The moment-Hessian inequality, its operator data, and $\CPaff\le\CMH$

Part of the moment-map mechanism, Chapter The moment map: CMH and the linear test; the reading order is on the full proofs page.

Overview. This dossier fixes the operator data of Route C, which answers Remark 15.2. It proves the Bochner identity Proposition 16.1, the endpoint reduction Theorem 16.1 ( $\CPaff\le\CMH$ ), the Hodge decomposition Proposition 16.2 with its consequence Corollary 16.1, and the algebraic countermodel Proposition 16.3. The work is done in the regular moment-map class. The reduction is a spectral duality argument. $\mathrm{CMH}(4)$ itself is not proved, and neither is a strict nonimplication between CMH and KLS. The passage beyond the regular class is only a conditional template, deferred to Proposition 15.1.

The Stein identity $\Div_\mu H=-p$ makes $L_\mu$ symmetric and nonpositive with Dirichlet form $\E\inner{H\nabla f}{\nabla g}$ (Lemma D30.1).
Bochner identity: $\E(L_\mu g)^2=\E[\inner{H\nabla g}{\nabla g}+\Tr(HD^2gHD^2g)]$ . It uses the total symmetry (D30.5) of the moment map (Proposition D30.1) and is not needed for step 3.
Endpoint: pairing $f$ with the spectrally truncated $\Aop^{-1}f$ and applying Cauchy–Schwarz gives $\CPaff\le\CMH$ (Theorem D30.1). This uses only step 1.
Hodge: $H\nabla g$ splits orthogonally into $\Sigma\nabla\Aop_1^{-1}h$ and a divergence-free $w$ (Proposition D30.2), using (D30.13). The first channel is exactly $\CPaff$ , and $w$ is the only possible gap (Corollary D30.1).
Linear test functions show that $\mathrm{CMH}(4)$ implies gate zero. Lemma D30.2 isolates the commutator. An explicit random $H\succeq0$ with $\E H=\Id$ satisfies the Letwin matrix bound but has $\lmax(\E H^2)>4$ for $m\ge18$ (Proposition D30.3). It is not claimed to be a moment-map Hessian, so Conjecture 16.1 is not refuted (Corollary D30.2).

Scope. This dossier discharges Remark 15.2 of Chapter The moment map: the deterministic inequality: it fixes every operator datum in the schematic Route C endpoint $\norm{\Sigma^{-1/2}H\nabla g}_2^2\le4\norm{-Lg}_2^2$ and proves the reduction to the affine Poincaré inequality. It then proves two structural facts about the resulting statement: that it dominates the affine Poincaré constant and has an additional solenoidal channel in dimension at least two (without proving a strict nonimplication), and that its linear-sector consequence is not implied by Letwin’s constant-matrix estimate through matrix algebra alone. It does not prove $\mathrm{CMH}(4)$ , which remains open (Conjecture 16.1 is only its necessary linear sector).

Standing hypotheses. $\mu$ is a centered, full-dimensional log-concave probability measure on $\R^n$ with density $\rho=e^{-V}/Z$ and covariance $\Sigma=\Cov_\mu\succ0$ , lying in the regular moment-map class of §Moment-map coordinates: the moment potential $\varphi$ of (4.1) is smooth and strictly convex, $\nabla\varphi$ is a diffeomorphism, and the integrations by parts below are justified for compactly supported fields or fields of vanishing normal flux, the general case following by completion. $H$ is the moment-map Hessian in target coordinates (4.19), so that $H=H^\top\succ0$ , $\Div_\mu H=-p$ , and $\E_\mu H=\Sigma$ . The final paragraph records a conditional limiting template. Existence of approximants with all the required uniformity, affine-support control, and limiting core density is the separate open Proposition 15.1, not a theorem of this dossier.

1. The operator data¶

Define the Stein generator and its nonnegative form

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

(D30.1)

That the same measure $\mu$ appears in the form and in the generator is exactly the Stein identity $\Div_\mu H=-p$ ; this is what makes Definition 16.1 canonical rather than a choice of reference measure.

2. The Bochner identity¶

Proof D30.2

We calculate in target coordinates, use Einstein summation, and first take $g$ in a smooth compactly supported core. The Stein identity and the generator formula are

\partial_i(\rho H_{ij})=-\rho p_j, \qquad L_\mu g=H_{k\ell}g_{k\ell}-p_k g_k.

