Centroid control implies KLS - The KLS theorem and its methods

Part of the fixed-cut archive, Chapter The fixed cut: the mass martingale and the Carleson estimate; the reading order is on the full proofs page.

Overview. This dossier proves two conditional implications. A stopped centroid bound controls the quadratic variation of the bounded mass martingale, hence the probability that a balanced cut leaves its window. The certified survival lemma then yields KLS. Separately, the all-cut assumption supplies exactly the premise of the certified tight-window consumption corollary at its coarse-window endpoint. Neither antecedent is discharged here.

Conventions. All horizons in the antecedents are strictly positive and all constants are finite. Let $\mu$ be isotropic log-concave on $\mathbb R^n$ , and let $E$ be a fixed measurable cut. Under the localization of the manuscript, write $p_t=\mu_t(E)$ , $q_t=1-p_t$ , $s_t=p_tq_t$ , and $\delta_t=m_t^E-m_t^{E^c}$ . The continuous process $p_t$ is a bounded martingale. Its stochastic differential and quadratic variation are

dp_t=s_t\delta_t\cdot dW_t, \qquad d[p]_t=s_t^2|\delta_t|^2\,dt.

(100.1)

Indeed the localization differential for the fixed indicator is $dp_t=\operatorname{Cov}_{\mu_t}(\mathbf1_E,X)\cdot dW_t$ ; the covariance is $p_t(m_t^E-a_t)=p_tq_t\delta_t$ . The identities hold locally, and the boundedness of $p_t$ makes its stopped increments square integrable. Thus the usual martingale isometry applies at every bounded stopping time. These are the mass-martingale conventions of (30.3).

Proof 100.1

It suffices, by Lemma 30.1, to obtain a universal balanced posterior event for every cut with $p_0=1/2$ . Set

\tau=\inf\{t\ge0:p_t\notin[1/3,2/3]\}.

(100.2)

For $T\le T_0$ , let $M_t=p_{t\wedge\tau}-p_0$ for $0\le t\le T$ . Continuity implies that on $\{\tau\le T\}$ the supremum of $|M_t|$ is at least $1/6$ . The nonnegative submartingale $M_t^2$ satisfies the weak maximal inequality; therefore

\mathbb P(\tau\le T) \le36\mathbb E M_T^2 =36\mathbb E\int_0^{T\wedge\tau}s_t^2|\delta_t|^2\,dt \le36CT.

(100.3)

The last inequality uses $s_t^2\le1$ and the assumed centroid estimate. In particular it requires no unproved integrability of a local martingale: $M$ is bounded, and the displayed occupation is finite by the antecedent. Choose

T_* =\min\{T_0,\,[72(C+1)]^{-1}\}>0.

(100.4)

Then $\mathbb P(\tau>T_*)\ge1/2$ . On that event both posterior masses are at least $1/3$ . The survival lemma applies with $c_0=1/2$ , $b_0=1/3$ and time $T_*$ . These quantities depend only on the universal antecedent constants, so its final assertion proves KLS.

Proof 100.2

Fix any isotropic log-concave $\mu$ and any measurable cut of mass $1/2$ . In Corollary 30.1 take $\eta=1/6$ . Its exit time $\tau_\eta=\inf\{t:|p_t-1/2|>\eta\}$ is precisely the coarse exit time $\tau$ , with the continuous-exit convention in the manuscript. The initial condition $|p_0-1/2|\le\eta/2$ holds. For each deterministic $0<T\le T_0$ , apply the all-cut assumption with $I=[0,T]$ . It gives

\mathbb E\int_0^{T\wedge\tau_\eta}S_t\,dt \le C_0T+C_1\mathbb E\int_0^{T\wedge\tau_\eta}r_t\,dt +\alpha\mathbb E\int_0^{T\wedge\tau_\eta}D_t\,dt,

(100.5)

which is exactly the premise of the certified corollary. That corollary gives a balanced survival event at a positive time depending only on $T_0,C_0,C_1,\alpha,\eta$ . Its conclusion, together with Lemma 30.1, proves the desired universal Cheeger lower bound. No extra stopping time has been inserted into the Carleson assumption.

Dependencies and applicability. The first implication uses Lemma 30.1 and assumes Assumption 30.1. The second uses Corollary 30.1 and Lemma 30.1, and assumes Assumption 28.1. The tight-window corollary is consumed as a certified result, rather than reproved here. In particular this dossier does not certify the analytic details of its existing dossier anew.

Fences respected. Neither target has a bounded_by edge. No universal centroid or Carleson estimate is proved here, and no trace upgrade or spectral occupation hypothesis is silently assumed. The universal KLS conclusions remain conditional on their displayed antecedents.