De Giorgi's conjecture in dimension eight

Proves De Giorgi's conjecture at its sharp dimension-eight endpoint: every entire C² solution $u:\mathbb R^8\to(-1,1)$ 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 $v:\mathbb R^7\to[-1,1]$ as constant wells or planar transitions, without an energy-growth assumption.