The moment map: exact cases - The KLS theorem and its methods

Chapter The moment map: CMH and the linear test fixed the CMH estimate and showed that it dominates the affine Poincaré constant (Theorem 16.1). This chapter computes it — or, in the fourth class, its linear sector — exactly, in the four classes where it is tractable. Two of them — the line and products — are the expected calibrations, and they already pin the constant: $\mathrm{CMH}(4)$ holds there and no smaller universal constant is possible. The third is the first genuinely nonproduct family on which the approach has an exact result: every log-concave Dirichlet law satisfies $\mathrm{CMH}(4)$ (Theorem 17.3). Its proof is a homogeneous lift to independent Gamma variables — a function on the simplex is rewritten as a function of independent Gamma variables, homogeneous of degree zero, where the generator is that of a product — followed by a sharp Hessian-row minimization; in the Dirichlet argument the log-concavity hypothesis is used only in the final scalar angular minimization.

The fourth class is solvable in a weaker but more pointed sense. The exponential cones of §Exponential cones: a second solvable non-product family attach a Gamma radial variable to an arbitrary centered base, and their moment map is explicit in terms of the base’s own (Proposition 17.1); unlike the first three classes they are not compactly supported and, for a general base, not affine images of products. What is computed exactly on them is the linear sector rather than the full constant: the axis value of the linear test is $1+n/\beta$ , so every cone with $\beta=n$ saturates the sharp linear test (Conjecture 16.2) in its axis direction (Proposition 17.2), and over any product of simplices the entire linear-test matrix is a closed form bounded by 2 (Proposition 17.3). They are the first non-product equality set the sharp linear sector has, which is what makes them a constraint on any argument for it.

The chapter closes at the exact product endpoint, the extreme case where the inequality holds with equality: centered one-sided exponentials saturate $\mathrm{CMH}(4)$ with zero slack. Whether a log-concave perturbation raises the full CMH Rayleigh quotient is a separate second-variation problem, Conjecture 17.1; changing its solenoidal term alone is not decisive (Corollary 17.5). The cone family bears on where to look for such a perturbation, and §Exponential cones: a second solvable non-product family reports what a first computation over it suggests.

The line¶

Let $\mu(\dd x)=\rho(x)\dd x$ be centered on an interval $(\ell,r)$ with variance $\sigma^2$ . Its canonical Stein kernel is the unique zero-flux solution of $(\tau\rho)'=-x\rho$ ,

\tau(x)=-\frac1{\rho(x)}\int_\ell^xt\rho(t)\dd t,

(17.1)

and it coincides with the moment-map Hessian in target coordinates. The Stein generator is $L_\mu f=\tau f''-xf'=\rho^{-1}(\tau\rho f')'$ , so that $h=-L_\mu f$ has

\tau(x)f'(x)=-\frac1{\rho(x)}\int_\ell^xh(t)\rho(t)\dd t.

(17.2)

Proof 17.1

Let $D$ be the closed maximal derivative on $L^2(\mu)$ and $D_\mu^*$ its no-flux adjoint, and let $\Aop=-L_\mu$ be defined by the natural weighted form $\int\tau f'q'\,d\mu$ . We compare the two constants by duality, allowing either to be infinite. The CMH numerator is the extended integral $\norm{\tau f'}_2^2/\sigma^2$ .

The compact flux fields $u=\phi/\rho$ , with $\phi\in C_c^\infty((\ell,r))$ , form a graph core for $D_\mu^*$ : the adjoint of $u\mapsto-\phi'/\rho$ on these fields is precisely the maximal derivative $D$ . Each field is attained by the bounded test

f_\phi(x)=\int_{x_0}^x\frac{\phi(t)}{\tau(t)\rho(t)}\,dt, \qquad \tau f_\phi'=u,\qquad \Aop f_\phi=D_\mu^*u=-\frac{\phi'}\rho.

(17.4)

Indeed, compactly supported integration by parts gives the weighted-form identity against every form test, proving $f_\phi\in\Dom(\Aop)$ . Thus, if $M=\sigma^2\CMH(\mu)<\infty$ , the bound $\norm u_2^2\le M\norm{D_\mu^*u}_2^2$ extends from this core to the whole adjoint domain. Its range is then closed and, since $\ker D$ consists of constants, equals $L^2_0(\mu)$ . For centered $v\in\Dom D$ , choose $D_\mu^*u=v$ ; the identity $\norm v_2^2=\inner{Dv}{u}$ gives $\norm v_2\le\sqrt M\norm{Dv}_2$ . Hence $\CP\le M$ , and in particular $\CP=\infty$ forces $\CMH=\infty$ .

