The moment map: CMH and the linear test - The KLS theorem and its methods

Chapter The moment map: the deterministic inequality said what normalizing the schematic estimate $\norm{\Sigma^{-1/2}H\nabla g}_2^2\le4\norm{-Lg}_2^2$ requires (Remark 15.2): every object in it fixed, so that the resulting statement implies a universal Poincaré bound. This chapter does so. The estimate is named, its operator data are fixed, the reduction to the affine Poincaré inequality is Theorem 16.1, and three structural consequences are recorded that were not visible while the endpoint was a schema:

an exact weighted Hodge decomposition showing that CMH dominates the affine Poincaré constant and contains an additional solenoidal channel in dimension at least two (Proposition 16.2, Corollary 16.1); universal $\mathrm{CMH}(4)$ implies KLS, and no converse is known (Chapter The moment map: the deterministic inequality);
a necessary condition, the linear test — the inequality tested on linear functions only, called gate zero in the statements of Corollary 16.2 and Proposition 17.2 — which is an operator-to-trace upgrade of exactly the kind catalogued in Remark 29.1, and which Theorem 4.2 does not supply by matrix algebra alone (Conjecture 16.1, Proposition 16.3). Its natural sharp form, the operator version of the Chen–Klartag trace bound, is Conjecture 16.2, and the constant 4 is not the natural one for this sector;
an exact resolution of that sector. The quadratic form of the linear-test matrix $\E H^2$ splits as one plus a quarter of the squared directional third moment plus a named high-mode remainder (Lemma 16.2), so any constant in the linear test already contains a directional third-moment bound (Corollary 16.2) and the sharp form is at least as strong as the sharp third-moment estimate. The spectral resolution of the three matrices of the linear sector is Lemma 16.1.

Throughout, $\mu$ is centered, full-dimensional and log-concave with covariance $\Sigma=\Cov_\mu\succ0$ , and $\varphi$ , $\nu$ , $H$ are the moment-map data of (4.1)–(4.3) in Chapter Family 4: moment maps, Monge–Ampère, and Stein kernels, transported to target coordinates as in (4.19). Regularity is assumed only to justify pointwise calculation; Appendix-level conventions for the weighted divergence, the closed forms, and the affine covariance are collected in §Conventions, domains, and affine covariance.

The Stein generator and the CMH constant¶

Recall the affine Poincaré constant

\CPaff(\mu) =\sup_{f\in H^1(\mu),\,f\not\equiv\mathrm{const}} \frac{\Var_\mu f}{\E_\mu\inner{\Sigma\nabla f}{\nabla f}},

(16.1)

so that Conjecture 0.1 is the assertion that $\CPaff$ is bounded by a universal constant over all dimensions and all log-concave $\mu$ .

Transporting the source diffusion (4.7) to the target gives the Stein generator

L_\mu g=\Div_\mu(H\nabla g)=\Tr(HD^2g)-p\cdot\nabla g, \qquad \Div_\mu u=\rho^{-1}\Div(\rho u),

(16.2)

where $\rho$ is the density of $\mu$ . With the sign convention that $\Aop=-L_\mu\ge0$ , the associated Dirichlet form is

-\E_\mu[fL_\mu g]=\E_\mu\inner{H\nabla f}{\nabla g}.

(16.3)

That $L_\mu$ is symmetric for $\mu$ — rather than for some auxiliary measure — is exactly the Stein-kernel identity $\Div_\mu H=-p$ of (4.20); it is what makes the following definition canonical rather than a choice.

Proof. One integration by parts in target coordinates against the Stein identity, on a smooth compactly supported core, followed by a graph-norm Cauchy argument that carries the identity to the operator-core closure. The calculation is carried out in Appendix The moment map: appendix.

For $\mu$ Gaussian, $H=\Id$ and (16.4) specializes to the familiar Ornstein–Uhlenbeck identity $\E(Lg)^2=\E[\abs{\nabla g}^2+\norm{D^2g}_{\HS}^2]$ . This is a calibration of the formula; the proof above is analytic and uses no numerical evidence.

