Proof of Pulling Back Intrinsic Test Functions along Tensor Powers

Each result cited is universally quantified over the data in its own statement. Results and notions of The Intrinsic Calculus on the Wasserstein Space: Standing Notation and of the settings on which it is layered are used at the particle dimension $d$ and, for measures on $\mathbb{R}^{dN}$, at the configuration level; results whose dimension is a parameter of their own statement are applied with the dimension named at the point of use. The results on particle blocks, tensor powers and one-particle marginals (Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points, Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts, The Tensor Power of a Probability Measure on Euclidean Space, Existence and Uniqueness of Tensor Powers, and the Average of the Block Marginals and Tensor Powers and One-Particle Marginals: Particle Laws, Product Integrals, Push-Forwards, Moments, Product Maps and Diagonal Shifts) are applied with their dimension parameter $q$ equal to $d$ or to $d+d$, as N-Particle Systems on the Wasserstein Space: Particles, Configurations and the Configuration Level §particles permits for $d$ and as they permit for every natural number; the block maps of $\mathbb{R}^{dN}$ are written $\mathfrak{p}_{k}$ and those of $\mathbb{R}^{(d+d)N}$ are written $\hat{\mathfrak{p}}_{k}$, and $b(k,i)=(k-1)d+i$ is the block index of Block Indices: Enumerating an Initial Segment of Length qN by Blocks and Positions for $q=d$. For $m\in\{d,dN\}$, $\mathrm{pr}_{1},\mathrm{pr}_{2}:\mathbb{R}^{m+m}\to\mathbb{R}^{m}$ are the coordinate projections of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pairs; for $z\in\mathbb{R}^{m+m}$ we write $x=\mathrm{pr}_{1}(z)$ and $y=\mathrm{pr}_{2}(z)$, and $c_{m}(z)=\lVert x-y\rVert^{2}$ is Borel and nonnegative by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions, so that $I(\pi)=\int c_{m}\,d\pi$ by Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost §cost. Composites of Borel maps are Borel and a map into a Euclidean space is Borel exactly when its components are (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps); sums, differences and products of Borel real functions are Borel (claims 2 and 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions); push-forwards and their change of variables are those of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward. Vector fields in the spaces $L^{2}$ are handled through Borel representatives, as the convention of that setting allows. For $m\in\{d,dN\}$, probability measures $\nu,\nu'\in\mathcal{P}_{2}(\mathbb{R}^{m})$, $\gamma\in\Pi(\nu,\nu')$, $q\in L^{2}(\nu;\mathbb{R}^{m})$ and $\eta\in L^{2}(\nu';\mathbb{R}^{m})$ we write

$$\Delta_{\gamma}(q,\eta)=\int_{\mathbb{R}^{m+m}}\lVert q(x)-\eta(y)\rVert^{2}\,\gamma(dz)$$

for their discrepancy along $\gamma$, a nonnegative real number not depending on the representatives by The Discrepancy of Two Square-Integrable Vector Fields Along a Coupling of Their Base Measures §well-defined. For a probability measure $\gamma$ on a Euclidean space and a bounded Borel map $F=(F_{1},\dots,F_{m})$ from that space into $\mathbb{R}^{m}$, each $F_{i}$ is bounded and Borel, hence integrable (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures), and $\int F\,d\gamma\in\mathbb{R}^{m}$ denotes the point with coordinates $\int F_{i}\,d\gamma$.

Step 1 (Three elementary facts). Let $\gamma$, $F$ be as just described and let $G$ be a second such map. (V1) For $c\in\mathbb{R}^{m}$, $c\cdot\int F\,d\gamma=\int c\cdot F\,d\gamma$ and $\int(F-G)\,d\gamma=\int F\,d\gamma-\int G\,d\gamma$: the dot product is the finite sum $\sum_{i}c_{i}F_{i}$, so this is Linearity of the Lebesgue Integral over a Finite Sum of Integrable Functions §linear and claim 2 of Linearity and Monotonicity of the Lebesgue Integral applied coordinatewise.

(V2) $\lVert\int F\,d\gamma\rVert^{2}\le\int\lVert F\rVert^{2}\,d\gamma$. Indeed each $F_{i}$ is a bounded, hence square-integrable, random variable on the probability space formed by $\gamma$, and so is the constant $1$, with $\mathbb{E}[1\cdot1]=1$; claim 1 of Cauchy-Schwarz and Triangle Inequalities for the Mean-Square Norm gives $|\int F_{i}\,d\gamma|\le\bigl(\int F_{i}^{2}\,d\gamma\bigr)^{1/2}$, and squaring (claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field) gives $(\int F_{i}\,d\gamma)^{2}\le\int F_{i}^{2}\,d\gamma$. Summing over $i\in[m]$ and using Linearity of the Lebesgue Integral over a Finite Sum of Integrable Functions §linear for $\lVert F\rVert^{2}=\sum_{i}F_{i}^{2}$ gives (V2).