Conversely, suppose $\CP<\infty$ and put $h=\Aop f$ for any $f\in\Dom(\Aop)$ . Poincaré bounds the functional $Dq\mapsto\inner hq$ by $\sqrt{\CP}\norm h_2\norm{Dq}_2$ . Riesz representation gives $w\in L^2(\mu)$ with $\inner w{Dq}=\inner hq$ and $\norm w_2\le\sqrt{\CP}\norm h_2$ . Test this identity and the weighted-form identity with bounded primitives of smooth compactly supported functions. Their derivatives are arbitrary interior tests, so $w=\tau f'$ almost everywhere. This establishes square integrability of the flux as well as $\norm{\tau f'}_2^2\le\CP\norm{\Aop f}_2^2$ . Together the two bounds prove (17.3), including infinite constants.

The one-dimensional specialization of the Kannan–Lovász–Simonovits bound Kannan et al., 1995, recorded for example in Cattiaux & Guillin, 2018, Eq. (2.25) with that attribution, is $\CP(\mu)\le4\Var(\mu)$ and hence gives $\mathrm{CMH}(4)$ .

For sharpness, let $X=Y-1$ with $Y\sim\mathrm{Exp}(1)$ , so that $X$ is centered, has density $e^{-(x+1)}\one_{x\ge-1}\dd x$ , and has variance 1. For $0<a<\tfrac12$ , the test function $f_a(x)=e^{a(x+1)}$ satisfies

\frac{\Var(f_a(X))}{\E\abs{f_a'(X)}^2} =\frac{(1-2a)^{-1}-(1-a)^{-2}}{a^2(1-2a)^{-1}} =\frac1{(1-a)^2}\longrightarrow4.

(17.5)

Thus $\CP(X)=4$ by the preceding upper bound, and (17.3) gives $\CMH(X)=4$ .

Theorem 17.1 is an identity, not an inequality: on the line CMH carries exactly the information of the affine Poincaré inequality and nothing more. This is Corollary 16.1 seen from the other side — the solenoidal channel is empty in dimension one.

Products¶

Let $\mu=\bigotimes_{i=1}^m\mu_i$ be a product of centered factors for which the canonical CMH data are defined. Then $\Sigma=\diag(\Sigma_1,\dots,\Sigma_m)$ , $H=\diag(H_1,\dots,H_m)$ , and $L_\mu=\sum_iL_i$ , where the $L_i$ are commuting nonpositive generators acting in their respective blocks.

Proof 17.2

Write $\nabla_i$ for the gradient in the $i$ th block. Conditioning on all other blocks and using the definition of $\CMH(\mu_i)$ gives

\E\inner{H_i\nabla_i g}{\Sigma_i^{-1}H_i\nabla_i g} \le \CMH(\mu_i)\,\E(L_i g)^2.

(17.7)

For $i\ne j$ , two integrations by parts give the block cross-term identity

\E[(L_ig)(L_jg)] =\E\Tr\!\left(H_iD^2_{ij}g\,H_jD^2_{ji}g\right) =\E\norm{H_i^{1/2}D^2_{ij}g\,H_j^{1/2}}_{\HS}^2\ge0,

(17.8)

whence $\sum_i\E(L_ig)^2\le\E\bigl(\sum_iL_ig\bigr)^2=\E(L_\mu g)^2$ . Summing the conditional estimates proves “ $\le$ ” in (17.6); testing on functions of a single block proves “ $\ge$ ”. The one-dimensional formula now follows from Theorem 17.1, and the final affine-image claim uses only the invertible affine invariance proved in §Conventions, domains, and affine covariance.

Note that Corollary 17.1 is a statement about $\CPaff$ , which is stable under noninvertible maps; the canonical CMH constant itself is claimed invariant only under invertible affine maps (§Conventions, domains, and affine covariance).