(D30.2)

Integrating once and differentiating $L_\mu g$ gives

\begin{aligned} \E_\mu(L_\mu g)^2 ={}&\E_\mu[H_{ij}g_i g_j] -\E_\mu[H_{ij}g_j(\partial_iH_{k\ell})g_{k\ell}]\\ &-\E_\mu[H_{ij}g_jH_{k\ell}g_{ik\ell}] +\E_\mu[H_{ij}g_jp_k g_{ik}]. \end{aligned}

(D30.3)

Integrating the third-derivative term in $p_k$ gives

\begin{aligned} -\E_\mu[H_{ij}g_jH_{k\ell}g_{ik\ell}] ={}&-\E_\mu[p_\ell H_{ij}g_jg_{i\ell}] +\E_\mu[H_{k\ell}(\partial_kH_{ij})g_jg_{i\ell}]\\ &+\E_\mu[H_{k\ell}H_{ij}g_{jk}g_{i\ell}]. \end{aligned}

(D30.4)

The first term on the right cancels the preceding $p_k$ term after relabelling, and the last one is $\Tr(HD^2g\,HD^2g)$ .

The remaining cancellation is exactly where the moment-map structure enters. If $p=\nabla\varphi(y)$ and $H_{k\ell}(p)=\varphi_{k\ell}(y)$ , then

H_{mj}\partial_jH_{k\ell} =H_{mj}(H^{-1})_{rj}\varphi_{k\ell r} =\varphi_{mk\ell},

(D30.5)

which is totally symmetric in $m,k,\ell$ . Hence the two residual terms are

\E_\mu[g_jg_{i\ell}\varphi_{\ell ij}] -\E_\mu[g_jg_{k\ell}\varphi_{jk\ell}]=0.

(D30.6)

This proves the identity on the smooth core. If $g_q$ is graph-norm Cauchy there, apply the identity to $g_q-g_r$ . Since its two right-hand terms are nonnegative, $H^{1/2}\nabla g_q$ and $H^{1/2}D^2g_qH^{1/2}$ are Cauchy in their respective $L^2$ spaces. Their limits define the closed right-hand side, and passage to the limit proves the identity on the operator-core closure.

For $\mu$ Gaussian, $H=\Id$ and the proposition 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.

3. The endpoint reduction¶

Proof D30.3

Write $C=\CMH(\mu)$ ; the assertion is automatic if $C=\infty$ . Let $\mathscr C$ be the restrictions to $\operatorname{supp}\mu$ of $\R+C_c^\infty(\R^n)$ and first take a centered $f\in\mathscr C$ . This class lies in both form domains even when $H$ is globally unbounded: the Stein normalization $\E_\mu H=\Sigma$ gives

\E_\mu\inner{H\nabla f}{\nabla f} \le \norm{\nabla f}_\infty^2\E_\mu\Tr H =\norm{\nabla f}_\infty^2\Tr\Sigma<\infty.

(D30.7)

For $0<\eps<R<\infty$ put

\Pi_{\eps,R}=\one_{[\eps,R]}(\Aop), \qquad g_{\eps,R}=\Pi_{\eps,R}\Aop^{-1}f\in\Dom(\Aop).

(D30.8)

Then $\Aop g_{\eps,R}=\Pi_{\eps,R}f$ . Since $f$ is in the $H$ -form domain and $g_{\eps,R}\in\Dom(\Aop)$ , the form–operator pairing and Lemma D30.1 give

\begin{aligned} \norm{\Pi_{\eps,R}f}_2^2 &=\inner{f}{\Pi_{\eps,R}f}_{L^2} =\inner{f}{\Aop g_{\eps,R}}_{L^2} =\E_\mu\inner{\nabla f}{H\nabla g_{\eps,R}}\\ &\le\Bigl(\E_\mu\inner{\Sigma\nabla f}{\nabla f}\Bigr)^{1/2} \Bigl(\E_\mu\inner{H\nabla g_{\eps,R}} {\Sigma^{-1}H\nabla g_{\eps,R}}\Bigr)^{1/2}\\ &\le C^{1/2} \Bigl(\E_\mu\inner{\Sigma\nabla f}{\nabla f}\Bigr)^{1/2} \norm{\Pi_{\eps,R}f}_2. \end{aligned}

(D30.9)

After division (unless the projected norm is zero, when the conclusion is immediate) and squaring,

\norm{\Pi_{\eps,R}f}_2^2 \le C\E_\mu\inner{\Sigma\nabla f}{\nabla f}.