(V3) For $n\in\mathbb{N}$ and $a_{1},\dots,a_{n}\in\mathbb{R}^{m}$, $\lVert\sum_{k=1}^{n}a_{k}\rVert^{2}\le n\sum_{k=1}^{n}\lVert a_{k}\rVert^{2}$. Indeed, let $A=[a_{1},\dots,a_{n}]\in\mathbb{R}^{mn}$ be the configuration of Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points §configuration (with $q=m$ and $n$ particles) and $v=\sum_{k}a_{k}$. The $k$-th particle of $A$ is $a_{k}$, so Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §diagonal gives $v^{\oplus}\cdot A=v\cdot v=\lVert v\rVert^{2}$ and $\lVert v^{\oplus}\rVert^{2}=n\lVert v\rVert^{2}$, and Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §inner-product gives $\lVert A\rVert^{2}=\sum_{k}\lVert a_{k}\rVert^{2}$. By The Cauchy-Schwarz Inequality in a Real Inner Product Space in $\mathbb{R}^{mn}$, $\lVert v\rVert^{2}\le\lVert v^{\oplus}\rVert\,\lVert A\rVert$, and squaring (claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field) gives $\lVert v\rVert^{4}\le n\lVert v\rVert^{2}\lVert A\rVert^{2}$. If $v=0$ the claim is clear; otherwise multiplying by $(\lVert v\rVert^{2})^{-1}>0$ (claim 7 of Elementary Order Arithmetic in an Ordered Field and claim 5 of Elementary Arithmetic in an Ordered Field) gives $\lVert v\rVert^{2}\le n\lVert A\rVert^{2}$.

Step 2 (Property (a): continuity). For $\rho\in\mathcal{P}_{2}(\mathbb{R}^{d})$ we have $\rho^{\otimes N}\in\mathcal{P}_{2}(\mathbb{R}^{dN})$ by Tensor Powers and One-Particle Marginals: Particle Laws, Product Integrals, Push-Forwards, Moments, Product Maps and Diagonal Shifts §moments. We first record:

(W) If $\nu,\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, $\theta\in\mathbb{R}$ is positive and $W_{2}(\nu,\mu)<\theta/\sqrt{N}$, then $W_{2}(\nu^{\otimes N},\mu^{\otimes N})<\theta$.

Indeed, $W_{2}(\nu,\mu)^{2}<\theta^{2}/N$ by claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, so $W_{2}(\nu^{\otimes N},\mu^{\otimes N})^{2}=N\,W_{2}(\nu,\mu)^{2}<\theta^{2}$ by Tensor Powers Scale the Wasserstein Distance by the Square Root of N, and the One-Particle Marginal is Lipschitz with Constant N^{-1/2} §tensor and claim 10 of Elementary Order Arithmetic in an Ordered Field ($N>0$), and claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives the assertion, both distances being nonnegative.

Let $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and let $\varepsilon>0$. By property (a) of $\Phi$ and Continuous Map Between Metric Spaces there is $\theta>0$ with $|\Phi(P')-\Phi(\mu^{\otimes N})|<\varepsilon$ for every $P'\in\mathcal{P}_{2}(\mathbb{R}^{dN})$ with $W_{2}(P',\mu^{\otimes N})<\theta$. The number $\theta/\sqrt{N}$ is positive (claims 5 and 7 of Elementary Order Arithmetic in an Ordered Field), and by (W) every $\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ with $W_{2}(\nu,\mu)<\theta/\sqrt{N}$ satisfies $|\varphi(\nu)-\varphi(\mu)|=|\Phi(\nu^{\otimes N})-\Phi(\mu^{\otimes N})|<\varepsilon$. Hence $\varphi$ is continuous, which is property (a) for $\varphi$.