The Dirichlet family: geometry and normalization¶

Let $P\sim\Dir(\alpha_1,\dots,\alpha_m)$ with $\alpha_i\ge1$ , which is exactly the log-concave parameter range, and put $A=\sum_i\alpha_i$ , $q_i=\alpha_i/A$ . Write

C(p)=\diag(p)-pp^\top, \qquad L_\alpha g=\Tr\bigl(C(p)D^2g\bigr)+\inner{\alpha-Ap}{\nabla g}

(17.9)

for the Wright–Fisher generator; both expressions are independent of the ambient extension of $g$ because $C(p)\one=0$ and $\sum_i(\alpha_i-Ap_i)=0$ .

On $\R^m/\mathrm{span}\{\one\}$ consider $\varphi(y)=A\log\bigl(\sum_ie^{y_i/A}\bigr)-q\cdot y$ . Its gradient is $p-q$ with $p_i=e^{y_i/A}/\sum_je^{y_j/A}$ , its Hessian is $C(p)/A$ , and the softmax Jacobian identifies the pushforward of $e^{-\varphi}$ with the centered law $P-q$ . Hence the canonical moment-map data are

H(p)=\tfrac1AC(p),\qquad L_\mu=\tfrac1AL_\alpha, \qquad \Sigma=\frac1{A(A+1)}\Bigl(\diag(\alpha)-\frac{\alpha\alpha^\top}A\Bigr).

(17.10)

For every tangent vector $v$ , meaning $\sum_iv_i=0$ ,

v^\top\Sigma^\dagger v=A(A+1)\sum_i\frac{v_i^2}{\alpha_i},

(17.11)

because $x=A(A+1)\diag(\alpha)^{-1}v$ satisfies $\Sigma x=v$ exactly, and the remaining $\ker\Sigma=\mathrm{span}\{\one\}$ ambiguity pairs to zero against $v$ .

Setting

u_i(p)=\bigl(C(p)\nabla g(p)\bigr)_i, \qquad d_\alpha(g)=\E\sum_i\frac{u_i^2}{\alpha_i}, \qquad n_\alpha(g)=\E(L_\alpha g)^2,

(17.12)

equations (17.10)–(17.11) turn Definition 16.1 into the statement $A(A+1)d_\alpha(g)\le4n_\alpha(g)$ .

The independent Gamma lift¶

Let $Y_i\sim\GammaLaw(\alpha_i,1)$ be independent, $S=\sum_iY_i\sim\GammaLaw(A,1)$ and $P_i=Y_i/S$ ; then $P\sim\Dir(\alpha)$ and $S\perp P$ . Lift $g$ homogeneously of degree zero by $G(Y)=g(Y/S)$ . The product-Gamma (Laguerre) generator is $\calL_\Gamma G=\sum_i\bigl(Y_iG_{ii}+(\alpha_i-Y_i)G_i\bigr)$ . Proposition 16.1, specialized to the Gamma moment Hessian $H_\Gamma=\diag(Y_i)$ , gives the integrated Bochner identity

N_\Gamma(G):=\E(\calL_\Gamma G)^2 =\E\Bigl[\sum_iY_iG_i^2+\sum_{i,j}Y_iY_jG_{ij}^2\Bigr].

(17.14)

Put $D_i(G)=\E[Y_i^2G_i^2]/\alpha_i$ and $D_\Gamma(G)=\sum_iD_i(G)$ .

Proof. One integration by parts for the Gamma law, applied coordinatewise and summed, which recovers $N_\Gamma$ and leaves an explicit negative multiple of $D_i$ . The calculation is carried out in Appendix The moment map: appendix.

Proof. Euler’s identity, differentiated once, leaves a single linear constraint; the row minimum is then the elementary minimum of a weighted sum of squares subject to one linear constraint. The calculation is carried out in Appendix The moment map: appendix.

Homogeneity also gives the exact dictionary between the lift and the simplex: $\calL_\Gamma G=S^{-1}L_\alpha g$ and $Y_iG_i=u_i(P)$ . With $A>2$ one has $\E S^{-1}=(A-1)^{-1}$ and $\E S^{-2}=z_A^{-1}$ where

z_A=(A-1)(A-2),

(17.18)

so independence of $S$ and $P$ yields

N_\Gamma(G)=\frac{n_\alpha(g)}{z_A}, \qquad D_\Gamma(G)=d_\alpha(g),

(17.19)

and, substituting $Y_i=SP_i$ and $G_i=u_i/(SP_i)$ into (17.17),

\E R_i\ \ge\ \E_P\,u_i^2 \Bigl[\frac1{z_AP_i}+\frac2{(A-1)(\alpha_i+1)}+\frac{P_i}{(\alpha_i+1)^2}\Bigr].

