Proof of Corollary 2. Jensen’s inequality in each coordinate gives \(\int|x|^2d\mu\le na^2\log2\), so \(\mu\in\mathcal P_2(\mathbb R^n)\). Only the case \(H(\nu\mid\mu)<\infty\) needs proof. For \(s>0\), put \(\gamma_s=N(0,s\mathrm{Id})\), \(\mu_s=\mu*\gamma_s\), and \(\nu_s=\nu*\gamma_s\). The density of \(\mu_s\) is smooth and strictly positive. For \(X\sim\mu\) and an independent standard Gaussian \(G\), its potential \(V_s\) satisfies \[D^2V_s(y)=s^{-1}\mathrm{Id}-s^{-2}\mathop{\mathrm{Cov}}(X\mid X+\sqrt sG=y)\ge0.\] Indeed, the posterior has the original convex potential plus \(|x-y|^2/(2s)\), so Lemma 3 bounds its covariance by \(s\mathrm{Id}\). Gaussian differentiation is valid without compact support. Thus \(\mu_s\) is centered and log-concave, with a smooth convex potential.

Set \(\beta_s=\sqrt{8s/3}\) and \(a_s=a+\beta_s\). For a scalar standard Gaussian \(Z\), \(\mathbb E\exp(sZ^2/\beta_s^2)=2\). The inequality \[\frac{(u+t)^2}{(a+\beta_s)^2} \le\frac{a}{a+\beta_s}\frac{u^2}{a^2} +\frac{\beta_s}{a+\beta_s}\frac{t^2}{\beta_s^2}\] and Hölder’s inequality show that \(\mu_s\) satisfies (1) with parameter \(a_s\). Theorem 1, with the Sobolev extension in Section 2, gives entropy coefficient \(Ca_s^2\). In the conventions of (Otto and Villani 2000, Definitions 1–2), \(\mathop{\mathrm{Ent}}(f^2)\le L\int|Df|^2\) is \(\mathrm{LSI}(2/L)\), whose transport conclusion is \(W_2^2\le LH\). Their Theorem 1 applies to the smooth convex potential of \(\mu_s\), and hence \[W_2(\nu_s,\mu_s)^2\le Ca_s^2H(\nu_s\mid\mu_s) \le Ca_s^2H(\nu\mid\mu).\] The last inequality is data processing for the common Gaussian kernel, as in Section 2. Coupling a vector with itself plus independent Gaussian noise gives \(W_2(\mu_s,\mu),W_2(\nu_s,\nu)\le\sqrt{ns}\). Letting \(s\downarrow0\) at fixed \(n\), with \(a_s\to a\), proves the claim. ◻

Proof. Choose laws and compact smooth tests for which the entropy-to-energy ratio divided by the squared marginal parameter tends to infinity. The tests may be chosen with positive energy: a smooth test of zero energy is constant on the connected interior of the convex support, and its entropy vanishes. Restriction of the law to a sufficiently large ball preserves each test quotient arbitrarily closely. Multiply the restricted density by \(e^{-\varepsilon|x|^2/2}\) and normalize, with \(\varepsilon>0\) sufficiently small to preserve that quotient again. Both operations preserve log-concavity, their combined density relative to the original law can be bounded by \(2\), and Lemma 3 gives a finite logarithmic Sobolev constant for the resulting compact law.

A bounded density increase changes the exponential-square expectation by at most that factor. If \(\mathbb Ee^{Z^2/A^2}\le4\), Jensen gives \(\mathbb Ee^{Z^2/(2A^2)}\le2\). Centering costs only another absolute factor: \(\mathbb EZ^2\le A^2\log4\), while \((Z-\mathbb EZ)^2\le2Z^2+2(\mathbb EZ)^2\); increase the denominator first to make the resulting exponential expectation bounded by an absolute constant, and once more by Jensen to make it at most \(2\). These arguments apply separately to every unit linear form with the same constants. Translation preserves the test quotient. Thus the centered compact laws still have \(R/A^2\to\infty\) with \(0<R<\infty\). Dilation by \(R^{-1/2}\) gives Equation (11).

For completeness, if a centered scalar \(Z\) obeys \(\mathbb Ee^{Z^2/a^2}\le2\), then \(\mathbb E|Z|^k\le2a^k\Gamma(k/2+1)\), by integrating its tail. Expansion of \(e^{tZ}\), with the linear term absent, therefore gives \(\log\mathbb Ee^{tZ}\le Ca^2t^2\) for \(a|t|\le c\). For \(a|t|\ge c\), use \(tZ\le Z^2/a^2+a^2t^2/4\) and absorb \(\log2\) into \(Ca^2t^2\). Apply this to \(Z=X\cdot\theta\). The second moment bound follows either from the exponential-square bound or by differentiating the centered moment generating function at zero. ◻

Proof. The heat equation for \(\mu_s\) is the continuity equation with velocity \(DV_s/2\). On a compact positive-time interval this velocity is globally Lipschitz and grows at most linearly, so its smooth flow is a diffeomorphism carrying \(\mu_s\) to \(\mu_t\). Its Jacobian satisfies \(\dot J=K_tJ/2\). By positivity of \(K_t\), the flow expands distances and its inverse is \(1\)-Lipschitz when \(t\ge s\). Transport a test by that inverse. Its entropy is unchanged and its Dirichlet energy cannot increase, proving \(R_t\ge R_s\). Equation (4) gives \(R_{s+h}\le R_s+h\) and \(R_s\le1+s\). For every compact smooth test, its entropy and energy converge to those under \(\mu\) as \(s\downarrow0\). Taking the supremum of its quotient gives \(\liminf_{s\downarrow0}R_s\ge R(\mu)=1\). Monotonicity now proves the remaining bound. ◻

Proof. Write \(r=d\nu/d\gamma_s\). The Gaussian kernel formula gives \(D\log r=b_s/s\), which is bounded, and hence \(|\log r(y)|\le C(1+|y|)\). For \(v=u\sqrt r\), \[\mathop{\mathrm{Ent}}_\nu(u^2)=\mathop{\mathrm{Ent}}_{\gamma_s}(v^2) -\int v^2\log r\,d\gamma_s, \qquad Dv=\sqrt r\,(Du+\tfrac12uD\log r).\] Gaussian integration by parts followed by Cauchy–Schwarz gives \[\int |y|^2v^2d\gamma_s =sn\|v\|_2^2+2s\int v\,y\cdot Dv\,d\gamma_s \le2sn\|v\|_2^2+4s^2\|Dv\|_2^2.\] Initially this calculation is for compact tests, and cutoffs extend it by lower semicontinuity. The linear-growth entropy correction is consequently bounded by \(\delta\|Dv\|_2^2+C_\delta\|v\|_2^2\) for every \(\delta>0\). Gaussian logarithmic Sobolev and the displayed derivative formula, with another arbitrarily small Cauchy–Schwarz parameter, prove Equation (16). The same moment estimate shows tightness of the \(L^2(\nu)\) mass of any bounded form sequence. On a fixed ball, the smooth density is bounded above and away from zero, so the ordinary local Rellich theorem gives \(L^2\) precompactness. A diagonal extraction and tightness prove compactness globally.

We give the tail argument for the sharp coefficient \(2s\) in Equation (17). Put \(v_{j,L}=(|u_j|-L)_+\) and \(m_{j,L}=\|v_{j,L}\|_2^2\). Strong \(L^2\) convergence implies \(\lim_{L\to\infty}\sup_jm_{j,L}=0\). The energies of \(v_{j,L}\) and \(|u_j|\wedge L\) sum to \(\|Du_j\|_2^2\). Lower semicontinuity for the clipped functions therefore gives \[ \lim_{L\to\infty}\limsup_j\|Dv_{j,L}\|_2^2 \le E_*-\|Du\|_2^2. \tag{18}\] For a nonnegative \(v\) of squared norm \(m\le1\), the negative part of \(v^2\log v^2\) has integral at most \(C\sqrt m\) (use \(-z\log z\le C\sqrt z\) on \(0\le z\le1\)). Since \(m\log m\le0\), Equation (16) thus gives \[\int v^2\log_+v^2d\nu \le2(s+\varepsilon)\|Dv\|_2^2 +C_\varepsilon m+C\sqrt m.\] If \(|u_j|>ML\), where \(M>1\) is fixed, then \(|u_j|\le c_M v_{j,L}\) with \(c_M=M/(M-1)\). Consequently \[u_j^2\log_+u_j^2 \le c_M^2v_{j,L}^2\log_+v_{j,L}^2 +c_M^2\log(c_M^2)v_{j,L}^2.\] The integrals of bounded truncations of \(z^2\log z^2\) pass to the limit by strong \(L^2\) convergence; its negative part does as well, being bounded and continuous. The last two bounds and Equation (18), with limits in the order \(j\), \(L\), \(M\), \(\varepsilon\), show that the lost positive entropy is at most \(2s(E_*-\|Du\|_2^2)\). The normalization terms are zero because \(\|u\|_2=\|u_j\|_2=1\). ◻

Proof. Take nonnegative unit-norm maximizing tests \(f_j\). Replacing a test by its absolute value preserves entropy and energy, so this choice loses nothing. Equation (16), with \(s+\varepsilon<R\), bounds their energies. We first exclude energies tending to zero; this is the only place the strict inequality \(R>P\) is required.

Suppose \(\varepsilon_j^2=\|Df_j\|_2^2\to0\), and write \(c_j=\mathbb E_\nu f_j\) and \(h_j=(f_j-c_j)/\varepsilon_j\). Poincaré gives \(\|h_j\|_2^2\le P\), while \(\|Dh_j\|_2^2=1\) and \(1-c_j^2=\varepsilon_j^2\|h_j\|_2^2\). Thus \(c_j\to1\). By compactness, after extraction, \(h_j\to h\) in \(L^2\), where \[\mathbb Eh=0,\qquad \|Dh\|_2^2\le1,\qquad \|h\|_2^2\le P\|Dh\|_2^2.\] For \(c>0\), \(\varepsilon>0\) and \(z\ge-c/\varepsilon\) define \[Q_{c,\varepsilon}(z)=\varepsilon^{-2} \left[(c+\varepsilon z)^2\log\frac{(c+\varepsilon z)^2}{c^2} -(c+\varepsilon z)^2+c^2\right].\] Normalization and \(-\log c_j^2-1+c_j^2\ge0\) give \[ \varepsilon_j^{-2}\mathop{\mathrm{Ent}}(f_j^2) \le\int Q_{c_j,\varepsilon_j}(h_j)\,d\nu. \tag{20}\] Here is the scalar estimate needed to control unbounded \(h_j\). For \(1/2\le c\le1\) and \(0<\varepsilon\le1/2\), \[ 0\le Q_{c,\varepsilon}(z) \le(2+2\log2)z^2+z^2\log_+z^2, \qquad Q_{c,\varepsilon}(z)\longrightarrow2z^2 \tag{21}\] uniformly on bounded \(z\)-intervals as \(c\to1\) and \(\varepsilon\to0\). To verify the bound, put \(t=\varepsilon z/c\). Then \(Q=z^2B(t)\), where Taylor’s integral formula gives \[B(t)=2+4\int_0^1(1-r)\log(1+rt)\,dr, \qquad -1\le t<\infty.\] For \(t\le0\), \(B(t)\le2\). For \(t\ge0\), use \(t\le z\) to obtain \(B(t)\le2+2\log(1+z)\le2+2\log2+\log_+z^2\). Continuity at \(t=0\) proves the compact convergence.

The tail proof in Lemma 9 did not use unit norm until its last sentence. In particular, for any bounded form sequence \(u_j\to u\) in \(L^2\) with energies tending to \(E_*\), it proves \[ \lim_{K\to\infty}\limsup_j \int_{|u_j|>K}u_j^2\log_+u_j^2d\nu \le2s(E_*-\|Du\|_2^2). \tag{22}\] Apply the compact convergence in Equation (21) on \(|h_j|\le K\) and its upper bound on the complement. Strong \(L^2\) convergence removes the quadratic tail as \(K\to\infty\); Equation (22) controls the entropy tail. Equation (20) consequently gives \[\limsup_j\varepsilon_j^{-2}\mathop{\mathrm{Ent}}(f_j^2) \le2\|h\|_2^2+2s(1-\|Dh\|_2^2) \le2\max(P,s)<2R.\] This contradicts the maximizing property.

We may therefore extract \(f_j\to f\) strongly in \(L^2\) with \(\|Df_j\|_2^2\to E_*>0\). Put \(E=\|Df\|_2^2\). The entropy defect and the sharp inequality give \[2RE_*\le\mathop{\mathrm{Ent}}(f^2)+2s(E_*-E) \le2RE+2s(E_*-E).\] Since \(R>s\) and \(E\le E_*\), equality forces \(E=E_*>0\). Thus \(f\) is a nonnegative, nonconstant optimizer.

The homogeneous functional \(\mathop{\mathrm{Ent}}(g^2)-2R\|Dg\|_2^2\) is nonpositive and is zero at \(f\). Its derivative along every compact smooth variation \(\zeta\) is \[4\int f\log f\,\zeta\,d\nu -4R\int Df\cdot D\zeta\,d\nu=0.\] The entropy normalization term cancels the additional linear term. Differentiation is legitimate also at zeros because \(z^2\log z^2\) has derivative zero at zero. This proves the Euler equation in Equation (19).

We have obtained a nonnegative optimizer with positive energy. To use its logarithmic derivative, we still need local regularity and strict positivity. We spell out the weak-solution regularity step, using the interior Newtonian-potential \(L^p\) and Schauder estimates; see (Viaclovsky 2004, Lecture 24, p. 2; Lecture 10, Proposition 1). Write \(\nu=e^{-V}\) and set \(v=e^{-V/2}f\ge0\). The function \(V\) is smooth, and direct differentiation of the weak equation gives \[-\Delta v=\lambda v\log v+ \left(\frac{\lambda V}{2}-\frac{|DV|^2}{4} +\frac{\Delta V}{2}\right)v.\] The conjugating factor and its reciprocal are bounded on each fixed ball. Since \(|z\log z|\le C_\delta(1+z^{1+\delta})\) for \(z\ge0\), local \(L^p\) integrability of \(v\) puts the right side in \(L^q_{\rm loc}\), where \(q=p/(1+\delta)>1\). Localize that right side on a larger ball and take its Newtonian potential with the sign solving \(-\Delta u=F\). The cited \(L^q\) estimate gives \(u\in W^{2,q}_{\rm loc}\); the difference \(v-u\) is distributionally harmonic and hence smooth by Weyl’s lemma (Viaclovsky 2004, Lecture 3, Theorem 2). This proves the required \(W^{2,q}\) gain directly for the weak solution.

For \(n\ge2\), start with \(p=2\) and choose \(0<\delta<\min(1,2/n)\). Before boundedness is reached, Sobolev embedding improves reciprocal integrability from \(1/p\) to \((1+\delta)/p-2/n\), a decrease greater than \(1/n\) because \(p\ge2\). At a critical exponent use any sufficiently large finite integrability exponent and continue once more. After finitely many steps \(q>n/2\), so \(v\), and thus \(f\), is locally bounded and Hölder continuous. In dimension one the same conclusion follows by twice integrating the local Poisson equation. The function \(z\log z\) is locally Hölder of every exponent below one on \([0,\infty)\); reducing the exponent if necessary, the Schauder estimate therefore gives \(f\in C^{2,\alpha}_{\rm loc}\) for some \(\alpha>0\). All constants in this paragraph concern one fixed law and dimension; none enters a uniform estimate along the counterexample sequence.

