Some hard math problems are hard to find the answer to; others are hard because every step must be airtight — so where does AI actually help?

This explores the difference between math problems that are hard because the answer is hidden somewhere in a huge space, and problems that are hard because you have to build a long, airtight argument. The corpus doesn't name this split directly, but several notes point to it, and they suggest it is one of the clearest ways to see where AI is actually useful in mathematics today.

The sharpest example is a counterexample to the Jacobian conjecture, a century-old open problem. Levent Alpöge used Fable 5 to find a three-dimensional polynomial that breaks it Can AI search find what human proof cannot?. A counterexample has an unusual shape of difficulty. Finding it can take a lifetime because the space of candidate polynomials is enormous. Once it's found, though, anyone can check it by plugging it in. All the difficulty is in the search, and almost none is in the verification. Terence Tao makes the general version of this point Can opaque machine learning models help prove new mathematics?: it matters less that a machine learning tool is a black box if what it proposes can be checked by something reliable, like a proof assistant or a numerical method. His example is a neural network that suggested candidate solutions for a fluid-dynamics blowup problem, which mathematicians then confirmed by rigorous perturbation arguments. Search-type problems are exactly where an untrustworthy but prolific guesser becomes useful.

Construction is a different situation. Every step has to hold, and the result is only as strong as its weakest link. The Erdős Problem 728 case shows what AI construction currently looks like Did an AI system truly solve Erdős Problem 728 autonomously?. An AI system produced a proof in Lean, a formal proof language that a computer can check line by line, so its correctness isn't in doubt. The open questions moved elsewhere: how autonomous the system really was, and whether humans can understand the argument. A checked construction still isn't automatically understood.

Here is the twist you may not expect. When LLMs do step-by-step reasoning, they turn out to be poor searchers in the systematic sense. One study describes reasoning models as "wandering explorers" Why do reasoning LLMs fail at deeper problem solving?. They revisit dead ends, skip branches, and take unnecessary steps, so their chance of success drops exponentially as a problem needs more steps. Other notes suggest that what looks like a model "working harder" is often familiarity in disguise. Longer reasoning traces reflect how close a problem is to the training data, not how hard it is Does longer reasoning actually mean harder problems?. Math accuracy also falls apart when only the numbers in a problem change Does LLM math reasoning truly generalize or just pattern match?. Models can even sense how hard a question is before they start, yet fail to act on that sense Can models recognize question difficulty before they reason?.

Putting this together: the AI math wins in this collection don't come from models reasoning through a long chain the way a mathematician would. They come from pairing a generator of candidates with an external checker, whether that checker is direct substitution, a numerical test, or Lean. So "search difficulty" is where AI is starting to beat humans, and "construction difficulty" is where it still needs a formal system to hold it to account. If you want one doorway in, start with Tao's note; it explains why the checker, not the model, is where trust comes from.