Machine learning for scientific discovery: the solvability frontier

A first pass, the Solver Gap Engine of 2026-09-30, ranked 1,364 open problems from the formal-conjectures repository by their nearest solved problem in another file and found 31 at cosine similarity 0.8 or above. The frontier is the second pass. We embed 5,209 problem statements across seven canon branches with a small sentence encoder, call a problem's reach its highest similarity to a solved problem, and draw a frontier at the 10th percentile of solved problems' reach, 0.762 on this set. Open problems are sorted into a reach class: 268 close to known results, 863 borderline and 954 that need a new idea.

A backtest at cutoffs 2005 and 2021, and frozen forecasts at 2005, 2015 and 2021, score the rule on the questions posed by those years under two outcome codings. Settled only: the inside rows settle at 24% against 5.9% outside at 2005 (161 and 68 rows, permutation p = 0.002, AUC 0.799), 22% against 3% at 2015 (p < 0.001, AUC 0.812) and 13% against 1% at 2021 (272 and 167 rows, p < 0.001, AUC 0.906); with the solved rows that have no resolved year removed, the 2005 gap holds (p = 0.020) and the 2021 gap shrinks but holds (p = 0.013); placing the 90 undecided 2005 rows outside reverses the 2005 comparison to 24% against 27%. Settled or advanced shows no gap: AUC 0.524, 0.554 and 0.543. The embeddings and stored neighbours come from the 2026 corpus and the embedding model postdates every cutoff, so the frozen forecasts are a retrospective check under that leak and no prospective test.