Definition 16.1 (The canonical moment-Hessian constant — def:cmh · Correction)

The canonical moment-Hessian constant of $\mu$ is

\CMH(\mu) =\sup_{g\in\Dom(\Aop)\setminus\ker\Aop} \frac{\E_\mu\inner{H\nabla g}{\Sigma^{-1}H\nabla g}}{\E_\mu(L_\mu g)^2}.

(16.5)

The assertion $\CMH(\mu)\le C$ is written $\mathrm{CMH}(C)$ . Here $L^2$ means $L^2(\mu)$ , $\Aop=-L_\mu$ is the nonnegative self-adjoint operator of the closed form (16.3), and every inverse is the pseudoinverse on $(\ker\Aop)^\perp$ . For a measure supported on a proper affine subspace — the simplex laws of Chapter The moment map: exact cases are the case of interest — $\Sigma^{-1}$ denotes the inverse on that affine tangent space, equivalently the Moore–Penrose inverse in ambient coordinates.

Writing $K=-\Div_\mu(H\Sigma^{-1}H\nabla\,\cdot\,)$ , the definition is the quadratic-form comparison $K\preceq C\Aop^2$ , equivalently the second-order Riesz-transform bound

\norm{\Sigma^{-1/2}H\nabla\Aop^{-1}}_{L^2(\mu)\to L^2(\mu;\R^n)}\le\sqrt C.

(16.6)

This is the precise content of the schema (15.2): $\Sigma$ is the covariance, $L$ is the Stein generator (16.2), the measure is $\mu$ itself, and the admissible class is $\Dom(\Aop)$ .

CMH implies the affine Poincaré inequality¶

Proof. Test the quotient on centered functions in $\R+C_c^\infty$ , a class that lies in both form domains because the Stein normalization $\E_\mu H=\Sigma$ bounds the $H$ -energy by $\Tr\Sigma$ ; then a two-parameter cut-off and a density step in $H^1_\Sigma(\mu)$ . The density step is where care is needed, and it never asserts that finite $\Sigma$ -energy forces finite $H$ -energy when $H$ is unbounded. The calculation is carried out in Appendix The moment map: appendix.

The proof uses no Bochner identity, no positivity of the Monge–Ampère remainder, and no property of $H$ beyond symmetry, positivity, and $\Div_\mu H=-p$ . That economy is the reason the constant transfers with no loss: the reduction is exactly one Cauchy–Schwarz.

The Hodge content: the affine channel and the solenoidal excess¶

Let

\Aop_1=-\Div_\mu(\Sigma\nabla\,\cdot\,)

(16.8)

denote the closed nonnegative covariance generator on $L^2(\mu)$ , and write $L^2_0(\mu)=(\ker\Aop_1)^\perp$ .

Proposition 16.2 (Weighted Hodge identity — Proved · agent review (gpt-6-astra, 2026-10-04) · prop:cmh-hodge · Correction)

Let $g\in\Dom(\Aop)$ be such that $u=H\nabla g\in L^2(\mu;\Sigma^{-1})$ , and set

h=\Aop g=-\Div_\mu u\in L^2_0(\mu),\qquad \psi=\Aop_1^{-1}h,\qquad w=u-\Sigma\nabla\psi,

(16.9)

where

\Aop_1^{-1}:L^2_0(\mu)\longrightarrow \Dom(\Aop_1)\cap L^2_0(\mu)

(16.10)

is the inverse on the centered subspace. Then $\Div_\mu w=0$ , the fields $\Sigma\nabla\psi$ and $w$ are orthogonal in $L^2(\mu;\Sigma^{-1})$ , and

\E_\mu\inner{u}{\Sigma^{-1}u} =\E_\mu\inner{\Sigma\nabla\psi}{\nabla\psi} +\E_\mu\inner{w}{\Sigma^{-1}w}.