If \(f(x_0)=0\), choose a ball \(B\) about \(x_0\) with \(0\le f\le b<1\) on its closure. Start the \(\nu\)-diffusion at \(x_0\) and stop at its exit time \(\tau\) from \(B\). For \(Z_t=f(Y_{t\wedge\tau})\), \(m(t)=\mathbb EZ_t\), and \(g(z)=-\lambda z\log z\), the stopped Itô formula gives, for almost every \(t\), \[m'(t)=\mathbb E[\mathbf 1_{t<\tau}g(Z_t)]\le\mathbb Eg(Z_t)\le g(m(t)), \qquad m(0)=0.\] Indeed \(g\) is nonnegative and concave on \([0,b]\); the first inequality adds its nonnegative stopped boundary contribution. Also \(m'\ge0\). If \(m(t_1)>0\), let \(t_a\) be the first time at which \(m=a\), for \(0<a<m(t_1)\). Integration gives \[\int_a^{m(t_1)}\frac{dz}{-\lambda z\log z} \le t_1-t_a\le t_1,\] contradicting divergence as \(a\downarrow0\). Hence \(m=0\). The killed nondegenerate diffusion has positive probability of reaching each open subset of \(B\) while staying in \(B\): one can follow a smooth interior path by requiring Brownian motion to remain in a small tube about its corresponding driving path, and use local Lipschitz continuity of the drift. Brownian tube probabilities are positive by its finite increment densities and bridge continuity. Consequently \(f\) vanishes throughout \(B\). Its zero set is both open and closed, impossible in connected \(\mathbb R^n\) for a unit-norm function. Thus \(f>0\), and smooth elliptic bootstrap makes \(f\) smooth everywhere. ◻

Proof. Since \(w=2Df/f\), \(I=4\|Df\|_{L^2(\nu)}^2\); optimizer equality gives \(I=2\lambda H(\eta\mid\nu)>0\). The chain rule applied to its Euler equation gives the first identity in Equation (23). Commuting one derivative with \(A_\nu\) gives \[A_\nu w=D(A_\nu h)-Kw =\lambda w+(Dw)w-Kw.\] Since \(A_\eta=A_\nu-w\cdot D\) and \(Dw\) is symmetric, the second identity follows distributionally. Its right side is in \(L^2(\eta)\), since \(K\) is bounded and \(w\in L^2(\eta)\). Lemma 4 therefore puts \(w\) in the operator domain; no unproved integrability of higher derivatives has been used.

Let \(p=g^2\eta\) have unit mass, where \(g\) is compactly supported and smooth. Direct expansion and compactly supported integration by parts give \[\begin{align*} \mathcal I(p\mid\nu) &=\mathcal I(p\mid\eta)+\int g^2|w|^2d\eta +2\int D(g^2)\cdot w\,d\eta\\ &=\mathcal I(p\mid\eta)+\int g^2\{|w|^2+2A_\eta h\}\,d\eta\\ &=\mathcal I(p\mid\eta)+2\lambda\int h\,dp. \end{align*}\] These formulas may be read in square-root form at zeros of \(g\). On the other hand \(H(p\mid\nu)=H(p\mid\eta)+\int h\,dp\). Thus \[ \mathcal I(p\mid\nu)-2\lambda H(p\mid\nu) =\mathcal I(p\mid\eta)-2\lambda H(p\mid\eta). \tag{25}\] Apply the inequality for \(\nu\) to the compact smooth function \(gf\). Its deficit is nonnegative, so the right side is nonnegative. Homogeneity and the Sobolev approximation of Section 2 prove \(R(\eta)\le1/\lambda\). Finally the domain identity gives \[\mathbb E_\eta|Dw|^2=\lambda I-\mathbb E_\eta\langle w,Kw\rangle,\] while Poincaré for \(\eta\) gives the lower bound \(\lambda(I-|\mathbb E_\eta w|^2)\). Subtract to obtain Equation (24). ◻

Proof. The centered linear moment generating functions under \(\mu_s\) obey \[\log\mathbb E_{\mu_s}e^{v\cdot Y}\le C_J|v|^2, \qquad \log\mathbb E_{\mu_s}e^{v\cdot b_s(Y)}\le Ca^2|v|^2.\] The first follows by independent Gaussian addition and Equation (12); the second follows by conditional Jensen from that equation. Entropy duality, optimized over the scalar multiple of each unit vector, gives \[ |p|\le C_J\sqrt{D_0},\qquad |\mathbb E_\sigma b_s(Y)|\le Ca\sqrt{D_0}. \tag{27}\] All expectations exist. More explicitly, compact support of \(X\) makes \(\mathbb E_{\mu_s}e^{c|Y|^2}<\infty\) for some \(c>0\); entropy duality then implies \(\mathbb E_\sigma|Y|^2<\infty\). The latter fact is used only for integrability, not for a uniform bound in dimension.

Let \(\sigma_u\) be the translate of \(\sigma\) by \(-up\), \(0\le u\le1\), and let \(D_u=H(\sigma_u\mid\mu_s)\). Change of variables in entropy and the Hessian bound give \[\begin{align*} D_u &=D_0+\mathbb E_\sigma[V_s(Y-up)-V_s(Y)]\\ &\le D_0-u|p|^2/s+u\,p\cdot\mathbb E_\sigma b_s(Y)/s +u^2|p|^2/(2s) \le C_JD_0. \end{align*}\] Thus Equation (27), applied to \(\sigma_u\), implies \(|\mathbb E_\sigma b_s(Y-up)|\le C_Ja\sqrt{D_0}\) uniformly in \(u\). Differentiating the exact entropy expression, and then integrating, gives \[ D_1=D_0-\frac{|p|^2}{2s}+O_J(aD_0). \tag{28}\] Indeed its derivative is \(-(1-u)|p|^2/s+p\cdot\mathbb E_\sigma b_s(Y-up)/s\).

Write \(r(Y)=D\log(d\sigma/d\mu_s)(Y)\) in the weak square-root sense. Its mean is \[ \mathbb E_\sigma r=\mathbb E_\sigma DV_s=(p-\mathbb E_\sigma b_s)/s. \tag{29}\] To justify this for finite Fisher information, the Lebesgue density \(q\) of \(\sigma\) has weak derivative \(Dq=q(r-DV_s)\in L^1\): \(r\) is square integrable and \(DV_s\) has linear growth. Integrating against expanding cutoffs gives \(\int Dq=0\) and hence Equation (29).

In the moving coordinates \(Y\), the score of \(\sigma_1\) relative to \(\mu_s\) is \[r(Y)-p/s+d(Y)/s,\qquad d(Y)=b_s(Y)-b_s(Y-p).\] Since \(0\le Db_s\le\mathrm{Id}\), integration along the segment gives \[|d(Y)|^2 =\left|\int_0^1M_s(Y-up)p\,du\right|^2 \le\int_0^1|M_s(Y-up)p|^2du \le p\cdot d(Y).\] The mean estimates at the two endpoints therefore imply \[ \mathbb E_\sigma|d|^2\le C_JaD_0. \tag{30}\] Expanding the score square and using Equation (29), \[\mathbb E_\sigma|r-p/s|^2 =F_0-|p|^2/s^2+2p\cdot\mathbb E_\sigma b_s/s^2.\] The remaining cross term is bounded by \(C_J\sqrt{aD_0}\,(\sqrt{F_0}+\sqrt{D_0})\), and the \(d\)-square term by \(C_JaD_0\). Consequently the translated information is finite and satisfies \[ F_1=F_0-|p|^2/s^2+O_J(\sqrt a\,(F_0+D_0)). \tag{31}\] These calculations require only the weak derivatives already justified, so they apply also when the density of \(\sigma\) vanishes. Apply logarithmic Sobolev to \(\sigma_1\), namely \(F_1\ge2\lambda D_1\), and combine Equations (28) and (31). Since \(2/3\le\lambda\le1\) on the time interval in question, this proves Equation (26). ◻

Proof. In the first alternative the identities and the last inequality are Lemma 11. Apply Lemma 12 to \(\sigma=\eta\), where \(F_0=I\) and \(D_0=I/(2\lambda)\), so the deficit vanishes. The coefficient \(s^{-2}-\lambda/s\) is bounded below by a positive constant on \(J\). Thus, with \(p=\mathbb E_\eta Y\), \[|p|^2\le C_J\sqrt a\,I, \qquad |\mathbb E_\eta b_s|\le C_Ja\sqrt I.\] Equation (29) applied to \(\eta\) now gives \(|\mathbb E_\eta w|^2\le C_J\sqrt a\,I\). Since \(K_s^2\le s^{-1}K_s\), the curvature-mean inequality gives the asserted estimate for \(K_sw\).

In the second alternative, compact form embedding supplies a minimizer of the Rayleigh quotient among mean-zero functions of unit norm. Its positive eigenvalue is \(1/R_s^P=\lambda\). The nullspace consists of constants, since zero weak gradient on \(\mathbb R^n\) implies constancy. Compactness therefore makes this a positive gap, and interior elliptic regularity makes its eigenfunction \(\varphi\) smooth. Commuting a derivative gives \(A_{\mu_s}D\varphi=(\lambda\mathrm{Id}-K_s)D\varphi\). Its right side is in \(L^2\), so Lemma 4 gives the operator-domain assertion. Taking the energy pairing and then Poincaré gives, exactly as before, \[\mathbb E\langle w,K_sw\rangle\le\lambda|\mathbb Ew|^2.\] Coordinates belong to the operator domain because both \(Y\) and \(A_{\mu_s}Y=(Y-b_s)/s\) belong to \(L^2(\mu_s)\). Integration by parts, or self-adjointness with these coordinate tests, yields \[\mathbb Ew=\lambda\mathbb E[\varphi Y] =\frac{\mathbb E[\varphi Y]-\mathbb E[\varphi b_s]}s, \qquad (1-\lambda s)\mathbb E[\varphi Y]=\mathbb E[\varphi b_s].\] Conditional Jensen gives \[\mathbb E_{\mu_s}b_sb_s^*\le\mathbb E_\mu XX^*\le Ca^2\mathrm{Id}.\] For every unit vector \(v\), Cauchy–Schwarz thus bounds \(|\mathbb E[\varphi\,v\cdot b_s]|\) by \(Ca\|\varphi\|_2\). Taking the supremum over \(v\) introduces no dimension factor. Since \(1-\lambda s\ge1/2\) and \(I=\lambda\|\varphi\|_2^2\), it follows that \(|\mathbb Ew|\le C_Ja\sqrt I\). The bound for \(K_sw\) follows from \(K_s^2\le s^{-1}K_s\) once again.

It remains to justify the coordinate assertion for the first alternative. Finite \(H(\eta\mid\mu_s)\) and the Gaussian tails of \(\mu_s\) imply \(\mathbb E_\eta|Y|^2<\infty\), by the entropy argument in the proof of Lemma 12. The distributional identity \[ A_\eta Y=(Y-b_s)/s-w \tag{34}\] has an \(L^2(\eta)\) right side, because \(b_s\) is bounded for the fixed law and \(w\in L^2(\eta)\). The maximal-domain assertion of Lemma 4 applies. ◻

Proof. Fix \(s\), and let \(\Phi_{s,t}\) be the heat continuity flow from Lemma 8, for \(t\) on either side of \(s\). Transport a form-domain test \(g\) by \(g_t=g\circ\Phi_{s,t}^{-1}\). Its distribution under \(\mu_t\) is fixed, so both its entropy and its variance are constant. The Jacobian equation \(\partial_tD\Phi_{s,t}=K_tD\Phi_{s,t}/2\) gives \[ \left.\frac{d}{dt}\right|_{t=s} \int|Dg_t|^2d\mu_t =-\int\langle Dg,K_sDg\rangle d\mu_s. \tag{37}\] For compact smooth \(g\) this follows by differentiating the inverse-transpose Jacobian in the gradient. For a form-domain \(g\), the change-of-variables formula still holds by Sobolev approximation. The flow derivatives and their time derivatives are uniformly bounded on a compact time interval, so the derivative of the integrand is dominated by \(C|Dg|^2\). Dominated convergence proves Equation (37) without a smoothness or compact-support restriction on \(g\).

In the strict alternative take \(g=f\) and set \(Q(t)=\mathop{\mathrm{Ent}}_{\mu_t}(f_t^2)/(2\int|Df_t|^2d\mu_t)\). The denominator is positive near \(s\), and \(Q(t)\le R_t\) for both signs of \(t-s\), with \(Q(s)=R_s\). At a differentiability point of \(R_s\), this two-sided contact implies \(Q'(s)=R_s'\). Equation (37) and \(w=2Df/f\) give Equation (36). In the gap alternative take \(g=\varphi\) and instead use \(Q(t)=\mathop{\mathrm{Var}}_{\mu_t}(\varphi_t)/\int|D\varphi_t|^2d\mu_t\). Now \(Q(t)\le R_t^P\le R_t\) and \(Q(s)=R_s^P=R_s\). The same contact argument gives the identical derivative formula, with \(w=D\varphi\).

Equation (33) bounds this derivative by \(C_J\sqrt a\) in either alternative, with constants independent of which alternative holds at each time. Lipschitz continuity makes \(R_s\) absolutely continuous. For fixed \(0<\delta<s<1/2\), integration on \([\delta,s]\) gives \[1\le R_s\le R_\delta+C_{[\delta,s]}\sqrt a\,(s-\delta) \le1+\delta+C_{[\delta,s]}\sqrt a\,(s-\delta).\] First take the counterexample-sequence limit and then let \(\delta\downarrow0\). This proves Equation (35). ◻

Proof. Choose an infinite subsequence with the same alternative at \(s=1/8\). Lemma 13 gives all structural identities and the curvature and mean bounds. Proposition 14 gives \(\lambda\to1\). Consequently \[(A-1)w=(\lambda-1)w-K_sw=o_{L^2}(\sqrt I).\] The energy pairing yields \(\mathbb E|Dw|^2=\lambda I-\mathbb E\langle w,K_sw\rangle=(1+o(1))I\). Poincaré with constant \(\lambda^{-1}\) is the asserted spectral gap. The coordinate and branch-specific identities have already been proved. In the gap alternative all estimates are homogeneous in \(\varphi\), so multiplying it by positive scalars can enforce \(I\to0\) without changing any normalized assertion. ◻

Proof. The first equality is conditional expectation: \[\mathbb E\langle w(Y),h_r(U_r)\rangle=\mathbb E|h_r(U_r)|^2.\] For fixed \(y\), Gaussian integration by parts identifies the order-\(j\) Hermite coefficient of \(h_r(y+\sqrt r\,G)\) as \(r^{j/2}D^jg_r(y)\). Its orthogonal projection onto that chaos is \[\frac{r^{j/2}}{j!}D^jg_r(y):\mathfrak h_j(G),\] where the vector output slot is not contracted. By Equation (80), the squared norm of this projection is \(r^j|D^jg_r(y)|^2/j!\). Bessel’s inequality, integrated in \(y\sim\eta\), proves the asserted sum.