Step 3 (Replacement maps). Let $q\in\{d,d+d\}$, write $\mathfrak{q}_{1},\dots,\mathfrak{q}_{N}$ for the block maps of $\mathbb{R}^{qN}$ (so $\mathfrak{q}_{k}=\mathfrak{p}_{k}$ if $q=d$ and $\mathfrak{q}_{k}=\hat{\mathfrak{p}}_{k}$ if $q=d+d$), and let $k\in[N]$. For $z\in\mathbb{R}^{q}$ let $E_{k}(z)\in\mathbb{R}^{qN}$ be the configuration (Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points §configuration) whose $k$-th particle is $z$ and whose other particles are $0_{\mathbb{R}^{q}}$, and for $u\in\mathbb{R}^{qN}$ let $Z_{k}(u)$ be the configuration whose $k$-th particle is $0_{\mathbb{R}^{q}}$ and whose $l$-th particle is $\mathfrak{q}_{l}(u)$ for $l\ne k$. By that clause and Block Indices: Enumerating an Initial Segment of Length qN by Blocks and Positions §bijection, each coordinate of $E_{k}(z)$ is a coordinate of $z$ or $0$, and each coordinate of $Z_{k}(u)$ is the coordinate of $u$ with the same index or $0$. Coordinate functions and constants are smooth (claim 2 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set), hence continuous (claim 3 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous) and Borel (claim 3(a) of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets); so $E_{k}$ and $Z_{k}$ are smooth by claim 1 of Coordinate Functions, the $C^k$ Hierarchy, and Partial Derivatives of a Smooth Map and Borel by claims 2 and 5 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets, and they are linear by Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §linear. With the coordinate projections $\mathrm{pr}^{q,qN}_{1},\mathrm{pr}^{q,qN}_{2}$ and the concatenation map $\iota^{q,qN}$ of $\mathbb{R}^{q+qN}$ (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pairs), let

$$r_{k}:\mathbb{R}^{q+qN}\to\mathbb{R}^{qN},\qquad r_{k}(w)=E_{k}\bigl(\mathrm{pr}^{q,qN}_{1}(w)\bigr)+Z_{k}\bigl(\mathrm{pr}^{q,qN}_{2}(w)\bigr),$$

a Borel map, and write $r_{k}(z,u)=r_{k}(\iota^{q,qN}(z,u))=E_{k}(z)+Z_{k}(u)$ for $z\in\mathbb{R}^{q}$ and $u\in\mathbb{R}^{qN}$ (Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections). For $q=d+d$ we write $\hat r_{k}$ instead of $r_{k}$.

(R1) By the linearity of block maps (Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §linear) and Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points §configuration, $\mathfrak{q}_{k}(r_{k}(z,u))=z$ and $\mathfrak{q}_{l}(r_{k}(z,u))=\mathfrak{q}_{l}(u)$ for $l\ne k$.

(R2) For every $\rho\in\mathcal{P}(\mathbb{R}^{q})$, $(r_{k})_{\#}(\rho\boxtimes\rho^{\otimes N})=\rho^{\otimes N}$. Indeed, let $B_{1},\dots,B_{N}\in\mathcal{B}(\mathbb{R}^{q})$, $C=\bigcap_{l}\mathfrak{q}_{l}^{-1}(B_{l})$, and let $B'_{l}=B_{l}$ for $l\ne k$ and $B'_{k}=\mathbb{R}^{q}$, $C'=\bigcap_{l}\mathfrak{q}_{l}^{-1}(B'_{l})$. By (R1) and Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections, $r_{k}^{-1}(C)=(\mathrm{pr}^{q,qN}_{1})^{-1}(B_{k})\cap(\mathrm{pr}^{q,qN}_{2})^{-1}(C')$, so Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §product and The Tensor Power of a Probability Measure on Euclidean Space §tensor give

$$(\rho\boxtimes\rho^{\otimes N})\bigl(r_{k}^{-1}(C)\bigr)=\rho(B_{k})\,\rho^{\otimes N}(C')=\rho(B_{k})\prod_{l=1}^{N}\rho(B'_{l})=\prod_{l=1}^{N}\rho(B_{l}),$$

the last equality by claims 2 and 3 of Properties of Finite Products, applied to the factors $\rho(B'_{l})$ and to the factors equal to $1$ for $l\ne k$ and to $\rho(B_{k})$ for $l=k$, using $\rho(\mathbb{R}^{q})=1$. The push-forward is a probability measure on $\mathbb{R}^{qN}$, so the uniqueness in Existence and Uniqueness of Tensor Powers, and the Average of the Block Marginals §tensor gives (R2).