(16.11)

Moreover,

\sup_{0\ne h\in L^2_0(\mu)} \frac{\E_\mu\inner{\Sigma\nabla\Aop_1^{-1}h} {\nabla\Aop_1^{-1}h}} {\norm h_2^2} =\CPaff(\mu).

(16.12)

Corollary 16.1 (The exact Hodge consequence — Proved · agent review (gpt-6-astra, 2026-10-04) · cor:cmh-hodge-comparison · Correction)

$\Sigma\nabla\psi$ is the least- $L^2(\Sigma^{-1})$ field with divergence $-h$ , and by Proposition 16.2 controlling it is the affine Poincaré inequality. CMH demands in addition that the solenoidal excess $w$ be paid for within the same budget.

In dimension one a square-integrable divergence-free field with the no-flux convention of §Conventions, domains, and affine covariance vanishes identically, so $w=0$ and CMH degenerates exactly to the Poincaré inverse-divergence problem — this is why Theorem 17.1 is an identity rather than an inequality. In every dimension the decomposition proves the exact comparison

\CMH(\mu)\ge\CPaff(\mu).

(16.13)

In dimension at least two the divergence-free subspace is nontrivial, so the identity displays an additional channel that CMH must control.

Whether the solenoidal channel is ever active is not known: no measure separating $\CMH$ from $\CPaff$ is known, and the products of one-sided exponentials, which saturate $\mathrm{CMH}(4)$ with zero slack (Corollary 17.5), turn the question into a perturbative test.

The linear test: the necessary linear-sector condition¶

Take $g(p)=a\cdot p$ in (16.5). Then $D^2g=0$ , $L_\mu g=-a\cdot p$ , so $\E(L_\mu g)^2=a^\top\Sigma a$ while the numerator is $a^\top\E[H\Sigma^{-1}H]a$ . Hence:

This is necessary for universal $\mathrm{CMH}(4)$ and is not a known consequence of KLS. It is the cheapest falsifiable consequence of the whole mechanism: a single matrix expectation, with no test function and no operator inverse. A measure with $\lmax(\Sigma^{-1/2}\E[H\Sigma^{-1}H]\Sigma^{-1/2})>4$ would disprove $\mathrm{CMH}(4)$ .

The trace bound is sharp — products of centered exponentials give $\Tr(\E H^2)=2n$ exactly — so the constant 4 in (16.14) is not the natural one for the linear sector. The natural statement is the operator form of the Chen–Klartag inequality.

Conjecture 16.2 refines Conjecture 16.1 and implies it, and its trace is exactly Theorem 4.3. Its equality set is not small: products of centered exponentials attain (16.15) in every direction, and every exponential cone measure of Section Exponential cones: a second solvable non-product family attains it in its axis direction (Proposition 17.2), with the constant strictly below 2 on the same family as soon as the radial exponent moves off the cone value. The constant 2 is half the CMH constant 4: on the line, $\E\tau^2=2$ for the centered exponential while $\CMH=4$ by Theorem 17.1, so the linear sector saturates at half the full constant. By Corollary 16.2 below, the sharp form would improve the directional third-moment bound of Proposition 26.1 from $\kappa_n\le2\sqrt2$ to the sharp $\kappa_n\le2$ ; it is therefore at least as strong as a sharp third-moment estimate, which calibrates its difficulty.

The three matrices of the linear sector can be resolved spectrally. In isotropic source coordinates write $Q_{\rm lin}(\mu):=\lmax(\mathsf N)$ for the linear CMH quotient, $\mathsf N=\int H^2\,d\eta$ , and write $(\mathrm{AB})_{\rho,\beta}$ for the inequality $\mathsf R\succeq\rho\mathsf N-\beta I$ , called the bootstrap in the lemma below, a lower bound on the remainder matrix $\mathsf R$ by a fraction of $\mathsf N$ , with $\mathsf R=\mathsf N-\mathsf D$ as in the lemma below; its sharp form is $(\rho,\beta)=(\tfrac12,0)$ , which is equivalent to Conjecture 16.2 together with the high-mode excess bound recorded in the lemma. In the lemma, $A=H\,(D^2V\circ\nabla\psi)\,H$ and $Q_{k\ell}=\Tr(H^{-1}\partial_kH\,H^{-1}\partial_\ell H)$ are the two nonnegative terms of (4.9), and $\sum_bM_{ab}u_b$ abbreviates $\sum_b(M_a)_{\cdot b}u_b$ .