These calculations require only \(\mathbb E|h_r(Y+\sqrt rG)|^2<\infty\). To justify the derivatives, on a compact set of starting points use positivity of \(\eta\) on a slightly larger compact set. Integrating translated Gaussian kernels over that larger set bounds each fixed differentiated kernel by a constant times the mixture density of \(U_r\). The enlargement absorbs the polynomial factors from differentiation. Thus the kernel integrals are classically smooth locally; alternatively all identities first hold distributionally. The Bessel bounds for orders \(j\) and \(j+1\) put \(D^jg_r\) and its weak derivative in \(L^2(\eta)\), hence in the form domain by cutoff and local smoothing. The local constants need not be uniform in the dimension: only the exact integrated Bessel inequality is used quantitatively. ◻

Proof. The map \(u\mapsto u\cdot Z\) from Euclidean space to scalar \(L^2\) has operator norm at most \(\sqrt c\). Apply its adjoint to each scalar coefficient of \(F\) and sum the squared bounds. This proves Equation (82) without a factor equal to the number of coordinates of either variable.

For the second assertion, sum the coefficient spectral measures of \(S-\mathbb ES\) and call the resulting measure \(\nu\). Then \[\int(x-\lambda)\,d\nu(x) =\|DS\|_2^2-\lambda\|S-\mathbb ES\|_2^2=o(I),\] where the left side is nonnegative. For every fixed \(\delta>0\), the spectral mass and the spectral energy outside \([1-\delta,1+\delta]\) are \(o(I)\): the part below \(1-\delta\) is absent eventually, and on \(x\ge1+\delta\) both \(1\) and \(x\) are bounded by a \(\delta\)-dependent constant times \(x-\lambda\). The constant component has squared norm \(o(I)\) as well.

On the remaining interval the multipliers \(e^{1-x}-1\) and \((x-1)e^{1-x}\) tend uniformly to zero as \(\delta\downarrow0\). Outside it, \((1+x)|e^{1-x}-1|^2\le C(1+x)\) and \((x-1)^2e^{2-2x}\le C\). Integrate these inequalities against the spectral measure, first take the sequence limit and then let \(\delta\downarrow0\). This gives Equation (83). The same functional calculus shows that \(e^{1-A}S\) belongs to every \(\mathop{\mathrm{Dom}}A^p\). ◻

Proof. Start with \(T_0=w\). Suppose the assertions have been constructed through level \(l\), and put \(L=l+1\) and \(V=DT_l\). Self-adjointness gives \[ \|V\|_2^2=\langle T_l,AT_l\rangle_2=(1+o(1))I. \tag{84}\] If \(\Pi\) is an admissible deterministic projection acting on an existing tensor slot, it commutes with \(A\) and differentiation. Consequently \[\|\Pi_1V\|_2^2 =\langle\Pi_1T_l,A\Pi_1T_l\rangle_2 \le\|\Pi_1T_l\|_2^2+ \|\Pi_1T_l\|_2\|(A-1)T_l\|_2=o(I).\] The error is uniform over projections of rank at most \(cI\) for fixed \(c\).

Fix \(0<r<1/4\). Set \(V_j=T_j\) for \(0\le j\le l\) and \(V_L=V\). Successive form integration by parts gives \[ I^{-1}\langle D^jg_r,V_j\rangle_2\longrightarrow e^{-r} \qquad(0\le j\le L). \tag{85}\] Indeed the assertion at \(j=0\) is Equation (81). For \(j\le l\), \[\langle D^{j+1}g_r,DT_j\rangle_2 =\langle D^jg_r,AT_j\rangle_2 =\langle D^jg_r,T_j\rangle_2+o(I).\] Replacing \(DT_j\) by \(T_{j+1}\) when \(j<l\) costs \(o(I)\). All test derivatives and their form energies are finite by Lemma 21; its bound controls every error for fixed \(r\) and \(j\). No uniform estimate as \(r\downarrow0\) has been used.

Here is the precise Bessel budget. Equation (85) and \(\|V_j\|_2^2/I\to1\) imply \[\liminf I^{-1}e^{2r}\|D^jg_r\|_2^2\ge1 \qquad(0\le j\le L).\] Multiply Equation (81) by \(e^{2r}/I\) and subtract these minimum contributions. The resulting bounds are \[\begin{align*} \limsup\frac{\|e^rD^Lg_r-V\|_2^2}{I} &\le\frac{L!}{r^L} \left(e^r-\sum_{j=0}^{L}\frac{r^j}{j!}\right)=O_L(r), \tag{86}\\ \limsup\frac{e^{2r}\|D^{L+1}g_r\|_2^2}{I} &\le\frac{(L+1)!}{r^{L+1}} \left(e^r-\sum_{j=0}^{L}\frac{r^j}{j!}\right)=1+O_L(r). \tag{87}\end{align*}\] For the first line, expand the squared difference and use the correlation at order \(L\); its normalized limit subtracts exactly one from the normalized squared derivative norm. Thus the exponential-series tail accounts for the entire excess, rather than merely bounding the norms.

Choose \(r\downarrow0\) sufficiently slowly along the sequence in these two estimates. The field \(S_0=e^rD^Lg_r\) then satisfies \[\|S_0-V\|_2=o(\sqrt I),\qquad \|DS_0\|_2^2\le(1+o(1))I.\] It is symmetric in its last \(L\) slots; \(V\) is symmetric in its first \(L\) slots. For \(L\ge2\), the permutations of those two sets of slots generate every permutation of the \(L+1\) slots. Indeed their adjacent transpositions together are all the adjacent transpositions. If a permutation fixes \(S_0\), its action changes \(V\) by at most \(2\|S_0-V\|_2\); a permutation fixing \(V\) costs zero. Expressing each permutation as a fixed finite word therefore shows that \(V\) is \(o(\sqrt I)\) from its full symmetrization. For \(L=1\), \(V=Dw\) is already symmetric because \(w\) is a gradient. Let \(S\) be the full symmetrization of \(S_0\). Averaging coefficient permutations is an orthogonal contraction and commutes with differentiation, so \[ \|S-V\|_2=o(\sqrt I),\qquad \|S\|_2^2=(1+o(1))I,\qquad \|DS\|_2^2\le(1+o(1))I. \tag{88}\] The preceding projection estimate for \(V\) now holds on its last slot too, since a coefficient-slot permutation moves it to an old slot with an \(o(\sqrt I)\) error uniform over projections.

To apply Poincaré sharply we still need to center \(V\). Write \(C_y=\mathop{\mathrm{Cov}}(T_l,y)\), \(C_b=\mathop{\mathrm{Cov}}(T_l,b)\), and \(C_\omega=\mathop{\mathrm{Cov}}(T_l,\omega)\), with the vector variable occupying the last slot. Testing the equation for \(AT_l\) against centered coordinates gives \[ \mathbb EV=C_y+E_l,\qquad |E_l|=o(\sqrt I),\qquad \mathbb EV=s^{-1}(C_y-C_b)-C_\omega. \tag{89}\] For clarity, the first exact identity before the error is \(\mathbb EV=\mathbb E[(AT_l)\otimes(y-\mathbb Ey)]\). Poincaré gives \(\mathop{\mathrm{Cov}}_\eta y\le\lambda^{-1}\mathrm{Id}\), and Equation (82) bounds its error by \(\lambda^{-1/2}\|(A-1)T_l\|_2\). This is the needed dimension-free estimate. The second identity follows by integration by parts using \(Ay=(y-b)/s-\omega\). These coordinate tests are legitimate in the form domain; their distributional \(A\)-images are in \(L^2\) by the stated second-moment assumptions, hence they are also in \(\mathop{\mathrm{Dom}}A\).

Project the last slot onto \(1-\Pi_\epsilon\). The two covariance terms involving \(b\) and \(\omega\) have norm at most \(C\sqrt{\epsilon I}\) by Equations (74) and (82). Subtracting the two identities in Equation (89), and using \(s\ne1\), therefore bounds this part of \(\mathbb EV\) by \(C_s\sqrt{\epsilon I}+o(\sqrt I)\). On the \(\Pi_\epsilon\) part, \[|(\Pi_\epsilon)_{L+1}\mathbb EV| \le\|(\Pi_\epsilon)_{L+1}V\|_2=o(\sqrt I)\] by the projected-energy estimate and \(\mathop{\mathrm{rank}}\Pi_\epsilon\le C_\epsilon I\). First take the sequence limit and then let \(\epsilon\downarrow0\). This proves \(|\mathbb EV|=o(\sqrt I)\), and thus \(|\mathbb ES|=o(\sqrt I)\).

Lemma 22 applies to \(S\). Define \[T_{l+1}=e^{1-A}S.\] It is symmetric, lies in every operator-power domain, is \(o(\sqrt I)\) from \(V\), and satisfies the strong near-eigenfunction estimate. Its mean is \(e\mathbb ES=o(\sqrt I)\). Finally, coefficient projections commute with the semigroup, and \(\|e^{1-A}\|_{2\to2}\le e\). Together with \(S-V=o_{L^2}(\sqrt I)\), this preserves the uniform projected-energy estimates. The induction is complete. Each step used only a finite collection of limits; diagonal choices give the assertion for all fixed levels. ◻

Proof. The operator-domain Itô formula from Lemma 4 gives, for each fixed \(d\), \[\begin{align*} e^vT_d(Y_v) ={}&T_d(Y_0)+\sqrt2\int_0^v e^uT_{d+1}(Y_u):dB_u\\ &+\int_0^v e^u(1-A)T_d(Y_u)\,du +\sqrt2\int_0^v e^u(DT_d-T_{d+1})(Y_u):dB_u. \end{align*}\] Stationarity, Cauchy–Schwarz for the time integral, and Itô isometry show that the last two terms are \(o(\sqrt I)\), uniformly for \(v\in[0,T]\) in squared-mean norm. Substitute this formula for the integrand at the next level, and repeat \(m+1\) times. For fixed \(l,m,T\), every replacement error remains \(o(\sqrt I)\); iterated isometry sums coefficient squares and introduces no dimension factor.

After multiplication by \(e^{-t}\), the remaining integral has coefficient \(2^{(m+1)/2}\), is over the ordered time simplex of length \(t\), and has innermost integrand \(e^{u-t}T_{l+m+1}(Y_u)\). Since \(u\le t\), its squared norm is at most \[2^{m+1}\frac{t^{m+1}}{(m+1)!} \|T_{l+m+1}\|_2^2 =(1+o(1))I\frac{(2t)^{m+1}}{(m+1)!}.\] This proves the remainder estimate and its uniform version. The other terms have coefficients evaluated at \(Y_0\). Symmetry in their Brownian slots turns their ordered iterated integrals into \(t^{j/2}\mathfrak h_j(N)/j!\). To verify the factor directly, expand the Brownian exponential martingale \(\exp(\xi\cdot B_t-t|\xi|^2/2)\) by iterated Itô integration and compare its coefficients with Equation (79). The additional \(\sqrt2\) at each stochastic integration gives exactly \((2t)^{j/2}/j!\) in Equation (90). ◻

Proof. We first prove the assertion for \(Q\). Couple the heat observations by \(U_v=Y_0+W_v\), where \(W\) is a standard Brownian motion independent of \(Y_0\sim\eta\), and retain \(h_v=\mathbb E[w(Y_0)\mid U_v]\). Put \(H_v=Dh_v(U_v)\) wherever this derivative is square integrable. The conditional covariance has the exact representation \[ C_r=\int_0^r\mathbb E[H_vH_v^*\mid U_r]\,dv, \qquad \int_0^r\mathbb E|H_v|^2\,dv=(1-m(r))I. \tag{93}\] To check the first identity and its normalization, let \(p_v\) denote the density of \(U_v\). The entries of \(p_vh_v\) solve the heat equation with generator \(\Delta/2\). Direct differentiation gives the matrix identity \[(\partial_v-\tfrac12\Delta)(p_vh_vh_v^*) =-p_vDh_v(Dh_v)^*.\] Integrate it against the later heat kernel and divide by \(p_r\). The initial term becomes \(\mathbb E[ww^*\mid U_r]\), and the final term is \(h_rh_r^*\), which proves the first formula. Taking the trace and expectation proves the second formula. For nonsmooth or unbounded initial \(w\), perform the calculation on scalar projections with convex truncations increasing to the square, and then use polarization. Spatial cutoffs and the square-integrability of \(w\) justify the positive-time passage. At time zero, approximation of \(w\) in \(L^2(\eta)\) by bounded continuous functions gives \(\mathbb E|w-\mathbb E[w\mid U_v]|^2\to0\). Thus there is no missing boundary term in Equation (93).

By Equation (72), \(\|Q\|_2^2=\langle w,Aw\rangle_2=(1+o(1))I\). For almost every \(v>0\), convolution differentiation and form integration by parts give \[ \mathbb E\langle Q(Y_0),H_v\rangle =\langle Dw,Dg_v\rangle_2 =\langle Aw,g_v\rangle_2=m(v)I+o(I). \tag{94}\] The error is uniform in \(v\): Jensen’s inequality gives \(\|g_v\|_2\le\sqrt I\), so it is bounded by \(\|(A-1)w\|_2\sqrt I\). The convolution derivative identity follows first with cutoffs and bounded functions, then by the square-integrable derivative bounds in Equation (93).

Fix a small \(z>0\) with \(2z<1/4\). Combining the last two equations and using \(0\le m(v)\le1\) yields the exact limiting discrepancy \[\begin{align*} \lim\frac1I\int_0^{2z}\mathbb E|Q(Y_0)-H_v|^2\,dv &=2z+1-e^{-2z}-2\int_0^{2z}e^{-v}\,dv\tag{95}\\ &=2z-1+e^{-2z}=O(z^2). \tag{96}\end{align*}\] Dominated convergence is applicable to \(m(v)\), so this calculation does not require uniform convergence of the prediction curve near zero.

We select a time in \([z,2z]\) while preserving a covariance-tail bound. For a fixed \(\kappa>0\), define \[a(r)=I^{-1}\mathbb E\mathop{\mathrm{tr}}\left(\frac{C_r}{r} \mathbf 1_{\{C_r/r>\kappa\}}\right).\] For fixed \(r\in[z,2z]\), Equation (75) gives \(a(r)\to0\), and \(a(r)\le1/z\) follows from \(\mathbb E\mathop{\mathrm{tr}}C_r\le I\). Consequently \(\int_z^{2z}a(r)\,dr\to0\). On the other hand, Equation (96) and Markov’s inequality show that the set of times satisfying \[\mathbb E|Q(Y_0)-H_r|^2\le C zI\] has measure at least \(z/2\), with an absolute \(C\) and for sufficiently far terms of the sequence. The set where \(a(r)\) exceeds a quantity tending to zero has measure tending to zero. Select \(r\in[z,2z]\) in the intersection. Thus \(r\) may vary with the sequence index, but \[ \mathbb E|Q(Y_0)-H_r|^2\le CzI,\qquad \mathbb E\mathop{\mathrm{tr}}\left(\frac{C_r}{r}\mathbf 1_{\{C_r/r>\kappa\}}\right)=o(I). \tag{97}\] All constants in this selection are for fixed \(z,\kappa\).

