The Kannan-Lovász-Simonovits conjecture

Claims the conjecture in full, for all isotropic log-concave measures: $C_P(\mu) \le C$ with $C$ universal. The route bounds the third cumulant, propagates the estimate to cumulants of every order by Eldan's stochastic localization, realizes derivatives of tilt averages as mixed cumulants through a suspension construction, and closes with the Song-Zhang criterion. The previous best bounds were Klartag's $C\sqrt{\log n}$ and Song and Zhang's $C \cdot 16^{\log^* n}$ (1 October 2026). Priority: Song and Zhang revised their preprint on 4 October 2026, the day this paper was posted, to claim the same $O(1)$ bound by their own route; the two are parallel claims of the same result. Consequences include the thin-shell and variance conjectures, already known to follow from KLS.

Claims the conjecture in full, for all isotropic log-concave measures: $C_P(\mu) \le C$ with $C$ universal. The route bounds the third cumulant, propagates the estimate to cumulants of every order by Eldan's stochastic localization, realizes derivatives of tilt averages as mixed cumulants through a suspension construction, and closes with the Song-Zhang criterion. The previous best bounds were Klartag's $C\sqrt{\log n}$ and Song and Zhang's $C \cdot 16^{\log^* n}$ (1 October 2026). Priority: Song and Zhang revised their preprint on 4 October 2026, the day this paper was posted, to claim the same $O(1)$ bound by their own route; the two are parallel claims of the same result. Consequences include the thin-shell and variance conjectures, already known to follow from KLS.