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 5,080 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, 0.781 on this set. Open problems are sorted into a reach class: 251 close to known results, 764 borderline, 911 that need a new idea and 103 in a branch with too few solved rows to judge.

A backtest at cutoffs 2005 and 2021 scores the rule on the questions posed by those years under two outcome codings, and the backtest is inconclusive. Settled only: at 2005 the inside rows settle at 21% against 6% outside (112 and 71 rows, permutation p = 0.006, AUC 0.800) and at 2021 at 12% against 1% (238 and 126 rows, p < 0.001, AUC 0.883), but 11 and 21 of the scored rows are solved rows with no resolved year, and without them the gaps are 13% against 6% (p = 0.131) and 3% against 1% (p = 0.269). Settled or advanced: 46% inside against 61% outside at 2005 (183 rows, p = 0.051, AUC 0.479) and 43% against 39% at 2021 (364 rows, p = 0.503, AUC 0.523). The verdict depends on how partial is coded, and the embeddings and stored neighbours come from the 2026 corpus, so the test is a check against known outcomes under that leak and no forecast.