The moment map: the deterministic inequality

Overview of the mechanism¶

The idea. Bound one deterministic quantity, the canonical moment-Hessian constant $\CMH$, and pass it to an affine Poincaré inequality. No stochastic localization anywhere.

What it would add. Beyond the three proofs: a deterministic argument for the Poincaré bound, with no stochastic localization, and the constant 4, which products of exponentials attain. The target inequality bounds the affine Poincaré constant with no loss of constant, and the passage to every log-concave law needs one further assumption:

The first arrow is Theorem 16.1. The second is Proposition 15.1, which assumes Assumption 15.1. Throughout this mechanism the endpoint is the inequality $\mathrm{CMH}(4)$ itself — the last statement of the chain, which everything else here serves to prove — and the endpoint reduction is the first arrow, from it to the affine Poincaré inequality.

The condition $\mathrm{CMH}(4)$ implies KLS, and no converse is known. By Proposition 16.2 it also charges a solenoidal excess, which vanishes identically in dimension one; no measure separating $\CMH$ from $\CPaff$ is known (Corollary 16.1).

What it builds on. From the literature: the fixed-matrix Hessian bound (Theorem 4.2) and the moment-Hessian trace bound (Theorem 4.3); moment-potential regularity from Berman & Berndtsson, 2013 and the Stein identity from Fathi, 2019. Set up in this manuscript: the endpoint’s operator data (Definition 16.1) and the statements of Chapters The moment map: CMH and the linear test–The moment map: exact cases.

What it gives. The endpoint is made precise and then computed. It is given operator data rather than left as a slogan (Definition 16.1); it dominates the affine Poincaré constant (Theorem 16.1); its Hodge content separates an affine channel from a solenoidal excess (Proposition 16.2); and it is evaluated exactly on the line, on products, and on every log-concave Dirichlet law (Chapter The moment map: exact cases). That last concerns a nontrivial family and is not a consistency check. The linear sector is resolved separately and exactly: it equals a directional third moment plus a named high-mode remainder (Lemma 16.2, Corollary 16.2), and the exponential cones of §Exponential cones: a second solvable non-product family give it an explicit non-product equality set with an explicit moment map (Proposition 17.1–Proposition 17.2). For product-simplex bases the full gate spectrum and its equality subspaces are determined by Proposition 17.3.

What blocks it. The mechanism splits into two layers, and the split is the point. A construction layer (Chapter The moment map: construction): Conjecture 18.1, deriving the target-flat Schur–Piola multiplier lift invariantly — the lift is the tensor through which a Haar multiplier, defined on one block of a Schur split of the Hessian, acts on the whole space, so far computed in coordinates and in low split dimensions only — and Conjecture 18.2, controlling the full Haar sum of $[N^{1/2},K_M]$ errors without double-spending the positive reservoir. A falsification layer: the linear test of Section The linear test: the necessary linear-sector condition (Conjecture 16.1), its sharp form (Conjecture 16.2), and the solenoidal perturbation test Conjecture 17.1. The first pair would complete the argument; a negative answer in the second would rule it out. Of the two linear tests, only the one with constant 4 is necessary for $\mathrm{CMH}(4)$.

What fails, and why. Proposition 16.3 is the decisive negative result: no matrix-moment argument supplies the linear test of the moment-Hessian inequality (Conjecture 16.1), so the fixed-matrix estimate cannot be leveraged into the linear sector. The linear test is itself an instance of the average-versus-uniform pattern of Section Obstacles for alternative arguments — it asks $\lmax(\E H^2)\le4$ where only $\Tr(\E H^2)\le2n$ is known — so this mechanism relocates that difficulty rather than escaping it, and Proposition 16.3 shows the relocation is not free. A separate saturation risk is discussed in Chapter The moment map: exact cases.

What would settle it. The mechanism succeeds if the construction layer is carried out and Assumption 15.1 is established, or replaced by the weaker recovery-envelope assumption Assumption 16.1, which suffices by Corollary 16.3. It fails if the linear test Conjecture 16.1 is false, or if some admissible log-concave perturbation of the saturating one-sided-exponential product has strictly positive second variation of $\CMH$ (Conjecture 17.1), which would push $\CMH$ above 4. The perturbative test first requires a two-sided admissible family in the precise source-potential ansatz; changing the target density potential is a different question (Section The saturation risk and the decisive test).

How to read it. In the order of the chapters. This one presents the mechanism and its first necessary condition; Chapter The moment map: CMH and the linear test makes the target precise; Chapter The moment map: exact cases computes it exactly where that is possible; Chapter The moment map: construction, the construction layer, attempts to prove it. The two layers share the target $\mathrm{CMH}(4)$ and almost no machinery. The moment-map family survey is Chapter Family 4: moment maps, Monge–Ampère, and Stein kernels. None of the localization apparatus of the shared technical foundations is used here; its longer calculations are in Appendix The moment map: appendix.

The endpoint and a first necessary condition¶

The starting point is a covariance–moment–Hessian estimate of the schematic form

A divergence-duality argument then gives $\CP(\mu)\le4$, the sharp plausible constant because a standard centered one-sided exponential has Poincaré constant 4.

This regular-class question is logically prior to the commutator calculation. Definition 16.1 fixes $\Sigma$ as the covariance, $L$ as the Stein generator $\Div_\mu(H\nabla\,\cdot\,)$, the $L^2$ space as $L^2(\mu)$, the admissible class as $\Dom(\Aop)$, and every inverse as the pseudoinverse on $(\ker\Aop)^\perp$; Theorem 16.1 is the reduction $\CPaff(\mu)\le\CMH(\mu)$, by a single Cauchy–Schwarz step, with a spectral truncation in place of an assumed gap.

The approximation closure Proposition 15.1 is conditional on Assumption 15.1. Centered Gaussian-convolution, Gaussian-tilt, and growing-ball approximants converge in ambient $W_2$, and the affine Poincaré inequality passes with no loss on the intrinsic closed covariance-form domain, including proper affine-support degeneration. The closure argument supplies no bound on $\CMH$: the uniform hypothesis $\sup_k\CMH(\mu_k)\le C$ is exactly what it assumes.

The explicit normalization gives the endpoint a cheap necessary condition, which bears directly on the construction of Chapter The moment map: construction. Testing $\mathrm{CMH}(4)$ on linear functions gives $\E[H\Sigma^{-1}H]\preceq4\Sigma$, the linear test of Section The linear test: the necessary linear-sector condition (Conjecture 16.1), and Proposition 16.3 shows by an exact countermodel that Theorem 4.2 does not imply it through matrix algebra alone. The obstruction there is the static commutator (16.21), $\Tr(B^2H^2)=\Tr(BHBH)+\tfrac12\norm{[B,H]}_{\HS}^2$ — the finite-dimensional shadow of the square-root commutator of Conjecture 18.2. An argument that controls $[N^{1/2},K_M]$ must in particular control $\E\norm{[B,H]}_{\HS}^2$; conversely, a counterexample to the linear test would rule out this approach without any Haar-tree analysis at all.