We compare \(B=H_rH_r^*\) with \(C=C_r/r\) in trace norm. For matrices \(U,V\), \[ \|UU^*-VV^*\|_1 \le |U-V|_{\mathrm{HS}}(|U|_{\mathrm{HS}}+|V|_{\mathrm{HS}}). \tag{98}\] Compare both \(B\) and \(C\) with \(\mathbb E[QQ^*\mid U_r]\). For \(B\), conditional Jensen’s inequality and the first bound in Equation (97) give an \(O(\sqrt z)I\) bound on the expected trace-norm difference. For \(C\), use the first identity in Equation (93), followed by Equation (98) and Cauchy–Schwarz in both probability and time. The integrated discrepancy is \(O(z^2)I\), whereas \[\int_0^r\mathbb E(|H_v|+|Q|)^2\,dv \le 2(1-m(2z))I+4z\|Q\|_2^2.\] Here we used \(r\le2z\) and the nonnegative integral in the second identity of Equation (93). The normalized limsup of the right side is \(O(z)\), by prediction at the fixed time \(2z\). No prediction limit at the selected varying time is needed. Division by \(r\ge z\) therefore also costs at most \(O(\sqrt z)I\). We have proved, in normalized limsup, \[ \mathbb E\|B-C\|_1\le O(\sqrt z)I. \tag{99}\]

Fix \(\delta>0\), take \(\kappa=\delta^2/2\) above, and set \(P=\mathbf 1_{\{B>\delta^2\}}\). Positivity of \(C\) implies \[\mathop{\mathrm{tr}}(BP)\le\frac{\delta^2}2\mathop{\mathrm{rank}}P +\mathop{\mathrm{tr}}\bigl(C\mathbf 1_{\{C>\delta^2/2\}}\bigr)+\|B-C\|_1.\] Since \(\delta^2\mathop{\mathrm{rank}}P\le\mathop{\mathrm{tr}}(BP)\), the first term is absorbed into the left side. Equations (97) and (99) bound the remaining high singular-value energy by \(O(\sqrt z)I+o(I)\). Clip \(H_r\) at \(\delta\); its squared distance from the operator-norm ball of radius \(\delta\) is at most this energy. The first estimate in Equation (97) then gives the same bound, up to constants, for the distance of \(Q\) from that ball. Let \(z\downarrow0\) after the sequence limit. The squared distance is \(o(I)\) for every fixed \(\delta>0\). Applying this distance assertion at cutoff \(\delta/2\) bounds the energy of singular values above \(\delta\) by four times the squared distance, proving Equation (92) for \(Q\). Since \(\|T_1-Q\|_2=o(\sqrt I)\), it also proves the assertion for \(T_1\).

It remains to propagate this estimate to higher levels. Fix \(l\ge1\) and \(t>0\). Apply Lemma 24 with starting level \(1\). Conditional on \(Y_0\), the Hermite chaoses are orthogonal even after applying an arbitrary measurable projection \(P(Y_0)\) to the first tensor slot. The chaos containing \(T_l(Y_0)\) has squared coefficient \[c_{l,t}=e^{-2t}\frac{(2t)^{l-1}}{(l-1)!}>0.\] Use the expansion to order \(m\ge l-1\), first take the sequence limit, and then let \(m\to\infty\) in its remainder bound. The resulting inequality is \[ \liminf\frac{ \mathbb E|P(Y_0)_1T_1(Y_t)|^2-c_{l,t}\mathbb E|P(Y_0)_1T_l(Y_0)|^2}{I}\ge0. \tag{100}\] The remainder estimates are uniform over such projections, since their operator norms are at most one.

Choose \(P(Y_0)\) to project onto the singular values greater than a fixed \(\epsilon>0\) of the first-slot flattening of \(T_l(Y_0)\). Then \[\mathbb E\mathop{\mathrm{rank}}P\le\epsilon^{-2}\|T_l\|_2^2 =(1+o(1))\epsilon^{-2}I.\] For every \(\delta>0\), let \(S_\delta\) be the singular-value clipping of \(T_1\) at \(\delta\). Pointwise, \[|P(Y_0)S_\delta(Y_t)|_{\mathrm{HS}}^2 \le\delta^2\mathop{\mathrm{rank}}P(Y_0).\] Stationarity controls the \(o(I)\) clipping error for \(T_1(Y_t)\). Hence the normalized limsup of the first expectation in Equation (100) is at most \(\delta^2/\epsilon^2\). Send \(\delta\downarrow0\) to obtain the tail claim at threshold \(\epsilon\) for \(T_l\). No independence between \(P(Y_0)\) and \(T_1(Y_t)\) was required. Tensor symmetry supplies every other one-slot flattening. ◻

Proof. For compactly supported smooth \(P\), commuting two first derivatives in weighted integration by parts gives \[ \|D^*P\|_2^2 =\sum_{i,j,\alpha} \langle\partial_jP_{i\alpha},\partial_iP_{j\alpha}\rangle_2 +\langle P,HP\rangle_2. \tag{103}\] Cauchy–Schwarz applied to the two arrays with \(i,j\) interchanged bounds the sum by \(\|DP\|_2^2\), without a dimension factor.

For the stated \(P\), membership in \(\mathop{\mathrm{Dom}}A\) gives \(P\in H^2_{\rm loc}\) by local elliptic regularity and \(DP\in L^2(\eta)\). Choose smooth cutoffs \(0\le\chi_R\le1\) tending to one with \(\|D\chi_R\|_\infty\to0\). Equation (103) applies to \(\chi_RP\) by local Sobolev approximation. Since \[\|D(\chi_RP)-DP\|_2\longrightarrow0, \qquad \langle\chi_RP,H\chi_RP\rangle_2 \longrightarrow\langle P,HP\rangle_2,\] its right side is bounded. The second convergence uses \(|\langle P,HP\rangle|\in L^1\), a consequence of \(P,HP\in L^2\). Weak compactness in \(L^2\) and distributional convergence of \(D^*(\chi_RP)\) identify the limit as \(D^*P\). Lower semicontinuity proves Equation (101). Only first derivatives of the cutoffs have entered the estimate.

For compactly supported smooth \(F\), differentiate the scalar operator: \[D(AF)=A(DF)+HDF.\] Pair this identity with \(P\), use symmetry of \(H\), and integrate by parts to obtain Equation (102). For general \(F\in\mathop{\mathrm{Dom}}A\), use the operator core from Lemma 4 to choose compact smooth \(F_j\) with \(F_j\to F\) and \(AF_j\to AF\) in \(L^2\). This convergence also holds in the form norm, since \[\|D(F_j-F)\|_2^2 =\langle F_j-F,A(F_j-F)\rangle_2\longrightarrow0.\] All terms in Equation (102) therefore converge: the three fixed factors are \(D^*P\), \(AP\), and \(HP\), each in \(L^2\). This proves the claimed global identity. ◻

Completion of the proof of Proposition 20. Lemma 23 supplies the fields and all assertions except Equation (77). Fix \(l\ge0\) and write \(F=T_l\). View \(DF\) as a matrix with its derivative slot as row index and all old tensor slots as column index. We construct a matrix field \(P\) satisfying \[ \|P-DF\|_2+\|(A-1)P\|_2=o(\sqrt I),\qquad \|DP\|_2=O(\sqrt I),\qquad \|QP\|_2=o(\sqrt I), \tag{104}\] where \(Q=Dw\) in the strict branch and \(Q=0\) in the gap branch.

In the strict branch, apply Lemma 25 to the corresponding flattening of \(T_{l+1}\). Choose a singular-value cutoff \(\delta\downarrow0\) sufficiently slowly that the clipped field \(C\) obeys \[\|C-T_{l+1}\|_2=o(\sqrt I),\qquad \|C(y)\|_{\mathrm{op}}\le\delta.\] Set \(P=e^{1-A}C=eP_1C\). Markov averaging and convexity of the operator-norm ball give \(\|P(y)\|_{\mathrm{op}}\le e\delta\). The spectral multipliers \(e^{1-x}\), \((x-1)e^{1-x}\), and \(\sqrt x\,e^{1-x}\) are bounded for \(x\ge0\). Moreover \(|e^{1-x}-1|\le C|x-1|\) there. These bounds, the approximation of \(C\), and the near-eigenfunction assertion for \(T_{l+1}\) give the first two estimates in Equation (104). Finally \(\|Q\|_2^2=(1+o(1))I\) and \[\|QP\|_2\le\|P\|_{\mathrm{op},\infty}\|Q\|_2=o(\sqrt I).\] The clipping is only used to construct \(P\), so it need not preserve tensor symmetry. In the gap branch take \(P=T_{l+1}\) directly. The first two estimates follow from Equation (76), and the last is immediate from \(Q=0\). No operator-tail argument is needed in this branch.

The drift hypothesis gives \(H=D^2(-\log\eta)=K-Q\) in either case. Both \(KP\) and \(QP\) are in \(L^2\), and the operator-domain assertion for \(P\) was retained by its construction. Lemma 26 therefore applies. In particular, \[\|D^*P\|_2^2 \le\|DP\|_2^2+s^{-1}\|P\|_2^2+\|P\|_2\|QP\|_2=O(I).\] Insert \(AF=F+(A-1)F\) and \(AP=P+(A-1)P\) into Equation (102). The terms \(\langle F,D^*P\rangle_2\) and \(\langle DF,P\rangle_2\) cancel. Consequently \[\begin{align*} \langle DF,KP\rangle_2 ={}&\langle(A-1)F,D^*P\rangle_2 -\langle DF,(A-1)P\rangle_2 +\langle DF,QP\rangle_2 =o(I). \end{align*}\] Since \(\|K\|_{\mathrm{op},\infty}\le s^{-1}\) and \(P-DF=o_{L^2}(\sqrt I)\), this gives \[\langle DF,KDF\rangle_2=o(I).\] Replacing \(DF\) by \(T_{l+1}\) costs \(o(I)\) by the derivative estimate. It proves Equation (77) at level \(l+1\) on the derivative slot, and hence on every slot by symmetry. Level zero was an initial assumption. All claims of the proposition follow. ◻

Proof. For rectangular matrices \(X,Y\), the Araki–Yamagami inequality (Araki and Yamagami 1981, Theorem 1) gives the following square-root bound. We give a finite-dimensional proof by self-adjoint dilation: \[ \|\sqrt{XX^*}-\sqrt{YY^*}\|_{\mathrm{HS}}^2 \le 2\|X-Y\|_{\mathrm{HS}}^2. \tag{107}\] Indeed the absolute value of the dilation of \(X\) has diagonal blocks \(\sqrt{XX^*}\) and \(\sqrt{X^*X}\). If \(A,B\) are self-adjoint and \(f\) is scalar Lipschitz, expansion in an eigenbasis of \(A\) and an eigenbasis of \(B\) shows that \(\|f(A)-f(B)\|_{\mathrm{HS}}\le\operatorname{Lip}(f)\|A-B\|_{\mathrm{HS}}\). Apply this with \(f(x)=|x|\). Difference quotients, followed by form approximation, prove that \(R\) is in the form domain and that \(\|DR\|_2^2\le2\|D\Psi\|_2^2\).

The curvature conclusion of the hierarchy yields \[\mathbb E\mathop{\mathrm{tr}}((1-M_s)R^2)=o(1).\] We also have \[ \mathop{\mathrm{tr}}((\mathbb EM_s)(\mathbb ER^2))=o(1). \tag{108}\] To see the latter assertion in the strict branch, fix \(\delta>0\) and let \(P_\delta\) be the spectral projection of \(\mathbb EM_s\) onto \((\delta,\infty)\). Lemma 19 gives \(\mathop{\mathrm{rank}}P_\delta=O_\delta(I)\); the deterministic-projection conclusion of Proposition 20 therefore gives \(\mathop{\mathrm{tr}}(P_\delta\mathbb ER^2)=o(1)\). Since \(0\le M_s\le1\), the left side of Equation (108) is at most \(o(1)+\delta\mathbb E\mathop{\mathrm{tr}}R^2\). Send \(\delta\) to zero after the sequence limit. In the gap branch use directly \(\|\mathbb EM_s\|_{\mathrm{op}}=o(1)\).

Let \(R'\) be evaluated at an independent point. Insert \(1-M_s+M_s\) between \(R\) and \(R'\) in \(\|\mathbb ER\|_{\mathrm{HS}}^2=\mathbb E\mathop{\mathrm{tr}}(RR')\). Hilbert–Schmidt Cauchy–Schwarz bounds the term containing \(1-M_s\) by \[\bigl(\mathbb E\mathop{\mathrm{tr}}((1-M_s)R^2)\bigr)^{1/2} \bigl(\mathbb E\mathop{\mathrm{tr}}R^2\bigr)^{1/2}=o(1).\] The other term is bounded by \[\bigl(\mathbb E\mathop{\mathrm{tr}}(M_sR^2)\bigr)^{1/2} \bigl(\mathop{\mathrm{tr}}((\mathbb EM_s)(\mathbb ER^2))\bigr)^{1/2}=o(1).\] Poincaré for \(\eta\), whose constant tends to one, now gives the lower bound on \(\|DR\|_2^2\) from Equation (106). ◻

Proof. For a component \(\psi_d\) with \(d\) slots, put \[a_d=\langle\psi_d,H_{(1)}^2\psi_d\rangle, \qquad c_d=\langle\psi_d,H_{(1)}H_{(2)}\psi_d\rangle.\] Symmetry and Cauchy–Schwarz give \(|c_d|\le a_d\), and direct expansion gives \[\|H_{\mathrm{av}}\psi\|^2 =\sum_d\left(\frac{a_d}{d}+\left(1-\frac1d\right)c_d\right) \le\sum_d c_d+\frac2k\sum_d a_d.\] It remains to prove \(|\sum_d c_d|\le\mathop{\mathrm{tr}}(RHRH)\). This correlation bound can also be deduced from the Wigner–Yanase case of Lieb’s trace-concavity theorem (Lieb 1973); we give a proof by weighted interpolation.

Define the linear map \(C\) by \[\mathop{\mathrm{tr}}(XC(Y))=\sum_d\langle\psi_d,X_{(1)}Y_{(2)}\psi_d\rangle.\] Flattening all remaining indices exhibits \(C(Y)\) as a sum of terms \(A_\alpha Y^{\mathsf T}A_\alpha^*\). Thus \(C\) is positive and \(C(1)=C^*(1)=\rho\). Restrict to the support of \(\rho\), on which inverse powers are defined, and set \[U(Y)=\rho^{-1/2}C(Y)\rho^{-1/2}.\] The map \(U\) is unital and contractive in operator norm: its preceding representation is an isometric compression of copies of \(Y^{\mathsf T}\). It also satisfies \(\mathop{\mathrm{tr}}(\rho U(Y))=\mathop{\mathrm{tr}}(\rho Y)\). For \(0\le\Re z\le1\) introduce \[U_z(Y)=\rho^{z/2} U(\rho^{-z/2}Y\rho^{-z/2})\rho^{z/2}.\] On \(\Re z=0\), multiplication on either side by imaginary powers of \(\rho\) is isometric, so \(U_z\) is operator-norm contractive. At \(z=1\), \[U_1(Y)=C(\rho^{-1/2}Y\rho^{-1/2}),\qquad U_1^*(1)=1.\] Its adjoint is a unital positive map with the same compression and transpose representation, hence is operator-norm contractive. Duality makes \(U_1\) trace-norm contractive; imaginary powers give the same conclusion on \(\Re z=1\).