Lemma 16.1 (Spectral resolution of the linear-sector matrices — Proved · agent review (2026-09-06) · lem:cmh-linear-spectral-resolution · Correction)

For a regular isotropic compact-target moment map with source potential $\psi$ , source density $e^{-\psi}$ , and Stein generator $\mathsf A_{\rm op}=-\mathcal L$ in source coordinates, where $\mathcal Lf=e^{\psi}\operatorname{div}(e^{-\psi}H^{-1}\nabla f)$ and $H=D^2\psi$ : the functions $u_b=\partial_b\psi$ are centered orthonormal eigenfunctions of $\mathsf A_{\rm op}$ at its Brascamp–Lieb spectral gap 1; the columns $Ha=\nabla\langle a,\nabla\psi\rangle$ lie in the form domain $\Dom(\calE)$ and satisfy the column equation in the weak form $\calE(f,(Ha)_i)=\langle f,(Ha)_i\rangle-\langle f,((A+Q)a)_i\rangle$ for every bounded smooth finite-energy $f$ , with $(A+Q)a\in L^1(\eta)$ orthogonal to every $u_b$ and $\int(A+Q)\,d\eta=I$ ; and, with the induced decomposition $Ha=a+\sum_bM_{ab}u_b+v$ , the linear-sector matrices $\mathsf N=\int H^2$ , $\mathsf D=\int H^{ab}(\partial_aH)(\partial_bH)$ , $\mathsf R=\mathsf N-\mathsf D$ resolve as

a^T\mathsf Na=1+\norm{M_a}^2+\norm v^2,\qquad a^T\mathsf Da=\norm{M_a}^2+\langle v,\mathsf A_{\rm op}v\rangle,\qquad a^T\mathsf Ra=1-\langle v,(\mathsf A_{\rm op}-1)v\rangle ,

(16.16)

where $\langle v,\mathsf A_{\rm op}v\rangle$ denotes the closed-form value $\calE(v,v)$ . Consequently $\mathsf R\preceq I$ , with equality in a direction exactly when the column fluctuation is spectrally pure at the gap; the bootstrap $(\mathrm{AB})_{\rho,\beta}$ is equivalent to the channel inequality $\langle v,(\mathsf A_{\rm op}-1)v\rangle+\rho(\norm{M_a}^2+\norm v^2)\le1+\beta-\rho$ for all unit $a$ ; $\mathsf R\succeq\mathsf N-nI$ holds unconditionally, so $Q_{\rm lin}\le n+1$ ; and $\mathsf R\succeq\mathsf D$ holds for every regular finite product of one-dimensional laws.

The gap-mode part of the column decomposition above has a description that needs no source coordinates at all. Write

T_3(\mu)=\bigl(\E_\mu[X_iX_jX_k]\bigr)_{i,j,k}, \qquad T_3(a)=\E_\mu\bigl[\inner Xa\,X\otimes X\bigr],

(16.17)

for the third-moment tensor and its contraction with $a\in\R^n$ , so that the directional third-moment parameter of (0.13) is $\kappa_n=\sup_{\mu,\theta}\norm{T_3(\theta)}_{\HS}$ .