(D30.10)

Because $H\succ0$ and the support is connected, the kernel of $\Aop$ consists of the constants. Thus $f\perp\ker\Aop$ , and the spectral theorem gives $\Pi_{\eps,R}f\to f$ in $L^2$ as $R\to\infty$ and then $\eps\downarrow0$ .

For completeness, let $H^1_\Sigma(\mu)$ be the closure of $\mathscr C$ for the norm $\norm f_2^2+\E_\mu\inner{\Sigma\nabla f}{\nabla f}$ . Since $\Sigma\succ0$ is constant, this is the usual $H^1(\mu)$ in the present regular setting. Approximate an arbitrary $f\in H^1_\Sigma(\mu)$ by functions in $\mathscr C$ and subtract their means. Variances and $\Sigma$ -energies converge, so the inequality passes to the limit. Crucially, this argument does not claim that finite $\Sigma$ -energy places $f$ in the $H$ -form domain when $H$ is unbounded.

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

Let

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

(D30.11)

be the closed nonnegative self-adjoint operator associated with the covariance form

\calE_\Sigma(f,k)=\E_\mu\inner{\Sigma\nabla f}{\nabla k}, \qquad \Dom(\calE_\Sigma)=H^1_\Sigma(\mu),

(D30.12)

and set $L^2_0(\mu)=(\ker\Aop_1)^\perp$ . The support is connected and $\Sigma\succ0$ , so $\ker\Aop_1$ consists of the constants. Moreover $\CPaff(\mu)<\infty$ for each fixed full-dimensional log-concave probability measure (only a dimension-free bound is open). Consequently the restriction of $\Aop_1$ to $L^2_0(\mu)$ has a bounded, everywhere-defined inverse

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

(D30.13)

Proposition D30.2 (= Proposition 16.2)

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.

(D30.14)

Then $\Div_\mu w=0$ ,

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

(D30.15)

and

\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).

(D30.16)

Proof D30.4

Put $\mathscr H_\Sigma=L^2(\mu;\Sigma^{-1})$ and let

G_\Sigma:H^1_\Sigma(\mu)\subset L^2(\mu)\longrightarrow\mathscr H_\Sigma, \qquad G_\Sigma f=\Sigma\nabla f.

(D30.17)

This is a densely defined closed operator, its adjoint is the weak weighted divergence $G_\Sigma^*=-\Div_\mu$ , and $\Aop_1=G_\Sigma^*G_\Sigma$ . Here the assertion $h=-\Div_\mu u\in L^2$ means precisely that $u\in\Dom(G_\Sigma^*)$ and $G_\Sigma^*u=h$ : the usual integration-by-parts identity is first valid on the smooth core and extends to $H^1_\Sigma(\mu)$ by closed-form density. Since $\psi\in\Dom(\Aop_1)$ and $\Aop_1\psi=h$ , we also have $G_\Sigma\psi\in\Dom(G_\Sigma^*)$ and $G_\Sigma^*G_\Sigma\psi=h$ . Therefore

w=u-G_\Sigma\psi\in\Dom(G_\Sigma^*), \qquad G_\Sigma^*w=h-h=0,

(D30.18)

which is the weak statement $\Div_\mu w=0$ . The required integration by parts is now the adjoint identity at exactly these domains:

\E_\mu\inner{\Sigma\nabla\psi}{\Sigma^{-1}w} =\inner{G_\Sigma\psi}{w}_{\mathscr H_\Sigma} =\inner{\psi}{G_\Sigma^*w}_{L^2(\mu)}=0.

(D30.19)

Thus $G_\Sigma\psi$ and $w$ are orthogonal in $\mathscr H_\Sigma$ , and expanding $u=G_\Sigma\psi+w$ gives the displayed Hodge identity. The same argument shows minimality: if $v\in\Dom(G_\Sigma^*)$ and $G_\Sigma^*v=h$ , then $v-G_\Sigma\psi\in\ker G_\Sigma^*$ and hence

\norm v_{\mathscr H_\Sigma}^2 =\norm{G_\Sigma\psi}_{\mathscr H_\Sigma}^2 +\norm{v-G_\Sigma\psi}_{\mathscr H_\Sigma}^2.

