Proof of The Centred Heat Gauge: Metric Properties, Comparison with the Wasserstein Distance, Behaviour Under Translations, the Squared Gauge as a Test Function, and the Polarisation Inequality

Each result cited is universally quantified over the data in its own statement. Throughout, $B$, $Q$ and the kernel pairing $\mathcal{K}$ are those of The Heat Gauge Form on Probability Measures: Algebra, Representation by Smoothed Densities, Positivity, Cauchy-Schwarz, Separation, the Triangle Inequality, the Wasserstein Bound and Derivative Bounds for the Potential and The Multiscale Gaussian Kernel: Definition by a Series, Regularity and Bounds, the Smoothed Potential of a Probability Measure and the Kernel Pairing of Two Measures §pairing in dimension $q=d$, so that $B(\mu,\nu;\mu',\nu')=\mathcal{K}(\mu,\mu')-\mathcal{K}(\mu,\nu')-\mathcal{K}(\nu,\mu')+\mathcal{K}(\nu,\nu')$ and $Q(\mu,\nu)=B(\mu,\nu;\mu,\nu)$ for $\mu,\nu,\mu',\nu'\in\mathcal{P}(\mathbb{R}^{d})$; $\varrho(\mu,\nu)=\sqrt{Q(\mu,\nu)}$ (The Heat Gauge on the Probability Measures on Euclidean Space §gauge), so $\varrho(\mu,\nu)^{2}=Q(\mu,\nu)$ and, for $\mu,\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$,

by The Centred Heat Gauge on the Wasserstein Space §gauge and Existence and Uniqueness of the Nonnegative Square Root, both summands being nonnegative (Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, claim 1; The Heat Gauge Form on Probability Measures: Algebra, Representation by Smoothed Densities, Positivity, Cauchy-Schwarz, Separation, the Triangle Inequality, the Wasserstein Bound and Derivative Bounds for the Potential §representation). For nonnegative reals $s\le t$ one has $\sqrt{s}\le\sqrt{t}$ (claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field applied to the roots) and $\sqrt{st}=\sqrt{s}\sqrt{t}$ (claim 5 of Real Powers Through the Exponential, and Elementary Asymptotic Tools: Monotonicity, Null Sequences of Negative Powers, Exponential Domination, Integer Rounding, and Square-Root and Exponential Inequalities); and $\lVert x\rVert^{2}=\sum_{i}x_{i}^{2}$, $\lVert x\rVert$ being the unique nonnegative root of that sum (claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$). The functions $K*\alpha$ for $\alpha\in\mathcal{P}(\mathbb{R}^{d})$ are of class $C^{3}$ with $\partial_{i}(K*\alpha)(x)=\int\partial_{i}K(x-y)\alpha(dy)$ and $|\partial_{i}(K*\alpha)|\le M_{1}$ (The Multiscale Gaussian Kernel: Definition by a Series, Regularity and Bounds, the Smoothed Potential of a Probability Measure and the Kernel Pairing of Two Measures §potential); their partial derivatives are continuous (clause 1 of C^k Maps on a Euclidean Open Set, claim 3 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous), hence Borel (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps), and a bounded Borel function is integrable against every probability measure with $|\int g\,d\alpha|\le\int|g|\,d\alpha\le\sup|g|$ (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures, claim 2 of Linearity and Monotonicity of the Lebesgue Integral, Simple Function and Its Integral). By claim 1 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set, $\partial_{i}\psi_{\mu,\nu}=\partial_{i}(K*\bar{\mu})-\partial_{i}(K*\bar{\nu})$, and $D\psi_{\mu,\nu}=D(K*\bar{\mu})-D(K*\bar{\nu})$ coordinatewise. The kernel $K$ is of class $C^{3}$, hence of class $C^{2}$ (clause 2 of C^k Maps on a Euclidean Open Set together with claim 2 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous applied to each $\partial_{i}K$), even, with $|K|\le M_{0}$, $|\partial_{i}K|\le M_{1}$, $|\partial_{j}\partial_{i}K|\le M_{2}$ (The Multiscale Gaussian Kernel: Definition by a Series, Regularity and Bounds, the Smoothed Potential of a Probability Measure and the Kernel Pairing of Two Measures §kernel); put $M=M_{0}+M_{1}+M_{2}$, so that all three bounds hold with $M$ (claims 2 and 3 of Elementary Arithmetic in an Ordered Field). Two elementary facts are used repeatedly. (W) Weak inequalities add: if $a\le b$ and $c\le d$ then $a+c\le b+d$, since $0\le(b-a)+(d-c)=(b+d)-(a+c)$ by claims 3 and 2 of Elementary Arithmetic in an Ordered Field. (Odd) $\partial_{i}K(-x)=-\partial_{i}K(x)$ for all $x\in\mathbb{R}^{d}$ and $i\in[d]$: for $h\ne0$, evenness gives $\bigl(K(x+he_{i})-K(x)\bigr)/h=-\bigl(K(-x+(-h)e_{i})-K(-x)\bigr)/(-h)$, where $e_{i}$ is the $i$th standard basis vector; given $\varepsilon>0$, let $\delta,\delta'$ be the radii of Partial Derivative on a Euclidean Open Set for $K$ at $x$ and at $-x$ with $\varepsilon/2$ in place of $\varepsilon$ (claim 8 of Elementary Order Arithmetic in an Ordered Field) and let $0<|h|<\min(\delta,\delta')$ (claim 9 there; $|-h|=|h|$ by claim 2 of Properties of the Absolute Value in an Ordered Field); then the left side lies within $\varepsilon/2$ of $\partial_{i}K(x)$ and the right side within $\varepsilon/2$ of $-\partial_{i}K(-x)$, so $|\partial_{i}K(x)+\partial_{i}K(-x)|<\varepsilon$ (claim 5 of Properties of the Absolute Value in an Ordered Field), and $\partial_{i}K(x)=-\partial_{i}K(-x)$ by Comparison of Real Numbers with Arbitrary Positive Slack §vanishing.