Lemma 16.2 (The linear sector and the third moment — Proved · agent review (2026-09-06) · lem:linear-sector-third-moment · Correction)

Let $\mu$ be an isotropic log-concave probability on $\R^n$ which is the moment measure of a convex $\varphi\in C^2(\R^n)$ with $\E_\nu\norm{D^2\varphi}_{\HS}^2<\infty$ , and let $\tau=\tau_\mu$ be its canonical Stein kernel (4.19). Then for every $a\in\R^n$ the column $\tau a$ decomposes orthogonally in $L^2(\mu;\R^n)$ as

\tau a=a+\tfrac12\,T_3(a)\,X+v_a, \qquad \E_\mu[v_a]=0,\qquad \E_\mu[v_a\otimes X]=0,

(16.18)

and consequently

\E_\mu\abs{\tau a}^2=\abs a^2+\tfrac14\norm{T_3(a)}_{\HS}^2+\E_\mu\abs{v_a}^2 .

(16.19)

Equivalently, $\E_\nu[\partial_{ij}\varphi\,\partial_k\varphi]=\tfrac12\,\E_\mu[X_iX_jX_k]$ ; when moreover $\varphi\in C^3$ with $D^3\varphi\in L^1(\nu)$ , this says that the gap-mode coefficient tensor $\E_\nu[\partial_{ijk}\varphi]$ of Lemma 16.1 is one half of the third-moment tensor.

The identity $\E_\nu[\partial_{ijk}\varphi]=\tfrac12\E_\mu[X_iX_jX_k]$ is Lemma 3.7 of Chen & Klartag, 2026, and the orthogonal decomposition above is the exact form of the Bessel step in the proof of their Theorem 1.2; what is added here is the directional statement with the remainder $v_a$ named. The compact-target regular class of Theorem 4.1 satisfies the hypothesis, since its Hessian is bounded Klartag, 2014; so does every exponential cone measure of Section Exponential cones: a second solvable non-product family, whose kernel (17.26) is the Gamma radial variable times a matrix bounded by the same result applied to the base. For a symmetric $\mu$ the third moment vanishes and the entire content of the linear test is the high-mode term $\E\abs{v_a}^2$ ; for a product of centered exponentials the high-mode term vanishes and the entire content is the third moment.

Corollary 16.2 (The linear test controls the directional third moment — Proved · agent review (2026-09-06) · cor:gate-zero-third-moment · Correction)

Under the hypotheses of Lemma 16.2, for every unit vector $a$ ,

\norm{T_3(a)}_{\HS}^2 \le4\bigl(a^\top\E_\mu[\tau^2]\,a-1\bigr) \le4\bigl(\lmax(\E_\mu\tau^2)-1\bigr).

(16.20)

In particular, on any class of such measures with $\E\tau^2\preceq c\,\Id$ , every directional third moment satisfies $\norm{T_3(a)}_{\HS}\le2\sqrt{c-1}$ : the linear test (16.14) gives $2\sqrt3$ and the sharp form (16.15) gives 2, which products of centered exponentials attain. Conversely, any isotropic law in the class with $\norm{T_3(a)}_{\HS}>2\sqrt3$ for some unit $a$ refutes the linear test Conjecture 16.1.

Summing the corollary over an orthonormal basis recovers $\norm{T_3(\mu)}_{\HS}^2\le4(\Tr\E\tau^2-n)\le4n$ , the chain in the proof of Theorem 1.2 of Chen & Klartag, 2026. Corollary 16.2 places the linear test relative to the literature input it does not use: the linear test at constant $c$ contains a directional third-moment bound at $2\sqrt{c-1}$ , so any proof of (16.15) proves a sharper constant than Theorem 25.1 supplies, and any proof of (16.14) must at least reproduce a bound of that type. The reverse channel is what makes the corollary a falsification tool: it is a lower bound on the linear-test matrix that needs no moment map, only third moments.

Why the constant-matrix estimate cannot supply the linear test¶

In isotropic position define the positive self-adjoint superoperator $\calT(B)=\E[HBH]$ on symmetric matrices with the Hilbert–Schmidt inner product. Theorem 4.2 says $\calT\preceq2\,\Id$ ; the linear test asks instead for $\calT(\Id)=\E H^2\preceq4\,\Id$ . The two differ by the order of the noncommuting factors, and the gap is exactly one commutator:

\Tr(B^2H^2)=\Tr(BHBH)+\tfrac12\norm{[B,H]}_{\HS}^2 \qquad(B,H\ \text{symmetric}).