The interpolation used here is the finite-dimensional weighted Schatten argument; compare (Beigi 2013, sec. III, Theorem 4). We give the details for our positive correlation map. Given matrices of Hilbert–Schmidt norm one, continue the singular values of the input by powers \(2z\), and those of the matrix paired with its output by powers \(2(1-z)\), keeping the unitary factors fixed. The bilinear trace pairing is analytic in the strip, bounded by one on both boundary lines by the preceding endpoint bounds and trace duality. The three-lines inequality bounds it by one at \(z=1/2\). Zero singular values follow by approximation. This proves \(\|U_{1/2}(Y)\|_{\mathrm{HS}}\le\|Y\|_{\mathrm{HS}}\). Take \(Y=\rho^{1/4}H\rho^{1/4}\). Then \[\mathop{\mathrm{tr}}(YU_{1/2}(Y))=\mathop{\mathrm{tr}}(HC(H))=\sum_d c_d, \qquad \|Y\|_{\mathrm{HS}}^2=\mathop{\mathrm{tr}}(RHRH).\] Cauchy–Schwarz proves the claim. Compression to the support leaves the correlation unchanged: every tensor slot lies in that support. Complexification, used only for interpolation, preserves the conclusion for real tensors. ◻

Proof. In an eigenbasis of \(R\) with eigenvalues \(r_a\), define \((H_i)_{ab}=2(D_iR)_{ab}/(r_a+r_b)\) when \(r_a+r_b>0\), and zero otherwise. The kernel–kernel block of \(D_iR\) vanishes almost everywhere. One way to check this for a form-domain positive matrix field is to use absolutely continuous line representatives: at a differentiability point, the quadratic form on every vector in the kernel has a local minimum zero, and polarization gives the assertion. The formula is measurable and gives \[\sum_i\|H_iR\|_{\mathrm{HS}}^2\le4|DR|^2.\] Set \[e=\|DR\|_2^2,\quad a=\sum_i\mathbb E\mathop{\mathrm{tr}}(R^2H_i^2),\quad c=\sum_i\mathbb E\|[H_i,R]\|_{\mathrm{HS}}^2,\quad d=\|D\Psi\|_2^2.\] The Sylvester equation implies the exact identities \[ e=a-\frac c4,\qquad \sum_i\mathbb E\mathop{\mathrm{tr}}(RH_iRH_i)=e-\frac c4. \tag{113}\] Differentiating \(R^2=\Psi\Psi^*\) in the weak form sense gives \[e=\sum_i\mathbb E\langle D_i\Psi,(H_i)_{(1)}\Psi\rangle =\sum_i\mathbb E\langle D_i\Psi,(H_i)_{\mathrm{av}}\Psi\rangle.\] For example, the first equality follows by pairing \(D_i(R^2)\) with \(H_i/2\); all products are integrable by the displayed bound. The second uses the symmetry of each \(D_i\Psi\) on its existing slots. Lemma 28 and Cauchy–Schwarz yield \[e^2\le d\left(e-\frac c4+\frac{2a}{k}\right).\] Since \(a\le4e\), \(d\to1\), and \(\liminf e\ge1\), this first forces \(e\to1\) and then \(c\to0\); Equation (113) gives \(a\to1\). Finally, \[\sum_i\|D_iR-H_iR\|_2^2=\frac c4, \qquad \sum_i\|D_i\Psi-(H_i)_{(1)}\Psi\|_2^2=d+a-2e.\] Both tend to zero. Symmetry gives the assertion for the other slots. ◻

Proof. We repeatedly use the dimension-free inequality \[ \|XX^*-YY^*\|_1 \le(\|X\|_{\mathrm{HS}}+\|Y\|_{\mathrm{HS}})\|X-Y\|_{\mathrm{HS}}. \tag{116}\] By Equation (112), the first sum in Equation (114) is close in expected trace norm to the one-slot reduction of \(D\Psi\), with the derivative index included among the column indices. The derivative coupling identifies this with \[\frac1{kI}\sum_{k\le l<2k}T_{l+1}T_{l+1}^*.\] The trace-norm error from the coupling is \(o(1)\) by Equation (116) and Equation (106). Shifting the band changes its reduction only by its two endpoints, whose expected trace-norm cost is \[\frac{\|T_{2k}\|_2^2+\|T_k\|_2^2}{kI}=O(k^{-1}).\] This proves the first balance equation. Comparing the arrays of matrices \((H_iR)_i\) and \((RH_i)_i\) proves the second by the same inequality and the summed commutator estimate.

For the original, possibly unbounded column write \(\mathbf H v=(H_iv)_i\), \(S=\mathbf H^*\mathbf H\), and \(J=(1+S)^{-1}\). Let \(\mathbf K=\mathbf HR-R^{\oplus n}\mathbf H\), so that \(\mathbb E\|\mathbf K\|_{\mathrm{HS}}^2=o(1)\). Algebra gives \[[S,R]=\mathbf H^*\mathbf K-\mathbf K^*\mathbf H, \qquad [R,J]=J[S,R]J.\] The bounds \[\|\mathbf HJ\mathbf H^*\|_{\mathrm{op}}\le1,\qquad \|\mathbf HJ\|_{\mathrm{op}}\le\tfrac12,\qquad \|J\|_{\mathrm{op}}\le1\] therefore imply \(\|\mathbf H[R,J]\|_{\mathrm{HS}}\le2\|\mathbf K\|_{\mathrm{HS}}\). For every fixed \(f(x)=a+b/(1+x)\), it follows that \[\mathbb E\|\mathbf H[f(S),R]\|_{\mathrm{HS}}^2=o(1).\] The squared norms of \(\mathbf Hf(S)R\) and \(\mathbf HRf(S)\) differ by \(o(1)\): the latter has bounded mean-square norm, and the preceding bound controls their difference. Testing the second balance equation against the bounded matrix \(f(S)^2\) consequently gives \[\mathbb E\mathop{\mathrm{tr}}\bigl(R^2(S-1)f(S)^2\bigr)=o(1).\] Use \(f=1,1+J,1-J\) and \((S-1)^2/(1+S)=(S-1)-2(S-1)J\) to obtain \[ \mathbb E\mathop{\mathrm{tr}}\left(R^2\frac{(S-1)^2}{1+S}\right)=o(1). \tag{117}\] All these expressions are integrable: their absolute values are bounded by a constant times \(\mathop{\mathrm{tr}}(R^2(1+S))\).

Let \(Q=\mathbf 1_{(4,\infty)}(S)\) and \(P=1-Q\). On this spectral region, \((S-1)^2/(1+S)\) dominates both a positive constant and a positive multiple of \(S\). Thus \[\mathbb E\mathop{\mathrm{tr}}(R^2Q)+\mathbb E\mathop{\mathrm{tr}}(R^2SQ)=o(1).\] The input truncation costs \(\sum_i\mathbb E\|H_iQR\|_{\mathrm{HS}}^2=\mathbb E\mathop{\mathrm{tr}}(R^2SQ)=o(1)\). The output truncation costs \[\sum_i\mathbb E\|QH_iR\|_{\mathrm{HS}}^2 =\mathbb E\mathop{\mathrm{tr}}\left(Q\sum_iH_i\rho H_i\right) \le\mathbb E\mathop{\mathrm{tr}}(Q\rho)+\mathbb E\left\|\sum_iH_i\rho H_i-\rho\right\|_1=o(1).\] Consequently replacing \(H_i\) by \(PH_iP\) has total right-\(R\) squared cost \(o(1)\). The new column obeys \(\sum_i(PH_iP)^2\le PSP\le4P\).

Here is the perturbation accounting needed both now and below. If symmetric \(\widetilde H_i\) have \[ \Delta=\sum_i\mathbb E\|(\widetilde H_i-H_i)R\|_{\mathrm{HS}}^2=o(1), \tag{118}\] then symmetry gives the same bound with \(R\) on the left. Each derivative error in Equation (112) changes in norm by at most \(\sqrt\Delta\), and the summed commutator norm changes by at most \(2\sqrt\Delta\). The two balance errors change in expected trace norm by at most \[\left(\Bigl[\sum_i\mathbb E\|H_iR\|_{\mathrm{HS}}^2\Bigr]^{1/2} +\Bigl[\sum_i\mathbb E\|\widetilde H_iR\|_{\mathrm{HS}}^2\Bigr]^{1/2}\right)\sqrt\Delta,\] by Equation (116), and this is \(o(1)\). The new summed energy is bounded by the old one plus the perturbation in norm. These statements apply to the compression just made.

For the bounded column, \([S,R]=\mathbf H^*\mathbf K-\mathbf K^*\mathbf H\) satisfies \(\|[S,R]\|_2\le4\|\mathbf K\|_2=o(1)\). Put \(T=S-1\), so \(\|T\|_{\mathrm{op}}\le3\). The second balance equation says \(\mathbb E\|RTR\|_1=o(1)\). Finally, \[\|TR\|_{\mathrm{HS}}^2 =\mathop{\mathrm{tr}}(TRTR)+\mathop{\mathrm{tr}}(T[T,R]R).\] The expectation of the first term is at most \(3\mathbb E\|RTR\|_1\) in absolute value; the second is at most \(3\|[T,R]\|_2\|R\|_2=o(1)\). This proves the last assertion. ◻

Proof. Equation (112) implies that moving \(H_i\) between any two existing slots costs \(o(1)\) in summed mean square. Three slots suffice to exchange two factors. In the following chain, each \(\simeq\) means that the difference has squared norm \(o(1)\) after expectation and summation over \(i,j\): \[\begin{align*} (H_i)_{(1)}(H_j)_{(1)}\Psi &\simeq (H_i)_{(1)}(H_j)_{(2)}\Psi = (H_j)_{(2)}(H_i)_{(1)}\Psi\\ &\simeq (H_j)_{(2)}(H_i)_{(3)}\Psi = (H_i)_{(3)}(H_j)_{(2)}\Psi\\ &\simeq (H_i)_{(3)}(H_j)_{(1)}\Psi = (H_j)_{(1)}(H_i)_{(3)}\Psi\\ &\simeq (H_j)_{(1)}(H_i)_{(1)}\Psi. \end{align*}\] The equalities use commutation on different slots. Each move costs at most the one-factor error multiplied by the column bound: for any tensor \(v\), \[\sum_i\|(H_i)_{(r)}v\|^2\le4\|v\|^2.\] The squared norm of the sum of the four errors is at most four times their summed squared norms. Thus the total commutator cost on \(\Psi\) is \(o(1)\). Its cost on \(\Psi\) equals its cost on \(R\), since both have reduction \(R^2\).

For the second conclusion, write \(\Psi=RJ_0\), where \(J_0J_0^*\) is the support projection of \(R\). The matrix-valued array \(Z_{ia,b}=(H_iR)_{ab}-(H_aR)_{ib}\) has its column index in that support, so \(\|ZJ_0\|_{\mathrm{HS}}=\|Z\|_{\mathrm{HS}}\). The entries of \(ZJ_0\) express the exchange of the derivative index and the first tensor index in \((H_i)_{(1)}\Psi\). Equation (112) and the approximation of \(D\Psi\) by fully symmetric \(T_{l+1}\) give the claim. For perturbed matrices, the derivative errors and the slot-moving errors have the bounds established after Equation (118); the same four moves use the new uniform column bound. The index-exchange norm itself changes by at most \(2\sqrt\Delta\). ◻

Proof. Choose an orthogonal eigenframe \(O\) for \(R\) at the initial point. Transform all three indices: in that frame the matrices are \[\widehat H_i=\sum_jO_{ji}O^*H_jO.\] This preserves the column bound, all the summed norms, and the index-exchange estimate, since these are Euclidean tensor contractions. Finite-dimensional spectral calculus and an ordered orthonormal basis completion give measurable choices, including at repeated eigenvalues.

Let \(h=\log(1+\epsilon)\) and take an independent \(\theta\) uniform on \([0,h)\). Group positive eigenvalues in \([e^{\theta+mh},e^{\theta+(m+1)h})\), \(m\in\mathbb Z\), and round those in each nonempty block down to \(\beta=e^{\theta+mh}\). The zero eigenspace is kept separate. Write \[\kappa=\sum_i\mathbb E\|[H_i,R]\|_{\mathrm{HS}}^2=o(1),\qquad a=\sum_i\mathbb E\|H_iR\|_{\mathrm{HS}}^2=O(1).\] For \(0<\delta<h\), a pair of positive eigenvalues with \(|\log(r_a/r_b)|\ge\delta\) satisfies \(r_b^2\le C\delta^{-2}(r_a-r_b)^2\), when \(\delta\le1\). If their logarithmic distance is smaller than \(\delta\), the probability that a block boundary separates them is at most \(\delta/h\). Consequently the right-\(R\) squared cost of deleting entries with row and column in different blocks is bounded by \[ C\delta^{-2}\kappa+\frac{\delta}{h}a+\kappa. \tag{123}\] The final term covers a zero row and positive column directly by the commutator. A zero column has no right-\(R\) cost. For example, choose \(\delta\to0\) slowly enough that \(\kappa/\delta^2\to0\), and then \(h\to0\) slowly enough that \(\delta/h\to0\).

Deleting entries whose outer index belongs to another block has vanishing cost as well. On the set where \(i\) and \(b\) have different block labels, \[|(H_i)_{ab}r_b|^2 \le2|(H_i)_{ab}r_b-(H_a)_{ib}r_b|^2 +2|(H_a)_{ib}r_b|^2.\] Sum over the indices. The first term is the second defect of Equation (119); the second is a row–column block exclusion of the kind already bounded in Equation (123). This includes outer indices in the zero eigenspace. Within each block the resulting matrices have the form \(P_\beta\widehat H_iP_\beta\) for \(i\in\beta\), and vanish otherwise. Thus \[\sum_{i\in\beta}(P_\beta\widehat H_iP_\beta)^2 \le P_\beta\left(\sum_i\widehat H_i^2\right)P_\beta\le4P_\beta.\] They are symmetric and their total change has the cost in Equation (118).

For every block matrix \(X\) one has the precise comparison \[ \tau(X^*X)\le\mathbb E\|XR\|_{\mathrm{HS}}^2 \le(1+\epsilon)^2\tau(X^*X). \tag{124}\] In particular \(\tau(1)\to1\). The perturbation bounds following Equation (118) prove the derivative, commutator, and balance conclusions for these matrices. The bounded-column argument at the end of Lemma 30 again gives \(\mathbb E\|(S-1)R\|_{\mathrm{HS}}^2=o(1)\), so Equation (124) proves the first tracial defect. Lemma 31 gives the second. For the third, all three indices now lie in one block, and the index-exchange defect is exactly weighted by \(r_b^2\); replacing this by \(\beta^2\) only decreases it. Matrix symmetry and exchange of \(i,a\) together give asymptotic symmetry under every permutation of the three indices.