Proof of claim 1.

Comparisons. By ($\rho$) and the nonnegativity of each summand, $\lVert m(\mu)-m(\nu)\rVert^{2}\le\rho(\mu,\nu)^{2}$ and $Q(\bar{\mu},\bar{\nu})\le\rho(\mu,\nu)^{2}$ (claim 3 of Elementary Arithmetic in an Ordered Field); taking roots, $\lVert m(\mu)-m(\nu)\rVert\le\rho(\mu,\nu)$ and $\varrho(\bar{\mu},\bar{\nu})\le\rho(\mu,\nu)$. By The Mean of a Square-Integrable Probability Measure, Its Lift, Its Centring, and Functions of the Mean and Centred Integrals as Test Functions §mean, $\lVert m(\mu)-m(\nu)\rVert^{2}\le W_{2}(\mu,\nu)^{2}$ (claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field); by The Heat Gauge Form on Probability Measures: Algebra, Representation by Smoothed Densities, Positivity, Cauchy-Schwarz, Separation, the Triangle Inequality, the Wasserstein Bound and Derivative Bounds for the Potential §wasserstein-bound and The Mean of a Square-Integrable Probability Measure, Its Lift, Its Centring, and Functions of the Mean and Centred Integrals as Test Functions §centring, $Q(\bar{\mu},\bar{\nu})\le C_{Q}W_{2}(\bar{\mu},\bar{\nu})^{2}\le C_{Q}W_{2}(\mu,\nu)^{2}$ (claim 5 of Elementary Arithmetic in an Ordered Field). Adding by (W), $\rho(\mu,\nu)^{2}\le(1+C_{Q})W_{2}(\mu,\nu)^{2}$, and taking roots, $\rho(\mu,\nu)\le\sqrt{1+C_{Q}}\,W_{2}(\mu,\nu)=C_{\rho}W_{2}(\mu,\nu)$.