(17.20)

The scalar angular inequality and the proof¶

Proof. Minimize in $p$ first — the minimizer is interior only when $\sqrt z>a+1$ — and then check monotonicity in $a$ separately on the two branches $z\le4$ and $z\ge4$ . The calculation is carried out in Appendix The moment map: appendix.

Proof of Theorem 17.3. Reduce $m=2$ to the line, then combine the Gamma completion, the row minimum and the integrated row bound: the total coefficient of $\E_P[u_i^2]$ is exactly $F_A(\alpha_i,P_i)$ , and Lemma 17.3 bounds that below. The calculation is carried out in Appendix The moment map: appendix.

Qualitative dimension-free KLS bounds for simplices and conservative Gamma models are prior art; see Kolesnikov & Milman, 2016, §1.2 as a secondary pointer to that literature. The specific content of this proof is the constant 4, the quantitative surplus, and the CMH-level estimate; no broader priority claim is intended.

Remark 17.2 (Closure and aggregation)

The proof is written for smooth $g$ with controlled boundary behavior. The closure argument takes polynomials on the simplex, lifts them after a radial cutoff $S\ge\eps$ , applies the Gamma identities and lets $\eps\downarrow0$ ; the required inverse Gamma moments are finite because $A\ge3$ throughout the Gamma branch. Polynomials form a core for $L_\alpha$ , so the closed forms extend (17.13) to $\Dom(L_\alpha)$ ; only the residual branch $m=2$ , $A<3$ uses the one-dimensional no-flux closure instead.

The estimate is also compatible with Dirichlet aggregation. If $Q\sim\Dir(\beta_1,\dots,\beta_N)$ , the coordinates are partitioned into blocks $G_i$ , and $P_i=\sum_{a\in G_i}Q_a$ , $\alpha_i=\sum_{a\in G_i}\beta_a$ , then for functions of $P$ alone the generator is the coarse one and $\E\bigl[\sum_{a\in G_i}Q_a^2/\beta_a\mid P\bigr] =P_i^2(\alpha_i+\abs{G_i})/(\alpha_i(\alpha_i+1))\ge P_i^2/\alpha_i$ , so CMH for the fine law implies CMH for its aggregation. This is a consistency check on Theorem 17.3, not an ingredient of it.

Remark 17.3 (Sharpness on simplex faces)

The constant 4 cannot be improved uniformly over the family. The face law $\mathrm{Beta}(1,b)$ is one dimensional, so Theorem 17.1 applies; its Neumann spectral problem reduces to Bessel’s equation and gives $\CMH(\mathrm{Beta}(1,b))=(b+1)^2(b+2)/\bigl(b\,j_{b/2,1}^2\bigr)$ with $j_{\nu,1}$ the first positive zero of $J_\nu$ . Since $j_{b/2,1}\sim b/2$ , this tends to 4; after rescaling $\mathrm{Beta}(1,b)$ converges to the one-sided exponential. The extremal mechanism on the simplex is therefore the same one-dimensional exponential boundary mechanism that fixes the constant in Theorem 17.1. Corollary 17.2 says the interior of the family is strictly safer.

Exponential cones: a second solvable non-product family¶

The Dirichlet family is solvable because the simplex lifts to independent Gamma variables. A lift in the opposite direction — attach a Gamma radial variable to a fixed base — produces measures on cones whose moment map is explicit in terms of the moment map of the base, and on which the linear sector of the moment-map inequality is computed exactly. Unlike products and Dirichlet laws these measures are not compactly supported, and for a general base they are not invertible affine images of products.

Definition 17.1 (Exponential cone measures — def:exponential-cone · Correction)

Let $n\ge2$ , let $K\subset\R^{n-1}$ be a convex body with barycenter at the origin, let

C_K=\bigl\{(x_1,x')\in\R\times\R^{n-1}:\ x_1>0,\ x'/x_1\in\operatorname{int}K\bigr\}

(17.23)

be the open cone over $K$ , and let $\beta\ge n$ . The exponential cone measure $\mu_{K,\beta}$ is the probability measure on $\R^n$ with density proportional to $x_1^{\beta-n}e^{-x_1}\one_{C_K}(x)$ , and $\bar\mu_{K,\beta}$ is its centering, the law of $X-\beta e_1$ for $X\sim\mu_{K,\beta}$ .