It remains to bound the map from the outer index to matrix coefficients. On each block let \[Fv=\sum_i v_iH_i,\qquad (Gv)_{ab}=(H_bv)_a.\] The map \(G\) has operator norm at most two because \(\|Gv\|_{\mathrm{HS}}^2=\langle v,Sv\rangle\le4|v|^2\). Total symmetry just established says \[\mathbb E\sum_\beta\beta^2\|F-G\|_{\mathrm{HS}(\mathbb R^{\dim\beta},\mathrm{HS})}^2=o(1).\] Clip the singular values of \(F\) to two, obtaining \(F'=FB\) for a real positive contraction \(B\) on the input space. Metric projection onto the operator-norm ball in Hilbert–Schmidt norm gives \(\|F-F'\|_{\mathrm{HS}}\le\|F-G\|_{\mathrm{HS}}\); this also follows immediately by singular-value decomposition and the variational characterization of singular values. Thus the block-weighted squared cost is \(o(1)\). Writing \(H'_i=\sum_j B_{ji}H_j\) shows that every \(H'_i\) remains symmetric. For each vector \(v\), \[\sum_i\|H'_iv\|^2\le\sum_j\|H_jv\|^2,\] so the column bound is preserved, as is the block support. The weighted cost controls the right-\(R\) cost by Equation (124). Therefore the derivative and balance errors have exactly the bounds stated after Equation (118); index exchange changes by at most twice the perturbation norm, and the four-move argument controls the commutators. Repeating the bounded-column last step and then Equation (124) proves all three tracial defects for \(H'_i\). The defining singular-value clipping proves Equation (122). ◻

Proof. We first prove a conditional moment bound on one fixed block, allowing a general column bound \(S\le C\mathrm{Id}\). We use the Gaussian trace-moment integration-by-parts argument of (Tropp 2018, Lemma 6.1 and Section 7); the weighted near-commutation limit will be proved afterward. Write \(D=\sum_iN_iH_i\). Gaussian integration by parts gives \[\mathbb E_N\mathop{\mathrm{tr}}D^{2p} =\sum_{j=0}^{2p-2}\mathbb E_N\sum_i \mathop{\mathrm{tr}}(H_iD^jH_iD^{2p-2-j}).\] In an eigenbasis of \(D\) with eigenvalues \(d_a\), the absolute value of a summand is bounded by \[\sum_{i,a,b}|(H_i)_{ab}|^2|d_b|^j|d_a|^{2p-2-j}.\] For \(p>1\), the scalar weighted arithmetic–geometric mean inequality bounds the product of powers by \[\frac{j}{2p-2}|d_b|^{2p-2} +\left(1-\frac{j}{2p-2}\right)|d_a|^{2p-2}.\] Since \(\sum_{i,b}|(H_i)_{ab}|^2=S_{aa}\le C\) and similarly with \(a,b\) exchanged, each summand is at most \(C\mathop{\mathrm{tr}}|D|^{2p-2}\). For \(p=1\) the assertion is \(\mathop{\mathrm{tr}}S\le C\mathop{\mathrm{tr}}1\). Induction proves \(\mathbb E_N\mathop{\mathrm{tr}}D^{2p}\le C^p(2p-1)!!\mathop{\mathrm{tr}}1\). After multiplying by the block weights, all fixed moments are uniformly bounded. Also \(e^{u|x|}\le2\cosh(ux)\), so the even-moment series gives \[\tau(e^{u|D|}) \le2\sum_{p\ge0}\frac{u^{2p}C^p(2p-1)!!}{(2p)!}\tau(1) =2e^{Cu^2/2}\tau(1).\] This proves Equation (126) with \(C=4\).

For clarity, the norm \(\|X\|_{p,\tau}=\tau(|X|^p)^{1/p}\) is formed on the direct sum of the blocks and then averaged over the classical random variables. Schatten Hölder followed by scalar Hölder gives \[\|X_1\cdots X_m\|_{p,\tau} \le\prod_{j=1}^m\|X_j\|_{p_j,\tau}, \qquad \sum_j\frac1{p_j}=\frac1p.\] These bounds use the actual weights \(\beta^2\), with no factor depending on a block dimension. The finite-matrix Schatten inequality can itself be obtained by the same singular-value three-lines argument as in Lemma 28, interpolating the endpoint bounds with one trace norm and the other norms operator norms, and then applying duality.

For independent \(N_r,N_s\), \[\|[D_r,D_s]\|_{2,\tau}^2 =\sum_{i,j}\tau(|[H_i,H_j]|^2)=o(1).\] All its higher fixed moments are bounded by Hölder and the moment bound just proved. Interpolation between \(L^2(\tau)\) and an arbitrarily high fixed \(L^p(\tau)\) therefore makes this commutator tend to zero in every fixed finite power norm. Applying Hölder to the factors on either side of a commutator shows that an adjacent transposition in a word of fixed length costs \(o(1)\) in \(L^2(\tau)\). A finite sequence of transpositions proves the reordering assertion.

Now expand a mixed moment by Gaussian pairings, conditional on the matrices. Pairings must join indices of the same Gaussian vector. For each pair introduce its own independent standard Gaussian vector, shared only between the two factors in that pair. This represents the contracted matrix word exactly as a conditional expectation. Reorder the word until each pair is adjacent; the error is \(o(1)\) even in \(L^2(\tau)\). Averaging the independent vectors then gives \(S^{m/2}\). For each fixed integer \(b\), boundedness of \(S\) gives \[\|S^b-1\|_{2,\tau} \le\left(\sum_{a=0}^{b-1}4^a\right)\|S-1\|_{2,\tau}=o(1).\] Each pairing consequently contributes \(\tau(1)+o(1)=1+o(1)\). Odd moments vanish. The number of admissible pairings is precisely the scalar independent-normal moment, proving Equation (125).

Finally expand the tensor Hermite polynomial using its defining exponential \(\exp(\langle v,N\rangle-|v|^2/2)\). A term consists of free \(N\) indices and a number of signed Kronecker-delta pairs. Represent each delta pair by its own independent Gaussian vector, as above, and move its two matrix factors together. Conditional averaging is contractive in \(L^2(\tau)\), so the reordering error remains \(o(1)\). The resulting factors \(S\) can be replaced by one, even between fixed powers of \(D_1\), by Hölder and interpolation applied to the bounded matrix \(S-1\). The resulting polynomial is \[\sum_{b=0}^{\lfloor j/2\rfloor} \frac{(-1)^b j!}{2^bb!(j-2b)!}D_1^{j-2b} =\mathfrak h_j^{(1)}(D_1).\] There are only finitely many terms for fixed \(j\), proving Equation (127). ◻

Proof. The proof has three steps: replace the added tensor indices in the hierarchy expansion by products of the initial matrices, identify their Wick contractions, and sum the resulting scalar Hermite series. All limits involving the degree of that series are taken after the counterexample-sequence limit, with \(t\) fixed.

The derivative-action estimate and the hierarchy coupling give \[ \frac1{kI}\sum_{k\le l<2k}\sum_i \|T_{l+1,i\,\cdot}-(H_i)_{(1)}T_l\|_2^2=o(1). \tag{130}\] It also holds when the band is shifted by any fixed integer \(a\ge0\). Indeed the same matrices have been constructed using levels \(k\) through \(2k-1\). Removing or adding at most \(a\) levels in the error sum costs \(O_C(a/k)\), since the unnormalized squared norm of each one-step error is at most \[2\|T_{l+1}\|_2^2+2C\|T_l\|_2^2=O_C(I),\] by the bounded column and the diagonal endpoint control. This also explains why the matrices may already include the block and input clipping modifications: their cost on \(\Psi\) is the right-\(R\) cost in Equation (118).

Iterating Equation (130), for each fixed \(j\) we may replace the extra-index tensor \(T_{l+j}\) by \[(i_1,\ldots,i_j)\longmapsto (H_{i_j})_{(1)}\cdots(H_{i_1})_{(1)}T_l\] in the normalized band mean-square norm, with error \(o(1)\). More explicitly, if \(e_a\) is the root mean-square error in the one-step estimate on the band shifted by \(a\), the root error after \(j\) steps is at most \[\sum_{a=0}^{j-1}C^{a/2}e_{j-1-a}.\] Here multiplication by a column of \(H_i\) costs at most \(\sqrt C\) in the summed coefficient norm. Each \(e_a=o(1)\) at fixed \(a\), so the displayed bound vanishes. Contraction with \(\mathfrak h_j(N)\) multiplies squared coefficient norm by at most \(j!\): it first symmetrizes the contracted indices, an orthogonal projection, and Gaussian Hermite orthogonality then gives the factor \(j!\).

Apply Lemma 24 to the band of levels in \(\Psi\). For each fixed \(m\), the preceding replacements and Equation (127) give \[\Psi(Y_t)=P_{m,t}(D)\Psi(Y_0)+\mathcal E_{m,t},\qquad D=H[N],\] where \[P_{m,t}(x)=e^{-t}\sum_{j=0}^m \frac{(2t)^{j/2}}{j!}\mathfrak h_j^{(1)}(x)\] and \[ \limsup\|\mathcal E_{m,t}\|_2 \le \frac{(2t)^{(m+1)/2}}{\sqrt{(m+1)!}}. \tag{131}\] To justify the conversion of a tracial polynomial error to the norm of its action on \(\Psi(Y_0)\), use \(\|X\Psi(Y_0)\|_{\mathrm{HS}}=\|XR\|_{\mathrm{HS}}\) and Equation (124). The band version of the remainder bound follows from the uniform finite-list diagonal fixed at the start of the section. Its right side tends to zero as \(m\to\infty\).

The spectral measures of \(D\) averaged with \(\tau\) have total mass tending to one, converge in moments to the standard Gaussian by Equation (125), and have uniformly bounded exponential moments at every fixed linear parameter by Equation (126). These facts imply weak convergence to the standard Gaussian and convergence of the integrals of every continuous function bounded by a polynomial times a fixed linear exponential. Here is a direct justification: the second moment gives tightness; the higher moment bounds give uniform integrability of each polynomial, so every subsequential limit has the Gaussian moments. Its exponential moments determine its characteristic function by its power series, hence determine the law. Using a larger exponential parameter makes the tails of any of the specified test functions uniformly small.

The scalar Hermite generating identity gives, with \(Z\) standard normal, \[e^{-t}\sum_{j\ge0}\frac{(2t)^{j/2}}{j!}\mathfrak h_j^{(1)}(Z) =\exp(\sqrt{2t}\,Z-2t)\] in squared mean. In fact its squared tail norm is \(e^{-2t}\sum_{j>m}(2t)^j/j!\). Define the matrix multiplier \[M=\exp(\sqrt{2t}\,D-2t\mathrm{Id}).\] The just-proved convergence of spectral integrals, applied with each fixed \(m\), yields \[ \lim\tau(|M-P_{m,t}(D)|^2) =e^{-2t}\sum_{j>m}\frac{(2t)^j}{j!},\qquad \tau(M^2)\longrightarrow1. \tag{132}\] Together with Equations (131) and (124), first sending the sequence index and then \(m\) to infinity, this proves \[\mathbb E\|\Psi(Y_t)-M\Psi(Y_0)\|_{\mathrm{HS}}^2=o(1).\] There is no operator-norm bound on \(M\) in this argument; only its weighted trace moments are used.

Write \(\Psi(Y_0)=RJ_0\), with \(J_0J_0^*\) equal to the support projection. On each block \((R-R_*)^2\le\epsilon^2R_*^2\), and \(R_*\) is scalar there. By cyclicity and positivity of trace, \[\mathbb E\|M(R-R_*)J_0\|_{\mathrm{HS}}^2 \le\epsilon^2\tau(M^2)=o(1).\] Thus \(\Psi(Y_t)\) is close in squared mean to \(MR_*J_0\). Its reduction is exactly \[(MR_*J_0)(MR_*J_0)^* =MR_*^2M=\bigoplus_\beta\beta^2 \exp(\sqrt{8t}\,H[N]-4t\mathrm{Id}_\beta)=\sigma.\] Finally Equation (107) transfers the approximation of these rectangular matrices to the approximation of their positive reduced square roots, proving Equation (129). ◻

Proof. Write \(a=1/L\) and \(b=1+1/L\), so \(b-a=1\). If \(r=\max(v,w)>0\), discard the two denominator factors associated with the smaller eigenvalue and substitute \(q=Tr^2\). This gives \[M_L(v,w)\le4\int_0^\infty \frac{dq}{(1+aq)(1+bq)} =4\log(b/a)=4\log(1+L).\] For the identity in the middle, integration up to \(Q\) gives \(\log(1+bQ)-\log(1+aQ)\), which tends to \(\log(b/a)\). If \(v,w\) are positive and comparable as in the statement, let \(r_0=\min(v,w)\). Discard the two factors with coefficient \(a\), and use \(b\ge1\). The remaining integral is at most \(\int_0^\infty(1+Tr_0^2)^{-2}\,dT=r_0^{-2}\). Since \(v+w\le3r_0\), this proves the second bound. ◻

Proof. Let \(p_T(z\mid y)\) denote the density of the centered Gaussian with covariance \(C_T(y)\), and let \(p_T(z)\) be its marginal density. For each output \(z\), the posterior density with respect to \(\eta\) is \(q_z(y)=p_T(z\mid y)/p_T(z)\). Applying \(2\lambda H\le\mathcal I\) to this density and then averaging in \(z\) gives \[ 2\lambda\mathsf I(Y_t;Z_T) \le\mathbb E\big|D_y\log p_T(Z_T\mid Y_t)\big|^2. \tag{139}\] Here and below the expectation on the right includes the observation noise. The posterior normalization does not depend on \(y\) and therefore does not contribute to the derivative.

We compute that derivative explicitly. For any positive covariance \(C\) and its derivative \(D_iC\), differentiation of its determinant and inverse yields \[D_i\log p(z\mid y) =\tfrac12\left( z^*C^{-1}(D_iC)C^{-1}z-\mathop{\mathrm{tr}}(C^{-1}D_iC)\right).\] Conditional on \(y\), set \(B_i=C^{-1/2}(D_iC)C^{-1/2}\) and \(g=C^{-1/2}z\). The vector \(g\) is standard Gaussian. Expanding its fourth moments gives \(\mathbb E(g^*B_ig-\mathop{\mathrm{tr}}B_i)^2=2\mathop{\mathrm{tr}}B_i^2\); indeed the three Gaussian pairings give \((\mathop{\mathrm{tr}}B_i)^2+2\mathop{\mathrm{tr}}B_i^2\) before subtracting the mean. Consequently Equation (139) becomes \[ \mathsf I(Y_t;Z_T)\le\frac1{4\lambda} \mathbb E\sum_i\mathop{\mathrm{tr}}\big(C_T^{-1}D_iC_T\,C_T^{-1}D_iC_T\big). \tag{140}\]

These calculations also apply to the Sobolev matrix field \(R\). To see this without assuming additional regularity, fix \(T,L\) and the finite dimension of one member of the sequence. The map \(R\mapsto C_T\) has bounded derivative on positive semidefinite matrices, and its eigenvalues lie in \([1,1+L]\). For each fixed \(z\), the map \(C\mapsto p(z\mid C)^{1/2}\) has bounded derivative on this covariance interval. The Sobolev chain rule therefore puts \(q_z^{1/2}\) in the form domain; \(p_T(z)>0\) and is a harmless fixed normalization. The Sobolev approximation at the opening of Section 2 extends the logarithmic Sobolev inequality to this form-domain function. The squared derivatives integrate in \(z\) to the nonnegative finite expression in Equation (140). Approximation in the form domain and Tonelli thus justify both that equation and the subsequent scale integration. The finite-dimensional bounds used for form-domain membership do not enter its numerical coefficient.