Metric. We verify Metric Space. Nonnegativity: $\rho(\mu,\nu)\ge0$ as a nonnegative square root. Symmetry: $\lVert m(\nu)-m(\mu)\rVert=\lVert(-1)(m(\mu)-m(\nu))\rVert=\lVert m(\mu)-m(\nu)\rVert$ (claim 5 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$) and $Q(\bar{\nu},\bar{\mu})=Q(\bar{\mu},\bar{\nu})$ (The Heat Gauge Form on Probability Measures: Algebra, Representation by Smoothed Densities, Positivity, Cauchy-Schwarz, Separation, the Triangle Inequality, the Wasserstein Bound and Derivative Bounds for the Potential §algebra), so $\rho(\nu,\mu)=\rho(\mu,\nu)$ by ($\rho$). Identity: $\rho(\mu,\mu)=0$ since $m(\mu)-m(\mu)=0_{\mathbb{R}^{d}}$ has norm $0$ (claim 3 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$) and $Q(\bar{\mu},\bar{\mu})=0$ (The Heat Gauge Form on Probability Measures: Algebra, Representation by Smoothed Densities, Positivity, Cauchy-Schwarz, Separation, the Triangle Inequality, the Wasserstein Bound and Derivative Bounds for the Potential §algebra); conversely, if $\rho(\mu,\nu)=0$ then its square, the radicand in ($\rho$), is $0$, and a sum $a+b$ of two nonnegative reals is $0$ only if both are ($a\le a+b=0$ by (W), and $0\le a$, so $a=0$ by the antisymmetry of $\le$ in Ordered Field; likewise $b=0$), so $m(\mu)=m(\nu)$ (claim 3 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$) and $\bar{\mu}=\bar{\nu}$ (The Heat Gauge Form on Probability Measures: Algebra, Representation by Smoothed Densities, Positivity, Cauchy-Schwarz, Separation, the Triangle Inequality, the Wasserstein Bound and Derivative Bounds for the Potential §separation), whence $\mu=(\tau_{m(\mu)})_{\#}\bar{\mu}=(\tau_{m(\nu)})_{\#}\bar{\nu}=\nu$ by The Mean of a Square-Integrable Probability Measure, Its Lift, Its Centring, and Functions of the Mean and Centred Integrals as Test Functions §centring. Triangle inequality: let $\mu,\nu,\sigma\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and put $a_{1}=\lVert m(\mu)-m(\nu)\rVert$, $a_{2}=\lVert m(\mu)-m(\sigma)\rVert$, $a_{3}=\lVert m(\sigma)-m(\nu)\rVert$, $b_{1}=\varrho(\bar{\mu},\bar{\nu})$, $b_{2}=\varrho(\bar{\mu},\bar{\sigma})$, $b_{3}=\varrho(\bar{\sigma},\bar{\nu})$, all nonnegative. Then $a_{1}\le a_{2}+a_{3}$ (claim 6 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, as $m(\mu)-m(\nu)=(m(\mu)-m(\sigma))+(m(\sigma)-m(\nu))$) and $b_{1}\le b_{2}+b_{3}$ (The Heat Gauge Form on Probability Measures: Algebra, Representation by Smoothed Densities, Positivity, Cauchy-Schwarz, Separation, the Triangle Inequality, the Wasserstein Bound and Derivative Bounds for the Potential §triangle), so $a_{1}^{2}+b_{1}^{2}\le(a_{2}+a_{3})^{2}+(b_{2}+b_{3})^{2}$ (claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field and (W)). In the Euclidean space $\mathbb{R}^{2}$, with the coordinatewise sum of Sum of Points of $\mathbb{R}^n$ and claims 1 and 6 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, $\sqrt{(a_{2}+a_{3})^{2}+(b_{2}+b_{3})^{2}}=\lVert(a_{2},b_{2})+(a_{3},b_{3})\rVert\le\lVert(a_{2},b_{2})\rVert+\lVert(a_{3},b_{3})\rVert=\sqrt{a_{2}^{2}+b_{2}^{2}}+\sqrt{a_{3}^{2}+b_{3}^{2}}$, so by ($\rho$) and the monotonicity of roots, $\rho(\mu,\nu)=\sqrt{a_{1}^{2}+b_{1}^{2}}\le\rho(\mu,\sigma)+\rho(\sigma,\nu)$. Hence $\rho$ is a metric on $\mathcal{P}_{2}(\mathbb{R}^{d})$.