The measure $\mu_{K,\beta}$ is log-concave because $\beta\ge n$ , and $X\sim\mu_{K,\beta}$ has the representation $X=S\,(1,U)$ with $S\sim\GammaLaw(\beta,1)$ and $U$ uniform on $K$ independent. Hence $\E X=\beta e_1$ and

\Sigma=\Cov(X)=\beta\,e_1e_1^\top\oplus\beta(\beta+1)\Cov(U).

(17.24)

For $\beta=n$ the density is $e^{-x_1}$ on the cone; that case is the cone construction of Lemma 4.2 of Chen & Klartag, 2026, whose conical integration formula they attribute to Klartag, 2018, Lem. 2.1, and the general family $\beta\ge n$ and its moment map are not treated there. When $K$ is a simplex and $\beta=n$ this is an affine image of a product of centered exponentials; for a square base it is not an invertible affine image of any product of one-dimensional laws, since its support has four pairwise non-parallel facets.

Proposition 17.1 (Cone lift of the moment map — Proved · agent review (2026-09-06) · prop:cone-moment-map · Correction)

Let $\lambda$ be the moment potential of the uniform probability on $\beta K$ , so that $\lambda$ is smooth and strictly convex on $\R^{n-1}$ and $\nabla\lambda$ is a diffeomorphism onto $\beta\operatorname{int}K$ by Theorem 4.1. Then the moment potential of $\bar\mu_{K,\beta}$ is

\varphi(y_1,y')=\exp\bigl(y_1+\lambda(y')/\beta\bigr)-\beta y_1+\log\Gamma(\beta),

(17.25)

$\nabla\varphi$ is a diffeomorphism of $\R^n$ onto $C_K-\beta e_1$ , and the canonical Stein kernel of $\bar\mu_{K,\beta}$ at the point $x-\beta e_1$ , $x=(x_1,x_1u)\in C_K$ , is

\tau(x-\beta e_1)=x_1 \begin{pmatrix}1&u^\top\\ u&uu^\top+\beta\,\tau_K(u)\end{pmatrix},

(17.26)

where $\tau_K$ is the canonical Stein kernel of the uniform probability on $K$ . In particular the first column is the position vector itself, $\tau(x-\beta e_1)\,e_1=x$ .

Proposition 17.2 (The linear sector of an exponential cone — Proved · agent review (2026-09-06) · prop:cone-linear-sector · Correction)

Let $\mu=\bar\mu_{K,\beta}$ , with $\Sigma$ and $\tau$ as in (17.24) and (17.26), and write $\bar x=x-\beta e_1$ for the centered coordinate.

(i) The field $x=\tau e_1$ satisfies $\Div_\mu x=-\bar x_1$ with zero boundary flux, and $x=\Sigma\nabla\psi$ for $\psi=\tfrac12\,x^\top\Sigma^{-1}x$ . Hence the first Stein column is a covariance gradient with no solenoidal part in the sense of Proposition 16.2, and

\sup_{f}\frac{\bigl(\E_\mu[\bar x_1f]\bigr)^2}{\E_\mu\inner{\Sigma\nabla f}{\nabla f}} =\E_\mu\bigl[x^\top\Sigma^{-1}x\bigr]=\beta+n,

(17.27)

the supremum over $f$ of finite covariance energy being attained at $f=\psi$ .

(ii) The linear-test matrix is exact in the axis direction:

\frac{e_1^\top\,\E_\mu[\tau\Sigma^{-1}\tau]\,e_1}{e_1^\top\Sigma\,e_1} =1+\frac n\beta\ \le\ 2,

(17.28)

with equality if and only if $\beta=n$ ; this is also the value of the CMH Rayleigh quotient (16.5) at the linear test function $g=x_1$ .

(iii) In isotropic coordinates $Z=\Sigma^{-1/2}\bar x$ , with kernel $\tau_Z=\Sigma^{-1/2}\tau\Sigma^{-1/2}$ , the axis column is exactly its gap-mode projection: $T_3(e_1)=\E[Z_1\,Z\otimes Z]=2\beta^{-1/2}\,\Id$ and $\tau_Ze_1=e_1+\tfrac12T_3(e_1)Z$ . Hence $\norm{T_3(e_1)}_{\HS}^2=4n/\beta$ and the high-mode term of Lemma 16.2 vanishes, $v_{e_1}=0$ , for every base $K$ and every $\beta\ge n$ .