(R3) Let $\rho\in\mathcal{P}(\mathbb{R}^{q})$. (i) For every Borel $F:\mathbb{R}^{qN}\to[0,\infty]$ the function $z\mapsto\int F(r_{k}(z,u))\,\rho^{\otimes N}(du)$ is Borel and

$$\int_{\mathbb{R}^{qN}}F\,d\rho^{\otimes N}=\int_{\mathbb{R}^{q}}\Bigl(\int_{\mathbb{R}^{qN}}F(r_{k}(z,u))\,\rho^{\otimes N}(du)\Bigr)\rho(dz).$$

(ii) If $F:\mathbb{R}^{qN}\to\mathbb{R}$ is Borel and integrable with respect to $\rho^{\otimes N}$, and $u\mapsto F(r_{k}(z,u))$ is integrable with respect to $\rho^{\otimes N}$ for every $z$ (for instance, if $F$ is bounded), then the inner integral is a Borel function of $z$, integrable with respect to $\rho$, and the same identity holds. For (i): by (R2) and change of variables, $\int F\,d\rho^{\otimes N}=\int F\circ r_{k}\,d(\rho\boxtimes\rho^{\otimes N})$, which equals $\int F\circ r_{k}\circ\iota^{q,qN}\,d(\rho\otimes\rho^{\otimes N})$ by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §product; the Tonelli part of Tonelli and Fubini Theorems gives the measurability and the iterated form. For (ii): the positive and negative parts $F^{\pm}$ of Integrable Function and the Lebesgue Integral are Borel and at most $|F|$; by (i) the functions $J^{\pm}(z)=\int F^{\pm}(r_{k}(z,u))\,\rho^{\otimes N}(du)$ are Borel with $\int J^{\pm}\,d\rho=\int F^{\pm}\,d\rho^{\otimes N}<\infty$, and they are finite at every $z$ by the integrability of the sections, so they are integrable real functions; $\int F(r_{k}(z,u))\,\rho^{\otimes N}(du)=J^{+}(z)-J^{-}(z)$ by Integrable Function and the Lebesgue Integral, and claim 2 of Linearity and Monotonicity of the Lebesgue Integral gives $\int(J^{+}-J^{-})\,d\rho=\int F^{+}\,d\rho^{\otimes N}-\int F^{-}\,d\rho^{\otimes N}=\int F\,d\rho^{\otimes N}$.

Step 4 (Tensor couplings). Let $\mu,\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and $\pi\in\Pi(\mu,\nu)$. Let $\pi^{\otimes N}\in\mathcal{P}(\mathbb{R}^{(d+d)N})$ be its tensor power (The Tensor Power of a Probability Measure on Euclidean Space §tensor with $q=d+d$), let $\mathrm{pr}_{1}^{\oplus},\mathrm{pr}_{2}^{\oplus}:\mathbb{R}^{(d+d)N}\to\mathbb{R}^{dN}$ be the product maps (Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points §product-map) of $\mathrm{pr}_{1},\mathrm{pr}_{2}:\mathbb{R}^{d+d}\to\mathbb{R}^{d}$, Borel by Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §product-map, let $\Theta=(\mathrm{pr}_{1}^{\oplus},\mathrm{pr}_{2}^{\oplus}):\mathbb{R}^{(d+d)N}\to\mathbb{R}^{dN+dN}$, Borel with $\mathrm{pr}_{i}\circ\Theta=\mathrm{pr}_{i}^{\oplus}$ by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §pairing and Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections, and put $\pi^{\boxtimes}=\Theta_{\#}\pi^{\otimes N}$.

(T1) $\mu^{\otimes N},\nu^{\otimes N}\in\mathcal{P}_{2}(\mathbb{R}^{dN})$ (Step 2), and by Tensor Powers and One-Particle Marginals: Particle Laws, Product Integrals, Push-Forwards, Moments, Product Maps and Diagonal Shifts §pushforward (with $q=d+d$, $p=d$) and $(\mathrm{pr}_{1})_{\#}\pi=\mu$, $(\mathrm{pr}_{2})_{\#}\pi=\nu$ (Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost §coupling),

$$(\mathrm{pr}_{1}^{\oplus})_{\#}\pi^{\otimes N}=\mu^{\otimes N},\qquad(\mathrm{pr}_{2}^{\oplus})_{\#}\pi^{\otimes N}=\nu^{\otimes N};$$

hence $(\mathrm{pr}_{i})_{\#}\pi^{\boxtimes}=(\mathrm{pr}_{i}^{\oplus})_{\#}\pi^{\otimes N}$ gives $\pi^{\boxtimes}\in\Pi(\mu^{\otimes N},\nu^{\otimes N})$.