Proof of claim 2. By The Mean of a Square-Integrable Probability Measure, Its Lift, Its Centring, and Functions of the Mean and Centred Integrals as Test Functions §mean and The Mean of a Square-Integrable Probability Measure, Its Lift, Its Centring, and Functions of the Mean and Centred Integrals as Test Functions §centring, $m((\tau_{a})_{\#}\mu)=m(\mu)+a$, $m((\tau_{b})_{\#}\nu)=m(\nu)+b$, $\overline{(\tau_{a})_{\#}\mu}=\bar{\mu}$ and $\overline{(\tau_{b})_{\#}\nu}=\bar{\nu}$, so ($\rho$) gives $\rho((\tau_{a})_{\#}\mu,(\tau_{b})_{\#}\nu)^{2}=\lVert m(\mu)+a-m(\nu)-b\rVert^{2}+Q(\bar{\mu},\bar{\nu})$, which is the claim since $Q(\bar{\mu},\bar{\nu})=\varrho(\bar{\mu},\bar{\nu})^{2}$.

Proof of claim 3.

Bounds. By The Heat Gauge Form on Probability Measures: Algebra, Representation by Smoothed Densities, Positivity, Cauchy-Schwarz, Separation, the Triangle Inequality, the Wasserstein Bound and Derivative Bounds for the Potential §potential-bounds applied to $\bar{\mu},\bar{\nu}\in\mathcal{P}(\mathbb{R}^{d})$, the function $\psi_{\mu,\nu}=K*\bar{\mu}-K*\bar{\nu}$ is of class $C^{3}$ and its value and partial derivatives up to order three are bounded by $N_{k}\sqrt{Q(\bar{\mu},\bar{\nu})}=N_{k}\,\varrho(\bar{\mu},\bar{\nu})\le N_{k}\,\rho(\mu,\nu)$ ($k=0,1,2,3$), the last step by claim 1 and claim 5 of Elementary Arithmetic in an Ordered Field. Each $\partial_{i}\psi_{\mu,\nu}$ is bounded and Borel (preamble), so $E_{\mu,\nu}$ is defined, and $|(E_{\mu,\nu})_{i}|\le\int|\partial_{i}\psi_{\mu,\nu}|\,d\bar{\mu}\le N_{1}\rho(\mu,\nu)$; hence $\lVert E_{\mu,\nu}\rVert^{2}=\sum_{i}(E_{\mu,\nu})_{i}^{2}\le dN_{1}^{2}\rho(\mu,\nu)^{2}$ (claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, claim 1 of Comparison and Absolute Value Bounds for Finite Sums of Real Numbers) and $\lVert E_{\mu,\nu}\rVert\le\sqrt{d}\,N_{1}\rho(\mu,\nu)$ by taking roots. Also $\psi_{\nu,\mu}=K*\bar{\nu}-K*\bar{\mu}=-\psi_{\mu,\nu}$ pointwise.

Fubini for a bounded kernel. Let $\alpha,\beta\in\mathcal{P}(\mathbb{R}^{d})$ and let $F:\mathbb{R}^{d+d}\to\mathbb{R}$ be Borel with $|F|\le C$. Then $F+C\ge0$ is Borel (claims 1 and 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions); by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §product, $(F+C)\circ\iota$ is measurable for the product $\sigma$-algebra with $\int(F+C)\,d(\alpha\boxtimes\beta)=\int(F+C)\circ\iota\,d(\alpha\otimes\beta)$, and by Tonelli (Tonelli and Fubini Theorems) the latter equals both iterated integrals $\int(\int(F+C)(\iota(x,y))\beta(dy))\alpha(dx)$ and $\int(\int(F+C)(\iota(x,y))\alpha(dx))\beta(dy)$, the sections and the inner integrals being measurable. All values are at most $2C$, so subtracting the constant $C$ from the inner and then from the outer integrals (claim 2 of Linearity and Monotonicity of the Lebesgue Integral, Simple Function and Its Integral) gives

each inner integral being a bounded Borel function of the outer variable.