In particular every exponential cone measure with $\beta=n$ saturates the sharp linear test (16.15) in its axis direction and attains $\norm{T_3(e_1)}_{\HS}=2$ .

At $\beta=n$ , part (iii) is the third-moment computation of Lemma 4.2 of Chen & Klartag, 2026, which gives $T(Y)_{ijk}=2\delta_{ij}/\sqrt k$ , $T(Y)_{ikk}=0$ and $T(Y)_{kkk}=2/\sqrt k$ with $k=\beta$ ; parts (i)–(ii) and the general $\beta$ are new here. Proposition 17.2 gives a family of equality cases of Conjecture 16.2 that are not products, and it gives them for every base: the axis direction of a cone sees only the Gamma radial law, which is why the value $1+n/\beta$ does not depend on $K$ . The transverse block does depend on $K$ , through the base kernel $\tau_K$ ; when the base is a cube it is explicit.

Corollary 17.4 (The cube cone, exactly — Proved · agent review (2026-09-06) · cor:cube-cone-gate-zero · Correction)

For $K=[-1,1]^{n-1}$ the base kernel is $\tau_K(u)=\tfrac12\operatorname{diag}(1-u_j^2)$ , and the normalized linear-test matrix $\Sigma^{-1/2}\,\E[\tau\Sigma^{-1}\tau]\,\Sigma^{-1/2}$ of $\bar\mu_{K,\beta}$ equals

\Bigl(1+\frac n\beta\Bigr)\ \oplus\ \frac{6\beta^2+11\beta+5n+4}{5\beta(\beta+1)}\,\Id_{n-1}.

(17.29)

Both eigenvalues are at most 2 for all $n\ge2$ and $\beta\ge n$ : every cube cone satisfies (16.15). Equality holds in the axis direction exactly when $\beta=n$ , and in the transverse directions only for $n=\beta=2$ , where the matrix is $2\,\Id_2$ and the measure is a product of two centered exponentials.

For an interval base, the smallest case is $n=\beta=2$ : the linear-test matrix is $2I_2$ , as expected for an affine image of two independent exponentials. A square base behaves differently. At $n=\beta=3$ , its axis value is 2 while each transverse value is $53/30$ . The following result explains this distinction for every product of simplices, with intervals counted as one-dimensional simplices.

Proposition 17.3 (Product-simplex cones and the sharp linear test — Proved · agent review (gpt-6-astra, 2026-10-04) · prop:product-simplex-cone-gate · Correction)

Let $q\ge1$ , let $k_1,\ldots,k_q\ge1$ be integers, let $n=1+\sum_{j=1}^q k_j$ , and let $\beta\ge n$ be real. In Definition 17.1, take the centered base $K$ to be a Cartesian product of simplices of dimensions $k_1,\ldots,k_q$ . Let $\Sigma$ and $\tau$ be the covariance and canonical Stein kernel of $\bar\mu_{K,\beta}$ , and put $G=\Sigma^{-1/2}\mathbb E[\tau\Sigma^{-1}\tau]\Sigma^{-1/2}$ . Then $G$ has axis eigenvalue $1+n/\beta$ and, on each transverse factor of dimension $k$ , the eigenvalue

\lambda_k(n,\beta)= \frac{a_k\beta^2+(1+2b_k)\beta+n-k+c_k}{\beta(\beta+1)}, \quad a_k=\frac{2(k+2)}{k+4},\qquad b_k=\frac{2(k+1)(k+2)}{(k+3)(k+4)},\qquad c_k=\frac{(k+2)(k^2+9k+2)}{(k+3)(k+4)}.

(17.30)

In particular $G\preceq2I$ . If $\beta>n$ , then $G\prec2I$ . If $\beta=n$ and $q=1$ , then $G=2I$ . If $\beta=n$ and $q>1$ , then the eigenspace at 2 is exactly the cone axis. Thus a nonzero transverse equality direction exists exactly when $q=1$ and $\beta=n$ . The same spectral and inequality conclusions hold after any invertible linear change of coordinates, with equality subspaces transported in covariance-normalized coordinates.

