OpenAI research collection · 1 manuscript · Source checked 2026-10-19
OpenAI's result summary
Proves De Giorgi's conjecture at its sharp dimension-eight endpoint: every entire C^(2) solution u:\mathbb R^8→(-1,1) of \Delta u=u^3-u that is strictly increasing in one direction depends on only one linear coordinate. A stronger theorem classifies all stable entire solutions v:\mathbb R^7→[-1,1] as constant wells or planar transitions, without an energy-growth assumption.
Read the manuscripts
Short excerpts below come from OpenAI's abstracts. The full abstract and original PDF preserve the paper's precise scope.
MANUSCRIPT 01
A positive resolution of De Giorgi's conjecture in dimension eight
We resolve De Giorgi's conjecture positively in dimension eight: every entire C^(2) solution u:\mathbb R^8→(-1,1) of \Delta u=u^3-u with an everywhere positive directional derivative is a planar heteroclinic. We prove that every stable solution v:\mathbb R^7→[-1,1] of this equation is a constant well or a planar heteroclinic,…
Full source abstract
We resolve De Giorgi's conjecture positively in dimension eight: every entire C^(2) solution u:\mathbb R^8→(-1,1) of \Delta u=u^3-u with an everywhere positive directional derivative is a planar heteroclinic. We prove that every stable solution v:\mathbb R^7→[-1,1] of this equation is a constant well or a planar heteroclinic, without an energy-growth assumption.
This exact PDF is not listed in the imported formalization catalogue. Family scope notes may contain further information.
New mathematical reasoning could help AI tackle harder scientific and computational problems. How reliably these skills transfer beyond mathematics will determine how much they contribute to general intelligence.