The mean-field identity. Let $\alpha,\beta\in\mathcal{P}(\mathbb{R}^{d})$ and $i\in[d]$. Apply (Fub) to $F(z)=\partial_{i}K(\mathrm{pr}_{1}(z)-\mathrm{pr}_{2}(z))$, Borel (composition of the Borel map $z\mapsto\mathrm{pr}_{1}(z)-\mathrm{pr}_{2}(z)$, Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections and claim 2 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, with the continuous $\partial_{i}K$) and bounded by $M_{1}$, with $F(\iota(x,y))=\partial_{i}K(x-y)$. The left side of (Fub) is $\int\partial_{i}(K*\beta)\,d\alpha$; on the right, (Odd) gives $\int\partial_{i}K(x-y)\alpha(dx)=-\int\partial_{i}K(y-x)\alpha(dx)=-\partial_{i}(K*\alpha)(y)$ (claim 2 of Linearity and Monotonicity of the Lebesgue Integral). Hence

(a real number equal to its negative is $0$: claim 1 of Elementary Order Arithmetic in an Ordered Field and claim 8 there). Now, by the preamble and claim 2 of Linearity and Monotonicity of the Lebesgue Integral, $(E_{\mu,\nu})_{i}=\int\partial_{i}(K*\bar{\mu})\,d\bar{\mu}-\int\partial_{i}(K*\bar{\nu})\,d\bar{\mu}=0+\int\partial_{i}(K*\bar{\mu})\,d\bar{\nu}$ by (O) with $\alpha=\bar{\mu}$, $\beta=\bar{\nu}$; and $\int\partial_{i}\psi_{\mu,\nu}\,d\bar{\nu}=\int\partial_{i}(K*\bar{\mu})\,d\bar{\nu}-\int\partial_{i}(K*\bar{\nu})\,d\bar{\nu}=\int\partial_{i}(K*\bar{\mu})\,d\bar{\nu}-0$. So the point with coordinates $\int\partial_{i}\psi_{\mu,\nu}\,d\bar{\nu}$ is $E_{\mu,\nu}$. Finally $(E_{\nu,\mu})_{i}=\int\partial_{i}\psi_{\nu,\mu}\,d\bar{\nu}=-\int\partial_{i}\psi_{\mu,\nu}\,d\bar{\nu}=-(E_{\mu,\nu})_{i}$, i.e. $E_{\nu,\mu}=-E_{\mu,\nu}$.

Proof of claim 4. Fix $\nu$ and write $n=m(\nu)$.

The uncentred functional. Define $\tilde{\Theta}:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$ by $\tilde{\Theta}(\sigma)=Q(\sigma,\bar{\nu})$. By the preamble and the symmetry $\mathcal{K}(\sigma,\bar{\nu})=\mathcal{K}(\bar{\nu},\sigma)$ (The Multiscale Gaussian Kernel: Definition by a Series, Regularity and Bounds, the Smoothed Potential of a Probability Measure and the Kernel Pairing of Two Measures §pairing),