The proof uses the independent vertex-permutation symmetries of the simplex factors. They kill mixed blocks and make each transverse block scalar. Three moments of the canonical simplex kernel, evaluated by the elementary simplex integral, give $a_k,b_k,c_k$ . Substituting these in the cone kernel of Proposition 17.1 yields the displayed spectrum. To see both the inequality and its equality cases, put $t=\beta-n$ and $d=n-k-1$ . Multiplication of $2-\lambda_k$ by the positive denominator $(k+3)(k+4)\beta(\beta+1)$ gives

4(k+3)t^2+[5k^2+27k+28+8(k+3)d]t +4d[(k+3)n+k+1].

(17.31)

For $t,d\ge0$ this vanishes exactly at $t=d=0$ : the radial parameter is at its endpoint and there is just one factor. With several positive-dimensional factors, only the axis attains equality. This determines the linear sector for the family, without deciding the universal gate inequality or nonlinear CMH.

The explicit cone kernels also permit polynomial tests of the full CMH quotient. In the finite-degree computations on cube cones, those quotients lie below the exponential-product values at equal degree. This is numerical evidence, not an upper bound on CMH. A strict quotient above 4 would instead require exact verification and a justification that the boundary model transfers to the regular class.

The saturation risk and the decisive test¶

Conjecture 17.1 (Second variation of CMH at products, where it equals 4 — Not settled here · conj:cmh-second-variation · Idea · Counterexample)

Take the product moment potential $\psi_0(s,t)=\phi(s)+t^2/2$ with $\phi$ the one-sided exponential moment potential, and perturb it by $\psi_\eps=\psi_0+\eps\,a(s)b(t)$ . Call the perturbation admissible if, for all sufficiently small $\abs\eps$ , $\psi_\eps$ is smooth and strictly convex and its moment measure $\mu_\eps$ is log-concave and belongs to the regular moment-map class on which Definition 16.1 is set. Then, for every admissible perturbation,

\limsup_{\eps\to0}\frac{\CMH(\mu_\eps)+\CMH(\mu_{-\eps})-2\CMH(\mu_0)}{\eps^2}\le0,

(17.32)

where $\CMH(\mu_0)=4$ by Corollary 17.5.

Since $\CMH(\mu_0)=4$ , a strictly positive second variation for one admissible perturbation would give $\CMH(\mu_\eps)>4$ for some small $\eps$ , a counterexample to $\mathrm{CMH}(4)$ ; as the remark above explains, the solenoidal part alone does not decide the sign.

The first task is admissibility. The conjecture perturbs the source moment potential linearly and uses an exponential times a Gaussian as its base. A linear perturbation of the target density potential is a different family: the change of variables through the moment map generally introduces higher-order terms in the parameter. Tests of two-sided convexity for a target-linear family therefore do not decide admissibility, or vacuity, of the displayed conjecture. Nor does an exponential times exponential calculation automatically test its stated ansatz. A nonlinear cone path can probe $\mathrm{CMH}(4)$ separately, provided its regularity and limiting passage are justified.

A positive second derivative of a fixed-degree quotient is also insufficient by itself. If that quotient starts below 4, the perturbation must overcome its initial deficit with a controlled remainder. Passing a scalar Poincaré bound to approximants does not supply continuity of CMH or its second variation.

Conjecture 17.1 is the sharpest available probe of the approach because it attacks the target inequality $\mathrm{CMH}(4)$ itself rather than the machinery built to prove it, and because both ingredients are already exact: the saturation value comes from Theorem 17.2 and the splitting from Proposition 16.2. Together with Remark 16.4 it forms the falsification layer of the moment-map approach; Conjecture 18.1 and Conjecture 18.2 form the construction layer.

References¶

Kannan, R., Lovász, L., & Simonovits, M. (1995). Isoperimetric Problems for Convex Bodies and a Localization Lemma. Discrete & Computational Geometry, 13(3–4), 541–559. 10.1007/BF02574061
Cattiaux, P., & Guillin, A. (2018). On the Poincaré Constant of Log-Concave Measures.
Kolesnikov, A. V., & Milman, E. (2016). The KLS Isoperimetric Conjecture for Generalized Orlicz Balls.
Chen, Y., & Klartag, B. (2026). Digesting the Proof of the Sharp Thin-Shell Inequality.
Klartag, B. (2018). Isotropic Constants and Mahler Volumes. Advances in Mathematics, 330, 74–108. 10.1016/j.aim.2018.03.008