(16.21)

Indeed $[B,H]$ is antisymmetric, so $\norm{[B,H]}_{\HS}^2=-\Tr([B,H]^2)=2\Tr(B^2H^2)-2\Tr(BHBH)$ . The first term of (16.21) is what Letwin’s theorem controls; the second is the static transverse commutator, and the following shows it is genuinely unconstrained by matrix moments.

Proof 16.2

Let $z$ be uniform on $S^{m-1}$ and set

d=\frac m{\sqrt{2m-1}},\qquad c=1-\frac dm,\qquad H(z)=\begin{pmatrix}1&\sqrt d\,z^\top\\\sqrt d\,z&c\,\Id_m+d\,zz^\top\end{pmatrix}.

(16.23)

The Schur complement of the upper-left entry is $c\,\Id_m\succ0$ because $c>0$ , so $H\succeq0$ ; since $\E z=0$ and $\E zz^\top=\Id_m/m$ we get $\E H=1\oplus(c+d/m)\,\Id_m=\Id_{m+1}$ .

Write $B=\left(\begin{smallmatrix}a&r^\top\\r&D\end{smallmatrix}\right)$ with $D$ symmetric. Using $\E(z^\top Dz)^2=\bigl((\Tr D)^2+2\Tr(D^2)\bigr)/(m(m+2))$ ,

\begin{aligned} \E\Tr(BHBH) &=a^2+\frac{2d}ma\Tr D+2\Bigl(c+\frac{2d}m\Bigr)\abs r^2\\ &\quad+\Bigl(c^2+\frac{2cd}m+\frac{2d^2}{m(m+2)}\Bigr)\Tr(D^2) +\frac{d^2}{m(m+2)}(\Tr D)^2. \end{aligned}

(16.24)

Writing $D=t\,\Id_m+D_0$ with $\Tr D_0=0$ splits the form into three $O(m)$ -invariant sectors that do not interact: the cross-vector sector, the traceless- $D$ sector, and the two-dimensional scalar sector $(a,t)$ . Comparing with $2\Tr(B^2)=2a^2+4\abs r^2+2\Tr(D^2)$ sector by sector: the vector coefficient is $2(c+2d/m)=2(1+d/m)\le4$ since $d\le m$ ; the traceless coefficient is $c^2+2cd/m+2d^2/(m(m+2))=1+(m-2)/((2m-1)(m+2))\le2$ ; and on the scalar sector, using $mc^2+2cd+d^2=m+d^2-d^2/m$ , the deficit is

2(a^2+mt^2)-\E\Tr(BHBH)=a^2-2dat+\Bigl[m-d^2\Bigl(1-\frac1m\Bigr)\Bigr]t^2=(a-dt)^2\ge0,

(16.25)

the last equality because $d^2(2-1/m)=m$ by the choice $d^2=m^2/(2m-1)$ . Hence $\E\Tr(BHBH)\le2\Tr(B^2)$ for every symmetric $B$ , with equality on the scalar ray $a=dt$ .

Finally $e_1^\top\E H^2e_1=1+d\abs z^2=1+d$ , and $1+d>4$ is equivalent to $m^2>9(2m-1)$ , i.e. $m^2-18m+9>0$ , which holds precisely for $m\ge18$ .

Conventions, domains, and affine covariance¶

Weighted divergence and boundary flux. $\Div_\mu u=\rho^{-1}\Div(\rho u)$ . All integrations by parts are justified first for compactly supported fields or fields with vanishing normal flux, and the closed forms are obtained by completion. The no-flux convention is essential in dimension one, where it is what rules out a nonzero constant divergence-free field and forces $w=0$ in Corollary 16.1.

Closed operators. $\calE_H(f,g)=\E_\mu\inner{H\nabla f}{\nabla g}$ is closable under the smooth moment-map hypotheses of §Moment-map coordinates; $\Aop=-L_\mu$ is its nonnegative self-adjoint operator, and $\Aop^{-1}$ always means the pseudoinverse on $(\ker\Aop)^\perp$ . The spectral truncation in the proof of Theorem 16.1 avoids assuming a spectral gap.