Here $\mathcal{K}(\sigma,\sigma)=\int K(\mathrm{pr}_{1}(z)-\mathrm{pr}_{2}(z))\,(\sigma\boxtimes\sigma)(dz)=\mathcal{K}_{K}(\sigma)$, the quadratic kernel functional of Linear and Quadratic Kernel Functionals of a Measure Are Test Functions on the Wasserstein Space §quadratic, a test function with intrinsic gradient the class of $x\mapsto2D(K*\sigma)(x)$ at $\sigma$. Next, $\mathcal{K}(\sigma,\bar{\nu})=\int K*\bar{\nu}\,d\sigma$ (The Multiscale Gaussian Kernel: Definition by a Series, Regularity and Bounds, the Smoothed Potential of a Probability Measure and the Kernel Pairing of Two Measures §pairing), and $f=K*\bar{\nu}$ is of class $C^{3}$, hence $C^{2}$ (as for $K$ in the preamble), with $|\partial_{i}f|\le M_{1}\le M$ and $|\partial_{j}\partial_{i}f|\le M_{2}\le M$ (The Multiscale Gaussian Kernel: Definition by a Series, Regularity and Bounds, the Smoothed Potential of a Probability Measure and the Kernel Pairing of Two Measures §potential), so by Linear and Quadratic Kernel Functionals of a Measure Are Test Functions on the Wasserstein Space §linear the function $\sigma\mapsto\mathcal{K}(\sigma,\bar{\nu})=u_{f}(\sigma)$ is a test function with intrinsic gradient the class of $x\mapsto D(K*\bar{\nu})(x)$ at $\sigma$. Finally $\kappa=\mathcal{K}(\bar{\nu},\bar{\nu})$ is a real number, and the constant function $f_{0}\equiv\kappa$ on $\mathbb{R}^{d}$ is smooth (claim 2 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set) with vanishing partial derivatives of all orders (its difference quotients in Partial Derivative on a Euclidean Open Set are $0$), so Linear and Quadratic Kernel Functionals of a Measure Are Test Functions on the Wasserstein Space §linear with $M$ shows that $\sigma\mapsto u_{f_{0}}(\sigma)=\kappa$ is a test function with intrinsic gradient the class of the zero map. By Sums, Real Multiples and Differences of Test Functions on the Wasserstein Space §linear and Sums, Real Multiples and Differences of Test Functions on the Wasserstein Space §difference, $\tilde{\Theta}=\mathcal{K}_{K}-2u_{f}+u_{f_{0}}$ is a test function with, at every $\sigma$,

(sums of classes being represented by pointwise sums, The Space of Square-Integrable Random Vectors §classes applied on $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}),\sigma)$, Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §l2mu; and claim 1 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set).

Centring. For every $\sigma$, $\Theta_{\nu}(\sigma)=\varrho(\bar{\sigma},\bar{\nu})^{2}=Q(\bar{\sigma},\bar{\nu})=\tilde{\Theta}(\bar{\sigma})$; that is, $\Theta_{\nu}=\tilde{\Theta}^{\circ}$ in the notation of The Mean of a Square-Integrable Probability Measure, Its Lift, Its Centring, and Functions of the Mean and Centred Integrals as Test Functions §centred. By that clause, $\Theta_{\nu}$ is a test function, $\Theta_{\nu}((\tau_{a})_{\#}\sigma)=\Theta_{\nu}(\sigma)$ for all $\sigma$ and $a$, its translation Hessian is $0_{d}$ at every $\mu$, and its intrinsic gradient at $\mu$ is the class of $x\mapsto\eta(x-m(\mu))-\int\eta\,d\bar{\mu}$ for any representative $\eta$ of $\nabla\tilde{\Theta}(\bar{\mu})$. Take the representative $\eta=2D(K*\bar{\mu}-K*\bar{\nu})=2D\psi_{\mu,\nu}$; its coordinates are $2\partial_{i}\psi_{\mu,\nu}$, so $\int\eta\,d\bar{\mu}$ has coordinates $2\int\partial_{i}\psi_{\mu,\nu}\,d\bar{\mu}=2(E_{\mu,\nu})_{i}$ (claim 2 of Linearity and Monotonicity of the Lebesgue Integral), i.e. $\int\eta\,d\bar{\mu}=2E_{\mu,\nu}$. Hence $\nabla\Theta_{\nu}(\mu)$ is the class of $x\mapsto2D\psi_{\mu,\nu}(x-m(\mu))-2E_{\mu,\nu}=2\bigl(D\psi_{\mu,\nu}(x-m(\mu))-E_{\mu,\nu}\bigr)$, as claimed.