In an eigenbasis of \(R\), write \(v,w\) for the row and column eigenvalues. The covariance function is \[c_T(v)=1+f_L(Tv^2) =\frac{1+bTv^2}{1+aTv^2}, \qquad a=1/L,\quad b=1+1/L.\] Its divided difference, including its derivative at \(v=w\), is \[\frac{c_T(v)-c_T(w)}{v-w} =\frac{T(v+w)}{(1+aTv^2)(1+aTw^2)}.\] Multiplying its square by \(c_T(v)^{-1}c_T(w)^{-1}\) and integrating against \(dT/T^2\) gives exactly the multiplier in Equation (136) for the squared entry \(|(D_iR)_{vw}|^2\). When \(v=w=0\), the covariance derivative is zero, which explains the value assigned to \(M_L(0,0)\).

It remains to obtain a bound independent of the fixed cap. For noncomparable pairs with at least one positive eigenvalue, Equation (134) gives entrywise \[|(D_iR)_{vw}|^2 =\frac{(v+w)^2}{4(v-w)^2} |[\widehat H_i,R]_{vw}|^2 \le\frac94|[\widehat H_i,R]_{vw}|^2.\] This includes a pair with exactly one zero eigenvalue. Their total expected derivative energy is therefore \(o(1)\). The kernel–kernel derivative is zero almost everywhere, as established in the Sylvester construction, and in any case has zero covariance multiplier. Lemma 35 now yields \[\int_0^\infty\mathsf I(Y_t;Z_T)\,\frac{dT}{T^2} \le\frac1{4\lambda} \left(9\mathbb E|DR|^2+4\log(1+L)\,o(1)\right).\] Take the sequence limit at this fixed \(L\), and use Equation (133). Stationarity makes the bound independent of \(t\). This proves the proposition. In particular, the cap-dependent bound on noncomparable pairs is used only before its vanishing error has been removed. ◻

Proof. Direct differentiation gives \[h_T'(v)=\frac{Tv} {(1+Tv^2/L)^{3/2}(1+(1+1/L)Tv^2)^{1/2}}.\] For \(v>0\), substitution \(q=Tv^2\) therefore shows \[\int_0^\infty|h_T'(v)|^2\,\frac{dT}{T^2}=K_L.\] Discarding two of the three factors \(1+q/L\) bounds \(K_L\) by the integral evaluated in Lemma 35, namely \(\log(1+L)\). If \(v\ge w\ge0\), the fundamental theorem of calculus and Cauchy–Schwarz give \[|h_T(v)-h_T(w)|^2 \le(v-w)\int_w^v|h_T'(r)|^2\,dr.\] Integrating in \(T\) and using Tonelli proves the scalar bound \(K_L(v-w)^2\). The value of the derivative integral at the single endpoint \(r=0\) does not affect this argument, so it covers \(w=0\).

Choose an orthonormal eigenbasis \((e_a)\) of \(A\) with eigenvalues \(v_a\) and an orthonormal eigenbasis \((f_b)\) of \(B\) with eigenvalues \(w_b\). The entry of their functional-calculus difference between these two bases is \[\langle e_a,(h_T(A)-h_T(B))f_b\rangle =(h_T(v_a)-h_T(w_b))\langle e_a,f_b\rangle.\] Multiply the scalar bound by \(|\langle e_a,f_b\rangle|^2\) and sum over \(a,b\). The sum on the right is \(K_L\|A-B\|_{\mathrm{HS}}^2\), proving Equation (141). Finally, conditional on all data except \(G\), Gaussian isometry gives \[\mathbb E_G|Z_T-\widetilde Z_T|^2 =\|h_T(R(Y_t))-h_T(\sigma^{1/2})\|_{\mathrm{HS}}^2.\] Apply the matrix bound and the squared-mean convergence in Proposition 34. The resulting factor \(K_L\) is fixed while the sequence limit is taken. ◻

Proof. Put \(q=1-\alpha^2\). Given a posterior law \(\pi(dn)\) of \(N\) at fixed \((Z,\mathcal D)\), the density at \(y\) after adding the OU noise is the integral of \((2\pi q)^{-d/2}\exp(-|y-\alpha n|^2/(2q))\) against \(\pi\). Differentiating this kernel and dividing by its integral gives \[D\log p(y\mid Z,\mathcal D) =-\frac{y-\alpha\mathbb E[N\mid N_u=y,Z,\mathcal D]}q.\] The score of the standard Gaussian is \(-y\). Subtracting it proves Equation (146). Kernel differentiation is justified first with \(N\) truncated; on compact positive time intervals its derivatives are bounded by Gaussian kernels times polynomials, and truncation can then be removed by conditional second moments.

The OU semigroup has generator \(\Delta-y\cdot D\). For a positive posterior density \(r_u\) relative to \(\gamma\), integration by parts gives \[\frac{d}{du}\int r_u\log r_u\,d\gamma =-\int\frac{|Dr_u|^2}{r_u}\,d\gamma.\] This is the entropy dissipation calculation of Lemma 5; positive-time smoothing, truncation of the density and spatial cutoffs justify it for finite-entropy posteriors. Averaging over \((Z,\mathcal D)\) and integrating over \(\mathcal J\) bounds its nonnegative right side by the initial conditional information. Substitution of the score formula proves Equation (147).

For the last assertion, observation noise is independent of the initial data and diffusion path, and its law depends on them only through \(Y_t\). Hence \((\mathcal D,N)\longrightarrow Y_t\longrightarrow Z_T\) is a Markov chain. The chain rule and data processing give \[\mathsf I(N;Z_T\mid\mathcal D) \le\mathsf I(\mathcal D,N;Z_T)\le\mathsf I(Y_t;Z_T).\] These informations are finite for each member of the sequence: the covariance interval \([\mathrm{Id},(1+L)\mathrm{Id}]\) bounds the output entropy from above by that of a Gaussian of covariance \((1+L)\mathrm{Id}\) and bounds its conditional entropy from below by that of a standard Gaussian. Thus \(\mathsf I(Y_t;Z_T)\le(d/2)\log(1+L)\) in dimension \(d\). This finiteness assertion is used only to justify the identities. ◻

Proof. For a block, consider the linear map \(\mathcal H:v\mapsto H[v]\) between its Euclidean coordinate space and its matrix space with the ordinary Hilbert–Schmidt norm. Its adjoint is \[(\mathcal H^*Q)_i=\mathop{\mathrm{tr}}(H_iQ),\] because \(H_i\) is real symmetric. Taking the adjoint in Equation (144) gives \[ \sum_i|\mathop{\mathrm{tr}}(H_iQ)|^2\le C_1\mathop{\mathrm{tr}}(Q^*Q). \tag{150}\] This step has no factor involving the dimension of the block.

At fixed \(u,T\), select precisely those positive blocks for which \(x=T\beta^2\in\mathcal X\). On a selected block set \[V_i=\mathop{\mathrm{tr}}_\beta \bigl(H_iP(u,T\beta^2,H[(N_u)_\beta],H[(Z_T)_\beta])\bigr), \qquad i\in\beta,\] and set all other components, including those in the kernel, equal to zero. This vector is measurable with respect to \((N_u,Z_T,\mathcal D)\). If \(m=\mathbb E[N-\alpha N_u\mid N_u,Z_T,\mathcal D]\), completing the square gives \[\mathbb E|m|^2\ge2\mathbb E(N-\alpha N_u)\cdot V-\mathbb E|V|^2.\] The coordinates of distinct selected blocks are orthogonal. Their costs therefore add, and Equation (150) bounds the cost by \(C_1\) times the sum of their unnormalized squared matrix norms. Their linear terms contract exactly to \[(N-\alpha N_u)\cdot V =\sum_{\beta\text{ selected}}\mathop{\mathrm{tr}}_\beta \bigl((H[N_\beta]-\alpha H[(N_u)_\beta])P\bigr).\] Substitute these two statements in Lemma 38 and integrate in \(T\).

For each realized positive block separately, the change of variable \(x=T\beta^2\) has the exact Jacobian \[ T=\frac{x}{\beta^2},\qquad dT=\frac{dx}{\beta^2},\qquad \boxed{\displaystyle\frac{dT}{T^2}=\beta^2\frac{dx}{x^2}}. \tag{151}\] Thus the block selection becomes \(x\in\mathcal X\), and the coefficient of each block trace becomes precisely \(\beta^2\). This proves Equation (149).

For completeness, all its polynomial terms are integrable on the compact parameter rectangle. The moment bounds of Lemma 33 control \(A_\beta\) and \(A_{u,\beta}\) under the weighted trace. Conditional on the initial data, diffusion path, and OU noise, \(B_{\beta,x}\) is a Gaussian matrix series whose vector covariance is at most \((1+L)\mathrm{Id}\). If its coefficients are \(K_j=\sum_i Q_{ij}H_i\) for a square root \(Q\) of that covariance, then for every vector \(v\) \[\sum_j|K_jv|^2\le(1+L)\sum_i|H_iv|^2\le C(1+L)|v|^2.\] The same Gaussian even-moment proof therefore gives, for every fixed integer \(p\ge1\), \[ \mathbb E_G\mathop{\mathrm{tr}}_\beta|B_{\beta,x}|^{2p} \le[C(1+L)]^p(2p-1)!!\mathop{\mathrm{tr}}_\beta\mathrm{Id}. \tag{152}\] Trace Hölder bounds each mixed word by its factors’ moments. After multiplying by \(\beta^2\) and summing, the right side is a constant times \(\tau(\mathrm{Id})\), which is finite and bounded along the sequence. The measure \(\kappa(u)\,du\,dx/x^2\) is finite on the chosen rectangle. Tonelli applies to the square terms, while these moment bounds justify Fubini for the linear terms. One can equivalently first truncate \(V\) in the conditional-mean inequality and then remove that truncation in this integrated squared norm. ◻

Proof. The flattening bound and the inverse substitution in Equation (151) give \[\begin{align*} &\mathbb E\sum_\beta\beta^2\int_{\mathcal X} \|B_{\beta,x}-\widetilde B_{\beta,x}\|_{\mathrm{HS}}^2\,\frac{dx}{x^2} \\ &\quad=\int_0^\infty\mathbb E\sum_{\beta:T\beta^2\in\mathcal X} \|H_\beta[(Z_T-\widetilde Z_T)_\beta]\|_{\mathrm{HS}}^2\,\frac{dT}{T^2} \\ &\quad\le C_1\int_0^\infty \mathbb E|Z_T-\widetilde Z_T|^2\,\frac{dT}{T^2}=o(1), \end{align*}\] by Lemma 37. All projected errors are summed before this comparison; no small block weight is divided out. Multiplication by the bounded weight \(\kappa(u)\) and integration in the compact interval \(\mathcal J\) preserve \(o(1)\).

To pass from this squared norm to arbitrary fixed products, include the parameter integrals in the finite tracial functional \[\mathcal T(U)=\int_{\mathcal J}\int_{\mathcal X} \kappa(u)\mathbb E\sum_\beta\beta^2\mathop{\mathrm{tr}}_\beta U_{\beta,u,x} \,\frac{dx}{x^2}\,du.\] Every matrix in the statement has bounded \(L^p(\mathcal T)\) norm for every fixed finite \(p\). For \(B\) this was proved in Equation (152); for \(\widetilde B\) the same proof applies because its covariance is also at most \((1+L)\mathrm{Id}\). The remaining matrices are covered by the Gaussian moment lemma. For any \(q>p\ge2\), interpolation with \(1/p=\theta/2+(1-\theta)/q\) gives \[\|B-\widetilde B\|_{L^p(\mathcal T)} \le\|B-\widetilde B\|_{L^2(\mathcal T)}^\theta \|B-\widetilde B\|_{L^q(\mathcal T)}^{1-\theta}=o(1).\] For lower powers, Hölder and boundedness of \(\mathcal T(\mathrm{Id})\) give the same conclusion. Expand the difference of two words of length \(r\) by replacing one factor at a time. Trace Hölder with all exponents \(r\) bounds each resulting trace by the product of its factors’ \(L^r(\mathcal T)\) norms. The changed factor tends to zero and the others remain bounded. There are only finitely many such terms. This proves the claim, including polynomial squared norms and linear terms with \(A-\alpha A_u\). ◻

Proof. Suppose a continuous linear functional on \(L^p(\nu)\) annihilates all polynomials. Its representing density \(F\) belongs to \(L^{p'}(\nu)\), where \(p'=p/(p-1)\) and \(p'=\infty\) if \(p=1\). Hölder and the hypothesis show that the finite signed measure \(F\,d\nu\) has exponential moments of every linear magnitude. For each \(\xi\in\mathbb R^k\), the function \[z\longmapsto\int e^{z\xi\cdot w}F(w)\,d\nu(w)\] is entire: on every compact set of complex \(z\), its derivatives are dominated by an integrable exponential. All its derivatives at zero vanish because \((\xi\cdot w)^j\) is a polynomial. Its Taylor series and hence the entire function are zero. Evaluating at \(z=\mathrm i\) proves that the Fourier transform of \(F\,d\nu\) vanishes. Fourier uniqueness for finite measures can here be seen by convolving with a centered Gaussian: the convolution has zero Fourier transform and an integrable Fourier inversion formula, so is zero for every positive Gaussian variance; letting that variance tend to zero shows the original measure is zero. Thus \(F=0\) almost everywhere. Duality, or equivalently separation of a proper closed subspace, proves the density assertion. ◻

Proof. First we prove the matrix comparison \[ H[g_x(A)G]-g_x(A)H[G]\longrightarrow0 \quad\hbox{in weighted squared Hilbert--Schmidt norm}. \tag{154}\] All notation in this calculation is on one block. Define \[J(v)_{ab}=(H_bv)_a, \qquad D(v)=H[v]-J(v), \qquad \Delta_{abi}=(H_i)_{ab}-(H_b)_{ai}.\] Put \(q=g_x(A)\) and \(B=H[G]\). The exact columnwise identity is \[ H[qG]-qB=D(qG)+K-qD(G), \qquad K_{\cdot b}=[H_b,q]G. \tag{155}\] Indeed the \(b\)th column of \(J(qG)-qJ(G)\) is \(H_bqG-qH_bG\). Conditional Gaussian averaging therefore bounds the square of the left side by \[ 6(1+L)\sum_{a,b,i}|\Delta_{abi}|^2 +3\sum_b\|[H_b,g_x(A)]\|_{\mathrm{HS}}^2. \tag{156}\] For the first and last terms in Equation (155), we used \(\|q\|_{\mathrm{op}}\le\sqrt{1+L}\) and the Gaussian isometry for the linear map \(D\).