(T2) $I(\pi^{\boxtimes})=N\,I(\pi)$, and $\int c_{d}\circ\hat{\mathfrak{p}}_{k}\,d\pi^{\otimes N}=I(\pi)<\infty$ for every $k$. Indeed $(\hat{\mathfrak{p}}_{k})_{\#}\pi^{\otimes N}=\pi$ by Tensor Powers and One-Particle Marginals: Particle Laws, Product Integrals, Push-Forwards, Moments, Product Maps and Diagonal Shifts §particle-laws, so change of variables and Couplings on Euclidean Space: Product Coupling, Swap, Finiteness of the Cost, Push-Forward Couplings, Modifying One Marginal, Quantisation, Gluing over a Finitely Supported Measure, and the Lipschitz Bound §cost-finite give the second assertion. For $w\in\mathbb{R}^{(d+d)N}$, Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §inner-product, Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §linear and $\mathfrak{p}_{k}\circ\mathrm{pr}_{i}^{\oplus}=\mathrm{pr}_{i}\circ\hat{\mathfrak{p}}_{k}$ (Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §product-map) give $c_{dN}(\Theta(w))=\sum_{k=1}^{N}c_{d}(\hat{\mathfrak{p}}_{k}(w))$; change of variables for $\Theta$ and claim 1 of Linearity and Monotonicity of the Lebesgue Integral give $I(\pi^{\boxtimes})=\sum_{k}\int c_{d}\circ\hat{\mathfrak{p}}_{k}\,d\pi^{\otimes N}=N\,I(\pi)$.

(T3) For $z\in\mathbb{R}^{d+d}$, $w'\in\mathbb{R}^{(d+d)N}$, $k\in[N]$ and $i\in\{1,2\}$,

$$\mathrm{pr}_{i}^{\oplus}\bigl(\hat r_{k}(z,w')\bigr)=r_{k}\bigl(\mathrm{pr}_{i}(z),\mathrm{pr}_{i}^{\oplus}(w')\bigr).$$

Indeed, by Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §product-map and (R1) (for $q=d+d$ and for $q=d$), the $l$-th particle of either side is $\mathrm{pr}_{i}(z)$ if $l=k$ and $\mathrm{pr}_{i}(\hat{\mathfrak{p}}_{l}(w'))=\mathfrak{p}_{l}(\mathrm{pr}_{i}^{\oplus}(w'))$ if $l\ne k$; a point of $\mathbb{R}^{dN}$ is the configuration of its particles (Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §linear).

Step 5 (Particle averages of configuration gradients). Let $\rho\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and let $\psi\in C_{c}^{\infty}(\mathbb{R}^{dN})$ be a test function at the configuration level. Being smooth (Test Functions on Euclidean Space, Their Gradient Maps and Laplacians §space), $\psi$ is continuous by claim 3 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous, and being compactly supported it is bounded by claim 1 of A Continuous Compactly Supported Function on $\mathbb{R}^n$ is Bounded and Integrable; its partial derivatives are bounded and there is $K\ge0$ with $\lVert\nabla\psi\rVert\le K$, by The Gradient of a Test Function is Bounded and Square-Integrable, and Its Laplacian Bounded and Integrable, Against Every Probability Measure §gradient. Fix a real $C\ge0$ with $|\psi|\le C$ and $|\partial_{j}\psi|\le C$ on $\mathbb{R}^{dN}$ for every $j\in[dN]$. Let $k\in[N]$, let $E_{k},Z_{k},r_{k}$ be those of Step 3 with $q=d$, and let $\lambda_{k}=(-Z_{k})_{\#}\rho^{\otimes N}\in\mathcal{P}(\mathbb{R}^{dN})$. By The Convolution of a Bounded Continuous Function with a Probability Measure on Euclidean Space: Continuity, Boundedness and Differentiation Under the Integral §derivative (with $dN$ in place of $q$, $\lambda_{k}$ in place of $\mu$ and $G=\psi$), $\psi*\lambda_{k}$ is of class $C^{1}$ on $\mathbb{R}^{dN}$ with $\partial_{j}(\psi*\lambda_{k})=(\partial_{j}\psi)*\lambda_{k}$, and $|(\partial_{j}\psi)*\lambda_{k}|\le C$ by The Convolution of a Bounded Continuous Function with a Probability Measure on Euclidean Space: Continuity, Boundedness and Differentiation Under the Integral §continuous; by change of variables, $(H*\lambda_{k})(v)=\int H(v+Z_{k}(u))\,\rho^{\otimes N}(du)$ for $H\in\{\psi,\partial_{j}\psi\}$ and $v\in\mathbb{R}^{dN}$. Put $\Psi_{k}=(\psi*\lambda_{k})\circ E_{k}:\mathbb{R}^{d}\to\mathbb{R}$, so that $\Psi_{k}(z)=\int\psi(r_{k}(z,u))\,\rho^{\otimes N}(du)$.