Affine covariance. If $T$ is invertible and $Y=TX$ , then $\Sigma_Y=T\Sigma_XT^\top$ and $H_Y(Tx)=TH_X(x)T^\top$ ; with $g_Y(y)=g_X(T^{-1}y)$ both the numerator and denominator of (16.5) are unchanged. Hence $\CMH$ is an invariant of the affine equivalence class, matching the affine invariance of $\CPaff$ . For a noninvertible map the transported Stein kernel need not be the canonical moment-map kernel of the image, so only the Poincaré-level statement of Corollary 17.1 is available there.

Conditional closure for general log-concave measures. The argument behind Proposition 15.1 runs at the Poincaré level as follows. For an arbitrary centered log-concave $\mu$ , centered Gaussian-convolution, Gaussian-tilt, and growing-ball approximants $\mu_q$ belong to the published compact-target regular class of Theorem 4.1 and converge to $\mu$ in ambient $W_2$ with second moments, hence $\Sigma_q\to\Sigma$ . If $\sup_q\CMH(\mu_q)\le C$ , then for every smooth compactly supported $f$ ,

\Var_{\mu_q}f\le C\int\inner{\Sigma_q\nabla f}{\nabla f}\,d\mu_q.

(16.26)

Both sides converge because $f$ and $\nabla f$ are bounded and continuous and the covariances converge. Intrinsic smooth-core density on $S=\operatorname{Ran}\Sigma$ extends the inequality to the closed covariance-form relaxation $H^1_\Sigma(\mu)$ , while $\Sigma$ and $\Sigma^+$ annihilate ambient normal derivatives. This gives constant-preserving closure, including singular covariance. It does not give the uniform premise and asserts no continuity of $\CMH$ .

Proposition 16.4 (CMH recovery calculus — Proved · agent review (2026-08-27) · prop:cmh-recovery-calculus · Correction)

For a centered log-concave probability $\mu$ in a declared ambient Euclidean space, define its CMH recovery envelope by

\mathfrak R_{\rm CMH}(\mu) :=\inf_{(\mu_k)} \liminf_{k\to\infty}\CMH(\mu_k),

(16.28)

where the infimum runs over centered, full-dimensional, compactly supported regular moment-map laws in that ambient space with $W_2(\mu_k,\mu)\to0$ . Then:

(i) $\CPaff(\mu)\le\mathfrak R_{\rm CMH}(\mu)$ ;

(ii) $\mathfrak R_{\rm CMH}$ is invariant under invertible linear maps, is nonincreasing under singular square linear limits in the same ambient dimension, and satisfies

\mathfrak R_{\rm CMH}(\mu_1\otimes\cdots\otimes\mu_m) \le\max_i\mathfrak R_{\rm CMH}(\mu_i);

(16.29)

(iii) an intrinsic recovery sequence on a proper affine support with envelope at most $C$ extends to an ambient full-dimensional recovery sequence with envelope at most $\max\{C,1\}$ by tensoring compact Gaussian normal factors whose scale vanishes.

Consequently the envelope equals 1 for centered Gaussian laws and equals 4 for finite products of one-dimensional log-concave laws containing a centered one-sided-exponential factor. It is at most 4 for invertible linear images, same-ambient singular square linear images, and affine-support embeddings of finite products built from one-dimensional log-concave laws and uniform-simplex blocks. This calculus does not assert a universal bound for arbitrary log-concave laws and therefore does not discharge Assumption 16.1.

References¶

Chen, Y., & Klartag, B. (2026). Digesting the Proof of the Sharp Thin-Shell Inequality.
Klartag, B. (2014). Logarithmically-Concave Moment Measures I. In Geometric Aspects of Functional Analysis (Vol. 2116, pp. 231–260). Springer. 10.1007/978-3-319-09477-9_16