The mean term. Let $\phi:\mathbb{R}^{d}\to\mathbb{R}$, $\phi(a)=\lVert a-n\rVert^{2}=\sum_{i}(a_{i}-n_{i})^{2}$. Each $a\mapsto a_{i}-n_{i}$ is smooth (claim 2 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set: coordinate functions and constants; claim 3: sums), so $\phi$ is smooth by claim 3 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set (products and sums), in particular of class $C^{2}$. For the coordinate function $\pi_{i}(a)=a_{i}$ and $j\in[d]$, the difference quotient of Partial Derivative on a Euclidean Open Set is $((a+he_{j})_{i}-a_{i})/h=\delta_{ij}$, so $\partial_{j}\pi_{i}=\delta_{ij}$, the entry of the identity matrix $I_{d}$ (Identity Matrix); constants have vanishing partials. By claim 1 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set (sums, products), $\partial_{i}\phi(a)=2(a_{i}-n_{i})$ and $\partial_{j}\partial_{i}\phi(a)=2\delta_{ij}$; thus $D\phi(a)=2(a-n)$ and $D^{2}\phi(a)=2I_{d}$ (Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation §derivatives, Hessian Matrix of a C^2 Function). By The Mean of a Square-Integrable Probability Measure, Its Lift, Its Centring, and Functions of the Mean and Centred Integrals as Test Functions §function-of-mean, $\Psi:\sigma\mapsto\phi(m(\sigma))=\lVert m(\sigma)-m(\nu)\rVert^{2}$ is a test function whose intrinsic gradient at $\mu$ is the class of the constant map $x\mapsto2(m(\mu)-m(\nu))$ and whose translation Hessian at $\mu$ is $2I_{d}$.

The squared gauge. By ($\rho$), $\Xi_{\nu}(\sigma)=\rho(\sigma,\nu)^{2}=\Psi(\sigma)+\Theta_{\nu}(\sigma)$ for every $\sigma$, so by Sums, Real Multiples and Differences of Test Functions on the Wasserstein Space §linear the function $\Xi_{\nu}$ is a test function with $\nabla\Xi_{\nu}(\mu)=\nabla\Psi(\mu)+\nabla\Theta_{\nu}(\mu)$, the class of $x\mapsto2(m(\mu)-m(\nu))+2(D\psi_{\mu,\nu}(x-m(\mu))-E_{\mu,\nu})=2\bigl(m(\mu)-m(\nu)-E_{\mu,\nu}+D\psi_{\mu,\nu}(x-m(\mu))\bigr)$, and $H_{\Xi_{\nu}}(\mu)=2I_{d}+0_{d}=2I_{d}$ (Real Matrices, Symmetric Matrices and the Semidefinite Ordering: Standing Notation §matrices). Since $\rho(\mu,\sigma)=\rho(\sigma,\mu)$ by claim 1, the function $\sigma\mapsto\rho(\mu,\sigma)^{2}$ with $\mu$ fixed is $\Xi_{\mu}$, to which the above applies with $\mu$ in the role of $\nu$.

Proof of claim 5. Write $\alpha=\bar{\mu}$, $\beta=\bar{\nu}$, $\hat{\alpha}=\bar{\hat{\mu}}$, $\hat{\beta}=\bar{\hat{\nu}}$, so $\psi=\psi_{\hat{\mu},\hat{\nu}}=K*\hat{\alpha}-K*\hat{\beta}$, which is bounded and Borel by The Heat Gauge Form on Probability Measures: Algebra, Representation by Smoothed Densities, Positivity, Cauchy-Schwarz, Separation, the Triangle Inequality, the Wasserstein Bound and Derivative Bounds for the Potential §potential-bounds; by the last assertion of that clause, $B(\hat{\alpha},\hat{\beta};\mu',\nu')=\int\psi\,d\mu'-\int\psi\,d\nu'$ for all $\mu',\nu'$, so