The coordinate of $E_{k}(z)$ with index $b(k,i')$ is $z_{i'}$ and every other coordinate is $0$ (Step 3). By Partial Derivative on a Euclidean Open Set, whose difference quotients are here constantly $1$ or $0$, the partial derivative with respect to the $i$th variable of the coordinate with index $b(k,i')$ is $1$ if $i'=i$ and $0$ otherwise, and that of every other coordinate is $0$. Since $E_{k}$ is smooth (Step 3), hence of class $C^{1}$ (Smooth Map on a Euclidean Open Set), claims 2 and 1 of A Composition of $C^k$ Maps Between Euclidean Open Sets is of Class $C^k$ show that $\Psi_{k}$ is of class $C^{1}$ on $\mathbb{R}^{d}$ with

$$\partial_{i}\Psi_{k}(z)=\sum_{j=1}^{dN}\partial_{j}(\psi*\lambda_{k})(E_{k}(z))\,\partial_{i}(E_{k})_{j}(z)=\partial_{b(k,i)}(\psi*\lambda_{k})(E_{k}(z))=\int_{\mathbb{R}^{dN}}\partial_{b(k,i)}\psi(r_{k}(z,u))\,\rho^{\otimes N}(du),$$

the middle equality by claim 7 of Properties of Finite Sums of Vectors (only the summand with $j=b(k,i)$ can be nonzero), and $|\partial_{i}\Psi_{k}|\le C$. The $i$th coordinate of $\mathfrak{p}_{k}(\nabla\psi(x'))$ is $\partial_{b(k,i)}\psi(x')$ by Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points §blocks and Test Functions on Euclidean Space, Their Gradient Maps and Laplacians §gradient, so the gradient $a^{\rho,\psi}_{k}=D\Psi_{k}$ of Gradient of a Real-Valued Function on a Euclidean Open Set is

$$a^{\rho,\psi}_{k}(z)=\int_{\mathbb{R}^{dN}}\mathfrak{p}_{k}\bigl(\nabla\psi(r_{k}(z,u))\bigr)\,\rho^{\otimes N}(du)\qquad(z\in\mathbb{R}^{d}),\tag{5.1}$$

the integrand being Borel in $u$ and bounded by $K$ (Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §inner-product). By Gradients of Functions with Bounded Derivatives Belong to the Tangent Space; Constants Are Tangent; the Score Identity; the Score Has Mean Zero; Translation of Tangent Fields §tangent (with $f=\Psi_{k}$ and $M=C$), $a^{\rho,\psi}_{k}$ is Borel, its class lies in $L^{2}(\rho;\mathbb{R}^{d})$, and $a^{\rho,\psi}_{k}\in T_{\rho}$. Put $V^{\rho,\psi}=\sum_{k=1}^{N}a^{\rho,\psi}_{k}$; since $T_{\rho}$ is a linear subspace (Basic Properties of the Tangent Space: Closed Subspace, the Identity Map Belongs to It, Second-Moment Limits, and Representation of Bounded Functionals on Gradients §closed), $V^{\rho,\psi}\in T_{\rho}$.

Step 6 (The projection of a gradient). Let $\rho$ and $\psi$ be as in Step 5. Then $\rho^{\otimes N}\in\mathcal{P}_{2}(\mathbb{R}^{dN})$, $(\rho^{\otimes N})^{[1]}=\rho$ by Tensor Powers and One-Particle Marginals: Particle Laws, Product Integrals, Push-Forwards, Moments, Product Maps and Diagonal Shifts §marginal-of-tensor, and

$$\Pi_{\rho^{\otimes N}}(\nabla\psi)=N^{-1}V^{\rho,\psi}.\tag{6.1}$$

Indeed, let $\chi\in C_{c}^{\infty}(\mathbb{R}^{d})$, let $K_{\chi}\ge0$ bound $\lVert\nabla\chi\rVert$ (The Gradient of a Test Function is Bounded and Square-Integrable, and Its Laplacian Bounded and Integrable, Against Every Probability Measure §gradient), and for $k\in[N]$ let $h_{k}(x')=\mathfrak{p}_{k}(\nabla\psi(x'))\cdot\nabla\chi(\mathfrak{p}_{k}(x'))$ for $x'\in\mathbb{R}^{dN}$, a Borel function bounded by $KK_{\chi}$ (Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions). By (5.1), (V1) with $c=\nabla\chi(z)$, and (R1),

