When an AI's discovery passes a test, that shows it works; a proof shows why it must be true.

This explores the difference between checking an AI's discovery by scoring it and checking it by proving it, and what each check actually certifies. In short, the corpus says scoring tells you a candidate works. Proof tells you why it must be true, and in mathematics it also carries understanding that a score can't give you. Most of today's discovery systems run on scoring because scoring is cheap enough to repeat thousands of times.

Scoring is what makes evolutionary discovery loops possible. AlphaEvolve-style systems generate candidates, run each through an automated evaluator, and keep the winners. That only works because verification is cheap and objective enough to run at scale Can machine feedback sustain discovery at test time?. The Darwin Gödel Machine makes the trade openly. The earlier Gödel Machine idea required a system to formally prove that each change to itself was an improvement. DGM drops that requirement and tests variants on coding benchmarks instead, and that switch is what let it improve itself in an open-ended way Can AI systems improve themselves through trial and error?. Proof was the bottleneck, and replacing it with measurement was what moved the field forward.

The cost shows up in what a score leaves out. FunSearch certifies each discovered program by its score. Its claims that humans can interpret those programs are only hedged, with no evidence that anyone actually understood them How does FunSearch actually verify its discovered programs?. Across AlphaEvolve's 67 problems, the evaluator reliably certified solutions, but the paper treats interpretation as a separate question that succeeded only some of the time. The weakness of the evaluator also became something to exploit: the system found loopholes in it Can automated scoring verify mathematical constructions without human understanding?. This is the key distinction. A proof can't be gamed the way a scoring function can. A score is only as trustworthy as the evaluator behind it, and optimizers will push on its edges.

The mathematicians' side of the corpus adds a twist: even formal proof doesn't settle everything. Terence Tao argues that an opaque ML model is fine as a source of ideas as long as something reliable checks its output, whether a proof assistant or numerical methods Can opaque machine learning models help prove new mathematics?. In his example, a neural network suggested a candidate solution to a hard fluid-dynamics equation (the Boussinesq equations), and rigorous perturbation arguments then confirmed it. Scoring finds the candidate and proof closes the case. The Leiden Declaration goes further. It holds that a proof does two jobs, establishing certainty and conveying understanding, and that machine verification secures only the first. That is why it assigns correctness and credit to human authors alone Can AI-generated proofs ever replace human mathematical understanding?.

The practical pattern is to split the roles. Spark-to-Paper separates the model's judgment from deterministic checks it can execute, and it requires authors to specify what evidence will count before they see results Can separating judgment from verification improve research paper reliability?. LLMs propose good candidates but can't reliably estimate their value, so external surrogates fitted to real experimental data have to do that part Can language models reliably judge their own candidate quality?. The same tension shows up in benchmarking: a final score alone can hide an agent that took a shortcut, which is why some operators now attach evidence of how the task was completed Can infrastructure evidence replace terminal scores in benchmark validation?. The unexpected lesson is that the line between scoring and proof is less about rigor than about what the result leaves you holding: a working artifact, or a reason you can pass to someone else.