency given the formal definition used. It does not confirm that this definition exactly captures the informal conjecture the community had been discussing for 30 years. The gap between the prose statement and the formal statement is, in itself, a classic source of errors in the formal verification of mathematics.
That nuance is exactly what the r/math thread is debating. Several commenters point out that the key lemma could be a reformulation of a result already known in the Russian optimization literature from the 1990s, published in journals rarely translated or indexed in Western databases. If that is confirmed, GPT-5.6’s merit would not be discovering a new theorem, but finding and reconstructing an argument lost in the literature, and formalizing it in a language no one had used for that before.
The PromptZone analysis notes that two independent groups have confirmed the key lemma using computer algebra systems. One verification used CVXPY; the second used a custom SDP solver in Julia. No counterexamples have surfaced. The original poster has not released the exact prompt text, citing ongoing peer review.
What this means for AI builders
The work pattern is more important than this specific result. A prompt that breaks a problem into lemmas, immediate formalization in Lean, iterating only on the lemma that fails to compile. That workflow is being replicated in other r/math threads to tackle different open problems. It is a reproducible method using tools anyone can install today.
GPT-5.6 required the least scaffolding of any recent AI math tool. AlphaProof required formal statements for IMO problems. Lean plus GPT-4o required tactic guidance. GPT-5.6 produced a verified proof from a single 1,200-token prompt. The verified output set is still small, but the workflow is now reproducible by any researcher with GPT-5.6 access.
The community debate over whether the result is genuinely new or a rediscovery of a lost lemma is itself a signal. If the lemma was already known in the Russian literature from the 1990s, the model did something arguably more useful than proving a new theorem: it found and reconstructed an argument that had effectively been lost, and formalized it in a language that makes it impossible to lose again.
The repository is open for community review. In the coming weeks, a convex optimization specialist who actually reads the Russian literature from the 1990s is expected to confirm or dismiss the suspicion of redundancy. Meanwhile, the workflow is being replicated. That is what matters more than this specific result.