$$a^{\rho,\psi}_{k}(z)\cdot\nabla\chi(z)=\int_{\mathbb{R}^{dN}}h_{k}(r_{k}(z,u))\,\rho^{\otimes N}(du),$$

so (R3)(ii) with $F=h_{k}$ gives $\int a^{\rho,\psi}_{k}\cdot\nabla\chi\,d\rho=\int h_{k}\,d\rho^{\otimes N}$. By Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §inner-product and $\mathfrak{p}_{k}\circ(\nabla\chi)^{\oplus}=\nabla\chi\circ\mathfrak{p}_{k}$ (Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §product-map), $\sum_{k}h_{k}=\nabla\psi\cdot(\nabla\chi)^{\oplus}$, and the class of the product map $(\nabla\chi)^{\oplus}$ is the product field of $\nabla\chi\in L^{2}(\rho;\mathbb{R}^{d})$ (Product Fields and the Projection onto One-Particle Tangent Fields §product-field). Summing over $k$ with Linearity of the Lebesgue Integral over a Finite Sum of Integrable Functions §linear,

$$\bigl\langle N^{-1}V^{\rho,\psi},\nabla\chi\bigr\rangle_{\rho}=\frac{1}{N}\sum_{k=1}^{N}\int_{\mathbb{R}^{dN}}h_{k}\,d\rho^{\otimes N}=\frac{1}{N}\bigl\langle\nabla\psi,(\nabla\chi)^{\oplus}\bigr\rangle_{\rho^{\otimes N}} .$$

As $N^{-1}V^{\rho,\psi}\in T_{\rho}$ (Step 5), the uniqueness in Product Fields and the Projection onto One-Particle Tangent Fields §projection gives (6.1).

Step 7 (Pairing along tensor couplings). Let $\mu,\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, $\pi\in\Pi(\mu,\nu)$ and $P=\mu^{\otimes N}$, with $\pi^{\boxtimes}\in\Pi(P,\nu^{\otimes N})$ as in Step 4. We show that for every $D\in T_{P}$

$$\mathcal{J}(D,\pi^{\boxtimes})=\mathcal{J}\bigl(N\,\Pi_{P}(D),\pi\bigr),\tag{7.1}$$

the left side being the displacement pairing at the configuration level and the right side that at the particle dimension, where $\Pi_{P}(D)\in T_{P^{[1]}}=T_{\mu}$ (Product Fields and the Projection onto One-Particle Tangent Fields §projection, Tensor Powers and One-Particle Marginals: Particle Laws, Product Integrals, Push-Forwards, Moments, Product Maps and Diagonal Shifts §marginal-of-tensor).

(a) Let $D=\nabla\psi$ with $\psi\in C_{c}^{\infty}(\mathbb{R}^{dN})$, and let $K$ be as in Step 5. By The Displacement Pairing of a Square-Integrable Vector Field Along a Coupling §pairing and change of variables for $\Theta$ (with $\mathrm{pr}_{i}\circ\Theta=\mathrm{pr}_{i}^{\oplus}$),

$$\mathcal{J}(\nabla\psi,\pi^{\boxtimes})=\int_{\mathbb{R}^{(d+d)N}}\nabla\psi\bigl(\mathrm{pr}_{1}^{\oplus}(w)\bigr)\cdot\bigl(\mathrm{pr}_{2}^{\oplus}(w)-\mathrm{pr}_{1}^{\oplus}(w)\bigr)\,\pi^{\otimes N}(dw).$$

By Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §inner-product, Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §linear and Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §product-map, the integrand is $\sum_{k=1}^{N}f_{k}(w)$ with

$$f_{k}(w)=\mathfrak{p}_{k}\bigl(\nabla\psi(\mathrm{pr}_{1}^{\oplus}(w))\bigr)\cdot\bigl(\mathrm{pr}_{2}(\hat{\mathfrak{p}}_{k}(w))-\mathrm{pr}_{1}(\hat{\mathfrak{p}}_{k}(w))\bigr).$$

