No Fixed Frame Is Enough - Nova Spivack

Mathematics is producing proofs faster than it has ever produced them, and mathematical progress has not accelerated. That fact is not a paradox. It is the clearest available evidence for a structural claim about artificial intelligence — one that can be stated precisely, proved in three independent ways, and, uncomfortably, does not say what most people want it to say.

The Puzzle in Front of Us

In the summer of 2026, mathematics is experiencing something it has no precedent for. In May, an OpenAI model disproved the unit distance conjecture, an eighty-year-old problem of Erdős in combinatorial geometry. DeepMind systems resolved nine further Erdős problems with substantial autonomy. In July, a model proved the cycle double cover conjecture, open for more than half a century. And on the twentieth of July, Levent Alpöge posted a single short polynomial map that refutes the Jacobian conjecture in every dimension above two — a problem posed by Ott-Heinrich Keller in 1939 and listed by Stephen Smale among the great problems for the twenty-first century.

The natural inference is that mathematics is accelerating. Terence Tao, watching more closely than almost anyone, reports that it is not. He decomposes mathematical work into three activities — generating a proof, verifying it, and digesting it, where digestion means understanding a result well enough to contextualize it, explain it, and build on it. Artificial intelligence and formal verification have accelerated the first two dramatically and left the third roughly where it was. The result is what he calls an impedance mismatch: mathematics has moved from an era of proof scarcity to an era of proof abundance, and its infrastructure and culture have not adapted. His most striking observation is the one I want to build on. The enormous acceleration in proof generation has not produced a corresponding acceleration in mathematical progress.

Why not? If proofs are the product, and proofs are now cheap, the frontier should be racing forward.

The answer, I will argue, is that the frontier is not made of proofs. It is made of the frames within which questions can be posed at all — and those frames are produced by digestion, which nothing has made cheap. This is a specific instance of a general structural fact about lawful systems, one that can be established formally rather than gestured at. The general fact is that no fixed explanatory standpoint is ever sufficient. Not for a mathematician, not for a model, not for any system whatsoever.

That last clause is where this essay parts company with most of what has been written on the subject, including some of what I have written myself. The formal results are real, they are stronger than the usual hand-waving about creativity, and they do not establish a human advantage. They establish something more useful.

The Deficit Is Not Where We Think It Is

The common intuition is that AI is good at working with what is known and bad at reaching into what is not. It constructs from knowns and deduces from knowns. It operates in positive space. It does not reach into negative space — the unformed region where a genuinely new idea has to come from.

The intuition is pointing at something real, but the diagnosis is one layer off, and the layer matters.

Generating novelty is trivial. Raise the sampling temperature on any model and you obtain an unbounded supply of things nobody has ever said. Almost all of it is garbage. The hard part was never producing the strange thing. The hard part is recognizing that this particular strange thing is worth two years of your life, before anything exists that could confirm it.

Once you state it that way, the evidence rearranges itself. Consider where AI has in fact gone deep into unexplored territory. AlphaGo’s move 37, which no strong human player would have made and which was correct. AlphaFold, which solved a problem that had resisted fifty years of structural biology. AlphaTensor, which found matrix multiplication algorithms better than the best known human constructions. FunSearch, which produced genuinely new mathematical objects through program search.

Every one of these has a property in common, and it is not architectural. Each operates in a domain with a cheap, automatic, unambiguous oracle. Go has win and loss. Protein structure has RMSD against ground truth. Matrix multiplication has arithmetic correctness. FunSearch has a program that either runs and scores or does not.

Where a cheap verifier exists, machines search negative space better than we do. Where it does not, they revert to interpolating what has already been written.

This reframes the whole question. The bottleneck is not novelty generation. It is novelty evaluation in the absence of a verifier . And that is precisely what the word "intuition" has always been used to name.

What "Intuition" Actually Names

Intuition is not one capacity. It is at least four, with...