Glossary of recurring terms - The KLS theorem and its methods

Each entry says what the object is and where it is fixed. Where a normalization is at stake the pointer is to the prf:definition that fixes it, because an exponent or a constant quoted without its convention is not a statement.

Alternative mechanisms: The three mechanisms for the Poincaré bound this manuscript develops besides the three proofs of Conjecture 0.1, each named by the object it keeps, each resting on a sufficient condition that implies KLS, with no converse known. The moment map is deterministic and asks for one inequality for the Hessian of the moment map, with no localization at all (Chapter The moment map: the deterministic inequality); the fixed eigenfunction follows a first eigenfunction under Eldan’s stochastic localization (Chapter The fixed eigenfunction: following one eigenfunction through localization); conditional fibers replace Euclidean directions by one test-independent isotropic frame of conditional line resamplings (Chapter Conditional fibers: inverse-variance frames of line resamplings). They are compared in Chapter Alternative mechanisms after KLS. The fixed cut, which follows one would-be bottleneck set under localization, is kept as an archive for its obstructions and counterexamples (Chapter The fixed cut: approach and lessons).
All-cut and near-Cheeger variants: The two variants of the fixed cut: the all-cut variant asks for the all-cut Carleson estimate for every balanced set (Chapter The fixed cut: the mass martingale and the Carleson estimate), the near-Cheeger variant only for near-minimizers of the isoperimetric profile, through the weighted near-Cheeger package (Chapter The fixed cut: the near-Cheeger variant). See Section The all-cut and near-Cheeger variants.
CMH: The canonical moment-Hessian constant $\CMH$ : the deterministic quantity the moment-map mechanism is built on, fixed with its operator data in Definition 16.1. It dominates the affine Poincaré constant $\CPaff$ by Theorem 16.1; it also charges a solenoidal excess (Proposition 16.2), discussed in Chapter The moment map: the deterministic inequality.
QCTS: Quadratic-chaos thin shell: the bound $\Var(X^TMX)\le C\norm M_{\HS}^2$ for isotropic log-concave $X$ and every symmetric matrix $M$ , defined in Definition 25.1; Letwin’s Theorem 25.1 gives $C=8$ . What it can supply to the fixed cut is limited by the anisotropic two-tail cut of Proposition 25.2 (Remark 25.3).
Two-color notation: The two-color covariance and its bookkeeping, which separate the contribution of the cut being followed from that of everything else, so that a source term can be told from a damping term. Fixed in Section Two-color notation, used throughout the fixed-eigenfunction and fixed-cut arguments.
Stein source and damping: In the scalar Riccati equation $\dd r_t=\dd M_t+(S_t-D_t)\dd t$ of Theorem 24.1, $S_t$ is the only positive source and $D_t$ the coercive damping. “Absorbing the source into the damping” is what every fixed-cut argument is trying to do; Section The Stein dictionary gives the Stein representation of the source.
Interface functional: The bridge object $\Xi_T(\mu)$ of the bootstrap comparison, defined at (28.8) and evaluated in Chapter The fixed cut: the bootstrap. It is this manuscript’s version of “replace $\log$ -trace-exp by an effective rank”; why the crude evaluation cannot suffice is Remark 33.2.
Linear test: The moment-Hessian inequality tested on linear functions only, its cheapest necessary condition: in isotropic position, $\lmax(\E H^2)\le4$ (Conjecture 16.1), where the literature gives the trace bound $\Tr(\E H^2)\le2n$ . Its sharp form is $\E H^2\preceq2\Id$ (Conjecture 16.2). Some statements and labels call it gate zero. Section The linear test: the necessary linear-sector condition.
Endpoint: The word is used in three senses. (1) Mostly, in the moment-map mechanism, the inequality $\mathrm{CMH}(4)$ itself: the last statement of the chain, which everything else in that mechanism serves to prove; the endpoint reduction is Theorem 16.1, from it to the affine Poincaré inequality (Chapter The moment map: the deterministic inequality). The product endpoint is the extreme case of that inequality, the product of centered one-sided exponentials, where it holds with equality (Corollary 17.5). (2) A statement of the same strength as the conclusion, usable as a final target but not as a first step: the $H^{-1}$ residual bound of Section The $H^{-1}$ endpoint. (3) For a time scale, the largest one a method can reach: $1/\log n$ is the natural endpoint of covariance-only control (Chapter The fixed cut: product stress test), because the covariance of a product of exponentials spikes to order $\log n$ at that time.