(D30.20)

Finally, for every $h\in L^2_0(\mu)$ ,

\E_\mu\inner{\Sigma\nabla\psi}{\nabla\psi} =\norm{G_\Sigma\psi}_{\mathscr H_\Sigma}^2 =\inner{\Aop_1\psi}{\psi}_{L^2(\mu)} =\inner{h}{\Aop_1^{-1}h}_{L^2(\mu)}.

(D30.21)

The spectral theorem for the positive self-adjoint restriction of $\Aop_1$ to $L^2_0(\mu)$ therefore gives

\sup_{0\ne h\in L^2_0(\mu)} \frac{\inner{h}{\Aop_1^{-1}h}}{\norm h_2^2} =\norm{\Aop_1^{-1}} =\left( \inf_{\substack{0\ne f\in H^1_\Sigma(\mu)\\ f\perp\ker\Aop_1}} \frac{\calE_\Sigma(f,f)}{\norm f_2^2} \right)^{-1} =\CPaff(\mu),

(D30.22)

where the last equality is exactly the variational definition (16.1). This also verifies that no integration by parts has been applied outside the form/operator domains declared above.

Corollary D30.1 (Exact consequence of the Hodge decomposition)

$\Sigma\nabla\psi$ is the minimal- $L^2(\Sigma^{-1})$ field with divergence $-h$ , and controlling it is precisely the affine Poincaré inequality. Hence $\CMH(\mu)\ge\CPaff(\mu)$ , and $\mathrm{CMH}(C)$ implies $\CPaff(\mu)\le C$ . The only possible gap in the Hodge identity is the solenoidal excess $\E\inner{w}{\Sigma^{-1}w}$ . In dimension one the no-flux convention forces $w=0$ , so $\CMH=\CP/\Var$ exactly. In dimension at least two the divergence-free subspace is nontrivial, but the decomposition alone neither proves that $w$ contributes at a CMH extremizer nor gives a measure for which $\CMH>\CPaff$ . Equivalence and strict nonimplication therefore remain open; in particular this corollary does not prove that $\mathrm{CMH}(4)$ can fail while KLS holds.

5. Gate zero and the algebraic countermodel¶

Testing (16.5) on $g(p)=a\cdot p$ (so $D^2g=0$ , $L_\mu g=-a\cdot p$ ) gives $\E(L_\mu g)^2=a^\top\Sigma a$ and numerator $a^\top\E[H\Sigma^{-1}H]a$ . Hence $\mathrm{CMH}(4)$ implies gate zero, $\E[H\Sigma^{-1}H]\preceq4\Sigma$ , i.e. $\E H^2\preceq4\Id$ in isotropic position (Conjecture 16.1).

Proof D30.6

Let $z$ be uniform on $S^{m-1}$ , $d=m/\sqrt{2m-1}$ , $c=1-d/m$ , and $H(z)=\left(\begin{smallmatrix}1&\sqrt d\,z^\top\\ \sqrt d\,z&c\Id_m+d\,zz^\top\end{smallmatrix}\right)$ .

Positivity. $c>0$ since $d/m=1/\sqrt{2m-1}<1$ , and the Schur complement of the $(1,1)$ entry is $c\Id_m+d\,zz^\top-d\,zz^\top=c\Id_m\succ0$ ; hence $H\succeq0$ .

Normalization. $\E z=0$ kills the off-diagonal blocks and $\E zz^\top=\Id_m/m$ gives $\E H=1\oplus(c+d/m)\Id_m=\Id_{m+1}$ .

The quadratic form. With $B=\left(\begin{smallmatrix}a&r^\top\\r&D\end{smallmatrix}\right)$ , expanding $\Tr(BHBH)$ and using $\E z=0$ , $\E zz^\top=\Id_m/m$ and $\E(z^\top Dz)^2=\bigl((\Tr D)^2+2\Tr(D^2)\bigr)/(m(m+2))$ ,

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

(D30.23)

Decompose $D=t\Id_m+D_0$ with $\Tr D_0=0$ . Then $\Tr D=mt$ and $\Tr(D^2)=mt^2+\Tr(D_0^2)$ , so the form splits into three $O(m)$ -invariant sectors that do not interact: the vector sector $\abs r^2$ , the traceless sector $\Tr(D_0^2)$ , and the two-dimensional scalar sector $(a,t)$ .