The function \(g_x\) is Lipschitz with a bound depending only on \(t,L\), uniformly for all \(x>0\). In fact, with \(a=\sqrt{8t}\) and \(r=xe^{av-4t}/L\), differentiation gives \[|g_x'(v)| =\frac{aLr}{2(1+r)^2g_x(v)}\le\frac{aL}{8}.\] In an eigenbasis of \(A\) with eigenvalues \(r_a\), the squared commutator norm is \[\|[H_b,g_x(A)]\|_{\mathrm{HS}}^2 =\sum_{a,c}|(H_b)_{ac}|^2|g_x(r_c)-g_x(r_a)|^2 \le\operatorname{Lip}(g_x)^2\|[H_b,A]\|_{\mathrm{HS}}^2.\] This is a scalar Lipschitz estimate for matrix entries in the Hilbert–Schmidt norm. Averaging \(A=\sum_iN_iH_i\) over \(N\) gives \[\mathbb E_N\sum_b\|[H_b,A]\|_{\mathrm{HS}}^2 =\sum_{b,i}\|[H_b,H_i]\|_{\mathrm{HS}}^2.\] Multiply Equation (156) by the block weight, sum, and average. The total symmetry defect and Equation (145) prove Equation (154), uniformly in \(x\) at fixed \(t,L\).

Every fixed higher moment of both sides of this comparison is bounded. Conditional on \(A\) and the initial data, \(H[g_x(A)G]\) is a Gaussian matrix series with coefficient-square sum at most \((1+L)S\), by the quadratic-form argument used in Equation (152). Also the Schatten ideal inequality gives \[\|g_x(A)H[G]\|_{p}\le\sqrt{1+L}\,\|H[G]\|_{p}\] on each block. Interpolation and telescoping products, exactly as in Lemma 40, show that these two matrices may be interchanged in every fixed polynomial trace.

It remains to identify traces containing \(g_x(A)H[G]\). Lemma 33 gives the joint scalar Gaussian moment law for \(A=H[N]\), \(H[G']\), and \(H[G]\). In particular, \(A_u=\alpha A+\sqrt{1-\alpha^2}H[G']\) has the required joint law with them. To include the bounded continuous function \(g_x(A)\), fix \(x\) and a mixed word of length \(r\). Choose a polynomial \(p\) approximating \(g_x\) in a sufficiently high finite \(L^q(\gamma_1)\) norm, where \(\gamma_1\) is the scalar standard Gaussian law. Choose \(q\) larger than all the Hölder exponents needed for this fixed word, and take the approximation error at most \(1\); then \(\|p\|_{L^q(\gamma_1)} \le\sqrt{1+L}+1\). Such approximations exist by the polynomial-density argument in Lemma 41, applied to one Gaussian coordinate. The scalar spectral measures of \(A\) under \(\tau\) converge to \(\gamma_1\) by the moment and exponential-moment bounds in Lemma 33. Their higher moments are uniformly bounded. Consequently, for even \(q\), \[ \tau\bigl(|g_x(A)-p(A)|^q\bigr) \longrightarrow\mathbb E|g_x(n)-p(n)|^q. \tag{157}\] To justify this last passage directly, cut the scalar spectral variable off at \(|v|\le M\), use weak convergence for the bounded continuous integrand there, and bound the complement by a higher uniform moment of \(A\). Since \(g_x\) is bounded and \(p\) has fixed degree, that tail tends to zero as \(M\to\infty\).

Replace each occurrence of \(g_x(A)\) by \(p(A)\). Telescoping the word and applying trace Hölder bounds the replacement error by a constant times the \(L^q\) error in Equation (157), with all other factors controlled by fixed higher moments. For factors \(p(A)\), the preceding \(L^q\) bound and moment convergence give a bound independent of how accurately \(p\) approximates \(g_x\). Thus the error constant stays bounded as that approximation is improved. The remaining word is polynomial in the three independent Gaussian matrix series, so its trace has the scalar Gaussian limit. Let the scalar approximation error tend to zero. This proves the claimed limit at each fixed \(u,x\); the scalar third coordinate is \(g_x(n)g\).

Finally, the moment bounds for the original, bounded function \(g_x\) are uniform on the compact parameter rectangle, as are the bounds on polynomial coefficients in \(u,x\) and on \(\alpha\). They dominate each mixed trace by an integrable constant. Dominated convergence thus permits parameter integration. This last argument uses the uniform bounds for \(g_x\) itself, and does not require choosing one polynomial approximant uniformly in \(x\). ◻

Proof. Fix compact positive intervals \(\mathcal J,\mathcal X\), and let \(n,g',g\) be independent scalar standard Gaussians. Write \(n_u=\alpha n+\sqrt{1-\alpha^2}\,g'\), and fix a scalar polynomial \(P(u,x,a,b)\). Give its matrix version any fixed ordering of the factors of each monomial. Lemmas 39, 40, and 42, in that order, imply \[\begin{align*} &\liminf\int_0^\infty\mathsf I(Y_t;Z_T)\,\frac{dT}{T^2} \\ &\quad\ge\int_{\mathcal J}\int_{\mathcal X}\kappa(u) \mathbb E\big[2(n-\alpha n_u)P(u,x,n_u,g_x(n)g) \\[-2pt] &\hspace{58mm}-C_1P(u,x,n_u,g_x(n)g)^2\big] \,\frac{dx}{x^2}\,du. \tag{159}\end{align*}\] If the matrix polynomial is not symmetric, its prelimit cost is \(\mathop{\mathrm{tr}}(P^*P)\) as in Equation (149). The mixed trace law takes this cost to the real scalar \(P^2\), so symmetry of the chosen ordering is unnecessary.

The sequence limit in Equation (159) has been taken for each fixed polynomial. We now optimize the tests in one squared-mean space containing the compact parameters. Give \((u,x)\) the finite measure \(\kappa(u)\,du\,dx/x^2\) on \(\mathcal J\times\mathcal X\), independently of the scalar Gaussians \(n,g',g\), and put \[ W=(u,x,n_u,g_x(n)g),\qquad r=n-\alpha n_u. \tag{160}\] The first two coordinates of \(W\) are bounded. At each fixed \(u,x\), the variable \(n_u\) is standard Gaussian, and \(|g_x(n)g|\le\sqrt{1+L}|g|\). Cauchy–Schwarz therefore gives all linear exponential moments for its finite law, uniformly on the compact parameter rectangle. Lemma 41 shows that polynomials in the four coordinates of \(W\) are dense in this \(L^2\) space.

Since \(\mathbb E[r^2\mid u,x]=1-\alpha^2\le1\), the conditional mean \(m(W)=\mathbb E[r\mid W]\) belongs to the same space. For the finite joint measure just defined, completion of the square gives \[\int\big(2rP(W)-C_1P(W)^2\big) =C_1^{-1}\int m^2-C_1\int(P-m/C_1)^2.\] Density makes the infimum of the final squared error zero, using single four-variable polynomials in the integrated space. Taking their supremum and then disintegrating with respect to \((u,x)\) turns the right side of Equation (159) into \[ C_1^{-1}\int_{\mathcal J}\int_{\mathcal X} \frac{\alpha^2}{(1-\alpha^2)^2} \mathbb E\left|\mathbb E[n-\alpha n_u\mid n_u,g_x(n)g]\right|^2 \,\frac{dx}{x^2}\,du. \tag{161}\] The conditional expectation here is taken at each fixed \(u,x\). All approximation constants used before this optimization may depend on the fixed polynomial, \(t,L\), and the compact intervals. The coefficient \(C_1\) depends only on Equation (150).

We verify the scalar information identity including its endpoints. Fix \(x>0\) and write \(z_x=g_x(n)g\). Its information is finite: conditional variance lies in \([1,1+L]\), so Gaussian entropy comparison gives \[ 0\le\mathsf I(n;z_x)\le\tfrac12\log(1+L). \tag{162}\] Indeed \(\mathbb Ez_x^2\le1+L\) bounds its entropy above by \(\frac12\log(2\pi e(1+L))\), while its conditional entropy is \(\frac12\log(2\pi e)+\mathbb E\log g_x(n)\) and is at least \(\frac12\log(2\pi e)\). Apply the scalar version of Lemma 38 and its entropy dissipation identity between times \(a\) and \(b\), where \(0<a<b<\infty\). This gives \[\begin{align*} \mathsf I(n_a;z_x)-\mathsf I(n_b;z_x) =\int_a^b\frac{\alpha^2}{(1-\alpha^2)^2} \mathbb E\left|\mathbb E[n-\alpha n_u\mid n_u,z_x]\right|^2\,du. \end{align*}\] As \(a\downarrow0\), the joint pair \((n_a,z_x)\) converges in law to \((n,z_x)\). Lower semicontinuity of relative entropy, applied to the joint law and the product of its marginals, gives \(\mathsf I(n;z_x)\le\liminf_{a\downarrow0}\mathsf I(n_a;z_x)\). The reverse inequality follows from the Markov chain \(z_x\longrightarrow n\longrightarrow n_a\) and data processing. Thus the initial endpoint is exactly \(\mathsf I(n;z_x)\). At the other endpoint, the same chain gives \[0\le\mathsf I(n_u;z_x)\le\mathsf I(n_u;n) =-\tfrac12\log(1-\alpha^2)\longrightarrow0.\] The equality follows by subtracting the conditional Gaussian entropy of variance \(1-\alpha^2\) from that of the standard Gaussian marginal. We have proved \[ \mathsf I(n;z_x)=\int_0^\infty \frac{\alpha^2}{(1-\alpha^2)^2} \mathbb E\left|\mathbb E[n-\alpha n_u\mid n_u,z_x]\right|^2\,du. \tag{163}\]

Finally exhaust both positive parameter axes by compact intervals in Equation (161). Its integrand is nonnegative. Monotone convergence and Equation (163) prove Equation (158). In particular, the polynomial optimization and the exhaustion take place after the counterexample-sequence limit, so neither requires a uniform approximation over unbounded parameters. ◻

Proof. The Gaussian moment generating formula gives \(\mathbb E\ell=1\). Let \(P_{x,n}\) be the conditional law of \(z_x\) given \(n\), and let \(Q_x\) be its marginal law. Thus \[\mathsf I(n;z_x)=\mathbb E_n H(P_{x,n}\mid Q_x).\] Set \(p_0=\mathbb P\{|g|\ge\sqrt2\}>0\). Suppose first that \(\ell\ge e^{2t}\), and choose a scale with \(h=x\ell\in[h_0,t]\). Since \(h\le t\), \[\mathop{\mathrm{Var}}(z_x\mid n)=1+\frac{h}{1+h/t}\ge h/2.\] It follows that the event \(A_h=\{|z|\ge\sqrt h\}\) has conditional probability \(p=P_{x,n}(A_h)\ge p_0\).

To bound its marginal probability, use an independent variable \(\ell'\) with the law of \(\ell\). On \(x\ell'\le1\), the conditional variance of the mixture component is at most \(2\), so its probability of \(A_h\) is at most \(2e^{-h/4}\) by the Gaussian exponential bound. On the complementary event, Markov’s inequality and \(\mathbb E\ell'=1\) give probability at most \(x\). Hence \[ q=Q_x(A_h)\le2e^{-h/4}+x \le2e^{-h/4}+te^{-2t}\le3e^{-h/4}. \tag{165}\] The last inequality holds for \(t\ge1\) and \(h\le t\), because \(te^{-2t}\le e^{-t/4}\le e^{-h/4}\).

Apply data processing to the event indicator. The binary relative entropy is bounded below by \[\begin{align*} H(P_{x,n}\mid Q_x) &\ge p\log(p/q)+(1-p)\log\big((1-p)/(1-q)\big)\\ &\ge p\log(1/q)-\log2\\ &\ge p_0(h/4-\log3)-\log2. \end{align*}\] Choose once and for all \[h_0\ge\max\{1,\,8\log3+8\log2/p_0\}, \qquad c=p_0/8.\] The preceding expression is then at least \(ch\) for \(h\in[h_0,t]\). This entropy estimate is made separately for each conditional law \(P_{x,n}\); the chosen event may therefore depend on \(n\) through \(h=x\ell\).

Nonnegativity of relative entropy permits restricting the scale integral to these choices of \(n\) and \(x\). Tonelli and the change of variable \(x=h/\ell\) yield \[\begin{align*} \int_0^\infty\mathsf I(n;z_x)\,\frac{dx}{x^2} &\ge\mathbb E\left[\mathbf 1_{\{\ell\ge e^{2t}\}} \int_{h_0/\ell}^{t/\ell} H(P_{x,n}\mid Q_x)\,\frac{dx}{x^2}\right]\\ &\ge c\mathbb E\left[\ell\mathbf 1_{\{\ell\ge e^{2t}\}} \int_{h_0}^t\frac{dh}{h}\right]\\ &=c\mathbb E\big[\ell\mathbf 1_{\{\ell\ge e^{2t}\}}\big]\log(t/h_0). \end{align*}\] Under the probability change of measure with density \(\ell\), completion of the square makes \(n\) a Gaussian of mean \(\sqrt{8t}\) and variance \(1\). Therefore \(\log\ell\) under that measure is Gaussian of mean \(4t\) and variance \(8t\). If \(\Phi\) is the standard Gaussian distribution function, then \[\mathbb E\big[\ell\mathbf 1_{\{\ell\ge e^{2t}\}}\big] =\Phi(\sqrt{t/2})\ge\tfrac12.\] This proves both inequalities in Equation (164). ◻

Proof of Theorem 1. Suppose that the asserted dimension-free bound fails. Lemma 7 gives compact centered counterexamples with normalized logarithmic Sobolev constant \(1\) and linear subgaussian parameter tending to zero. Proposition 14 shows that their positive-time regularized constants converge to \(1\). Proposition 15 supplies the stationary laws and near-eigenfunction fields at the fixed regularization time \(s=1/8\), in either the strict entropy branch or the spectral gap branch. Proposition 18 gives the exact prediction curve in both cases. Proposition 20 then gives the symmetric tensor hierarchy, and Proposition 34 constructs its matrix square root and frozen block description. These are precisely the inputs used in the present section.

For each fixed \(t>0\) and \(L\ge1\), combine Propositions 36 and 43. They give \[ \int_0^\infty \mathsf I\left(n; \sqrt{1+f_L(xe^{\sqrt{8t}\,n-4t})}\,g\right) \,\frac{dx}{x^2}\le C_0C_1, \tag{166}\] where \(C_0C_1\) is independent of both fixed parameters. Choose an arbitrarily large but finite \(t\ge h_0\) and the fixed cap \(L=t\). Lemma 44 makes the left side at least \((c/2)\log(t/h_0)\), which exceeds \(C_0C_1\) for a sufficiently large choice of \(t\). This is a contradiction.

All counterexample limits used to obtain Equation (166) were taken with this \(t,L\) fixed. Only afterward was their numerical size chosen. Thus no uniform estimate on the intermediate errors as \(t,L\to\infty\) is needed. The contradiction excludes an unbounded ratio \(R(\mu)/a^2\) over the measures in the theorem. Finally the convention \(\mathop{\mathrm{Ent}}_\mu(f^2)\le2R(\mu)\int|Df|^2\,d\mu\) absorbs the factor \(2\) into a universal constant and gives the asserted inequality for every smooth compactly supported real test function. ◻