Lift: Two unrelated senses. (1) The homogeneous lift: a function on the simplex rewritten as a degree-zero homogeneous function of independent Gamma variables, which replaces the Dirichlet law by a product of Gamma laws (Section The independent Gamma lift); the cone lift goes the other way, attaching a Gamma radial variable to a base so that the cone’s moment map is built from the base’s (Proposition 17.1). (2) The invariant multiplier lift of Conjecture 18.1: the tensor through which a Haar multiplier, defined on one block of a Schur split of the moment-map Hessian, acts on the whole space; it is computed in coordinates and in low split dimensions, and the open question is to identify it intrinsically in every split dimension.
Full-damping occupation: The hypothesis Conjecture 21.1 of the fixed eigenfunction. Occupation is the source the eigenfunction accumulates over localization time, $\E\int_0^t\norm{H_s}_{\HS}^2\dd s$ ; full damping means it may be charged against the whole damping term $\E\int_0^t2g_s^TA_sg_s\dd s$ with coefficient one, where the fixed cut needs a fraction $\alpha<1$ of its damping. With coefficient one the two cancel exactly in the equation for $\E\abs{g_t}^2$ , which is why no surplus is needed (Proposition 21.1).
Exponential cone measure: The law $\bar\mu_{K,\beta}$ with density $\propto x_1^{\beta-n}e^{-x_1}$ on the cone over a centered convex body $K$ , centered; Definition 17.1. Its moment map is explicit in terms of the base’s (Proposition 17.1), and at $\beta=n$ it saturates sharp linear test along its axis, which makes the family the first non-product equality set of that statement. Section Exponential cones: a second solvable non-product family.
Covariance spike: The obstruction of Proposition 0.1: products of centered exponentials are dimension-free by tensorization, yet their conditional covariance spikes. It is why no uniform operator-norm bound on $\norm{A_t}_\op$ can exist, and it is the single sharpest constraint on what a correct argument may look like.
Appell coefficients: For a regular measure $\nu$ , $c_k(\nu)=\sqrt{K_k(\nu)}/k!$ , where $K_k(\nu)$ is the largest variance of a degree- $k$ Appell polynomial $\langle T,\mathcal A_k^\nu\rangle$ with $\|T\|_{\mathrm{HS}}=1$ . Defined in Section Notation and the smallest cases; KLS is equivalent to $c_k\le A^k$ with one universal $A$ (Proposition 7.1).
Compatible tensor field: A symmetric tensor field whose distributional derivative is fully symmetric, so that its entries satisfy the curl-free relations in Definition 11.2. BK’s argument integrates these fields repeatedly after choosing centered primitives. Its Hodge estimate controls the loss when weighted divergence is projected back onto compatible fields (Lemma 11.2); it is distinct from the moment-Hessian comparison Corollary 16.1.
Common integration prefactor: The factor $Q_D(B)$ in Lemma 11.3, independent of the power $k$ in $\|R^k\|^2\le B^kQ_D(B)$ . The BK proof uses finitely many polynomial observations to obtain this bound simultaneously for all powers. Replacing it by separate bounds on each integration would multiply the prefactor repeatedly and discard the gain.
Curvature profile: A function $F$ with $\CP(\nu)\le F(a)$ for every regular isotropic measure of curvature $a$ ( $aI\preceq D^2W$ ), in every dimension. Bakry–Émery gives $F(a)=1/a$ ; Theorem 7.3 gives iterated logarithms, and Theorem 7.4 turns a profile into a bound for every isotropic log-concave measure by localizing to curvature of order $1/\log(en)$ . That transfer gives the first-version bound Theorem 7.5 and the intermediate Theorem 9.2; the final argument of the second version, Theorem 9.3, does not use it, but transfers coefficient caps (Proposition 10.1) and passes to general measures by regular approximation. Section Notation and the smallest cases.
Conditional theorem: A statement that holds under a hypothesis which is itself not settled here, such as Proposition 21.1. Once proved it is a theorem, and it brings Conjecture 0.1 exactly as close as its hypothesis does: whether a statement is true and whether it applies are separate questions.