By The Heat Gauge Form on Probability Measures: Algebra, Representation by Smoothed Densities, Positivity, Cauchy-Schwarz, Separation, the Triangle Inequality, the Wasserstein Bound and Derivative Bounds for the Potential §algebra (the symmetry $B(\mu,\nu;\mu',\nu')=B(\mu',\nu';\mu,\nu)$, the antisymmetry $B(\nu,\mu;\cdot)=-B(\mu,\nu;\cdot)$, the additivity $B(\mu,\sigma;\cdot)=B(\mu,\nu;\cdot)+B(\nu,\sigma;\cdot)$, and the expansion $Q(\mu,\sigma)=Q(\mu,\nu)+2B(\mu,\nu;\nu,\sigma)+Q(\nu,\sigma)$):

by symmetry, additivity (with $\hat{\alpha},\hat{\beta},\beta$ in the first pair) and symmetry again, $B(\alpha,\hat{\alpha};\hat{\alpha},\beta)=B(\hat{\alpha},\beta;\alpha,\hat{\alpha})=B(\hat{\alpha},\hat{\beta};\alpha,\hat{\alpha})+B(\hat{\beta},\beta;\alpha,\hat{\alpha})=B(\hat{\alpha},\hat{\beta};\alpha,\hat{\alpha})+B(\alpha,\hat{\alpha};\hat{\beta},\beta)$; and by symmetry and antisymmetry, $B(\hat{\alpha},\hat{\beta};\hat{\beta},\beta)=B(\hat{\beta},\beta;\hat{\alpha},\hat{\beta})=-B(\beta,\hat{\beta};\hat{\alpha},\hat{\beta})=-B(\hat{\alpha},\hat{\beta};\beta,\hat{\beta})$. Substituting, and using $Q(\hat{\beta},\beta)=Q(\beta,\hat{\beta})$,

By The Heat Gauge Form on Probability Measures: Algebra, Representation by Smoothed Densities, Positivity, Cauchy-Schwarz, Separation, the Triangle Inequality, the Wasserstein Bound and Derivative Bounds for the Potential §cauchy-schwarz, $2B(\alpha,\hat{\alpha};\hat{\beta},\beta)\le2\sqrt{Q(\alpha,\hat{\alpha})}\sqrt{Q(\hat{\beta},\beta)}\le Q(\alpha,\hat{\alpha})+Q(\beta,\hat{\beta})$, the second step because $0\le(s-t)^{2}=s^{2}-2st+t^{2}$ for the two roots $s,t$ (Nonnegativity of Squares in an Ordered Field, claims 2 and 5 of Zero Products and Elementary Identities in a Field), with $s^{2}=Q(\alpha,\hat{\alpha})$ and $t^{2}=Q(\hat{\beta},\beta)=Q(\beta,\hat{\beta})$. Inserting this bound into (Pol) by (W) and replacing the two $B$-terms by the integrals displayed above gives exactly the asserted inequality, since $\varrho(\cdot,\cdot)^{2}=Q(\cdot,\cdot)$. If $\mu=\hat{\mu}$ and $\nu=\hat{\nu}$, then $\alpha=\hat{\alpha}$ and $\beta=\hat{\beta}$, so $Q(\alpha,\hat{\alpha})=Q(\beta,\hat{\beta})=0$ (The Heat Gauge Form on Probability Measures: Algebra, Representation by Smoothed Densities, Positivity, Cauchy-Schwarz, Separation, the Triangle Inequality, the Wasserstein Bound and Derivative Bounds for the Potential §algebra), $|B(\hat{\alpha},\hat{\beta};\alpha,\hat{\alpha})|\le\sqrt{Q(\hat{\alpha},\hat{\beta})}\sqrt{Q(\alpha,\hat{\alpha})}=0$ and likewise $B(\hat{\alpha},\hat{\beta};\beta,\hat{\beta})=0$ (The Heat Gauge Form on Probability Measures: Algebra, Representation by Smoothed Densities, Positivity, Cauchy-Schwarz, Separation, the Triangle Inequality, the Wasserstein Bound and Derivative Bounds for the Potential §cauchy-schwarz), and $B(\alpha,\hat{\alpha};\hat{\beta},\beta)=0$ by the same bound; so both sides of (Pol) reduce to $Q(\hat{\alpha},\hat{\beta})$, the right side of the asserted inequality equals $\varrho(\bar{\hat{\mu}},\bar{\hat{\nu}})^{2}=Q(\hat{\alpha},\hat{\beta})=Q(\alpha,\beta)$, and equality holds.