Vector sector. Coefficient $2(c+2d/m)=2(1+d/m)$ , to be compared with 4 from $2\Tr(B^2)$ . Since $d\le m$ (equivalently $\sqrt{2m-1}\ge1$ ), it is at most 4.

Traceless sector. Coefficient $c^2+2cd/m+2d^2/(m(m+2))$ . Substituting $c=1-d/m$ and $d^2=m^2/(2m-1)$ gives $1+\frac{m-2}{(2m-1)(m+2)}$ , which is $\le2$ for all $m\ge1$ ; the comparison value is 2.

Scalar sector. With $r=0$ , $D=t\Id_m$ , the above reduces to $\E\Tr(BHBH)=a^2+2dat+t^2\bigl(mc^2+2cd+d^2\bigr)$ , and $mc^2+2cd+d^2=m-2d+\tfrac{d^2}m+2d-\tfrac{2d^2}m+d^2=m+d^2-\tfrac{d^2}m$ . Hence the deficit is

2(a^2+mt^2)-\E\Tr(BHBH)=a^2-2dat+\Bigl[m-d^2\Bigl(1-\tfrac1m\Bigr)\Bigr]t^2 .

(D30.24)

The choice $d^2=m^2/(2m-1)$ is exactly $d^2(2-\tfrac1m)=m$ , i.e. $m-d^2(1-\tfrac1m)=d^2$ , so the deficit equals $(a-dt)^2\ge0$ — a perfect square, vanishing on the ray $a=dt$ .

Combining the three sectors, $\E\Tr(BHBH)\le2\Tr(B^2)$ for every symmetric $B$ .

Failure of gate zero. $(H^2)_{11}=1+d\abs z^2=1+d$ deterministically, so $e_1^\top\E H^2e_1=1+d$ . Now $1+d>4\iff d>3\iff m^2>9(2m-1)\iff m^2-18m+9>0$ , whose positive root is $9+6\sqrt2\approx17.49$ ; hence the inequality holds exactly for integers $m\ge18$ .

Obstructions respected. Route C carries no bounded_by edges: the six obstruction statements of the manuscript are scoped by their own statements to the fixed-cut Eldan program. The one with a method-level reach beyond it, rem:projection-ceiling, forbids proving quadratic-chaos thin shell from radial or projection information alone; nothing above uses projection tests — the endpoint reduction is a duality argument and the countermodel is exact finite-dimensional algebra. Remark 16.1 records that gate zero is an operator-to-trace upgrade of the same family as conj:trace-upgrade and conj:product-alignment; per the single-owner discipline of CLAUDE.md, no claim of equivalence with those nodes is made here.

Numerical status. The cmh-gate-zero numerics target computes $\lmax(\Sigma^{-1/2}\E[H\Sigma^{-1}H]\Sigma^{-1/2})$ in closed form on one-dimensional laws, products, and from exact moment matrices on the Dirichlet family, and regression-tests the sector identities of Proposition D30.3. Only an exact rational certificate may refute Conjecture 16.1; floating eigenvalues and Galerkin optima are directional. The current battery lies inside already-proved classes and contributes nothing to the analytic theorems above.

Conditional template beyond the regular class. Let $\mu_q$ be the regular approximants from §Sources, chosen so that $\mu_q\to\mu$ with second moments and therefore $\Sigma_q\to\Sigma$ . If $\sup_q\CMH(\mu_q)\le C$ , Theorem D30.1 applied to a fixed $f\in C_c^\infty$ gives

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

(D30.25)

The two sides converge because $f$ and $\nabla f$ are bounded and continuous and the second moments converge. Subject to a limiting $H^1_\Sigma(\mu)$ core-density theorem, the endpoint argument then extends the inequality to the full Sobolev domain. No continuity of $\CMH$ is used or claimed. A complete proof must still construct approximants with the asserted moment and affine-support behavior and justify this density passage; those obligations are precisely Proposition 15.1.