Each $f_{k}$ is Borel, and $|f_{k}|\le\frac{1}{2}(K^{2}+c_{d}\circ\hat{\mathfrak{p}}_{k})$ by the last inequality of Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions; the right side is integrable by (T2), so $f_{k}$ is integrable (claim 1 of Linearity and Monotonicity of the Lebesgue Integral and Integrable Function and the Lebesgue Integral), and Linearity of the Lebesgue Integral over a Finite Sum of Integrable Functions §linear gives $\mathcal{J}(\nabla\psi,\pi^{\boxtimes})=\sum_{k}\int f_{k}\,d\pi^{\otimes N}$. Fix $k$ and $z\in\mathbb{R}^{d+d}$. By (R1) for $q=d+d$, $\hat{\mathfrak{p}}_{k}(\hat r_{k}(z,w'))=z$, and by (T3),

$$f_{k}\bigl(\hat r_{k}(z,w')\bigr)=\mathfrak{p}_{k}\Bigl(\nabla\psi\bigl(r_{k}(x,\mathrm{pr}_{1}^{\oplus}(w'))\bigr)\Bigr)\cdot(y-x),$$

a bounded Borel function of $w'$. By (V1), change of variables for $\mathrm{pr}_{1}^{\oplus}$ with (T1), and (5.1) for $\rho=\mu$,

$$\int_{\mathbb{R}^{(d+d)N}}f_{k}\bigl(\hat r_{k}(z,w')\bigr)\,\pi^{\otimes N}(dw')=\Bigl(\int_{\mathbb{R}^{dN}}\mathfrak{p}_{k}\bigl(\nabla\psi(r_{k}(x,u))\bigr)\,\mu^{\otimes N}(du)\Bigr)\cdot(y-x)=a^{\mu,\psi}_{k}(x)\cdot(y-x).$$

Hence (R3)(ii), with $q=d+d$, $\rho=\pi$ and $F=f_{k}$, gives $\int f_{k}\,d\pi^{\otimes N}=\int a^{\mu,\psi}_{k}(x)\cdot(y-x)\,\pi(dz)$. Summing over $k$ (Linearity of the Lebesgue Integral over a Finite Sum of Integrable Functions §linear) and using The Displacement Pairing of a Square-Integrable Vector Field Along a Coupling §pairing, The Displacement Pairing of a Square-Integrable Vector Field Along a Coupling §linear and (6.1),

$$\mathcal{J}(\nabla\psi,\pi^{\boxtimes})=\int_{\mathbb{R}^{d+d}}V^{\mu,\psi}(x)\cdot(y-x)\,\pi(dz)=\mathcal{J}(V^{\mu,\psi},\pi)=\mathcal{J}\bigl(N\,\Pi_{P}(\nabla\psi),\pi\bigr).$$

(b) Let $D\in T_{P}$ and $\varepsilon>0$. By Basic Properties of the Tangent Space: Closed Subspace, the Identity Map Belongs to It, Second-Moment Limits, and Representation of Bounded Functionals on Gradients §closed at the configuration level, the gradients of test functions are dense in $T_{P}$, so there is $\psi\in C_{c}^{\infty}(\mathbb{R}^{dN})$ with $\lVert D-\nabla\psi\rVert_{P}<\varepsilon$. The linearity of the pairing (The Displacement Pairing of a Square-Integrable Vector Field Along a Coupling §linear) and of $\Pi_{P}$ (The Projection onto One-Particle Tangent Fields: Pairing with Product Fields, Contraction, and Product Fields of Tangent Fields §contraction) and part (a) give

$$\mathcal{J}(D,\pi^{\boxtimes})-\mathcal{J}\bigl(N\Pi_{P}(D),\pi\bigr)=\mathcal{J}(D-\nabla\psi,\pi^{\boxtimes})-\mathcal{J}\bigl(N\Pi_{P}(D-\nabla\psi),\pi\bigr).$$

The contraction $N\lVert\Pi_{P}(E)\rVert_{\mu}^{2}\le\lVert E\rVert_{P}^{2}$ of The Projection onto One-Particle Tangent Fields: Pairing with Product Fields, Contraction, and Product Fields of Tangent Fields §contraction gives $\lVert N\Pi_{P}(E)\rVert_{\mu}^{2}\le N\lVert E\rVert_{P}^{2}$, hence $\lVert N\Pi_{P}(E)\rVert_{\mu}\le\sqrt{N}\lVert E\rVert_{P}$ (claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field); also $\sqrt{I(\pi^{\boxtimes})}=\sqrt{N}\sqrt{I(\pi)}$ by (T2) and claim 3 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field. So the bound of The Displacement Pairing of a Square-Integrable Vector Field Along a Coupling §bound, applied at both levels, gives

$$\bigl|\mathcal{J}(D,\pi^{\boxtimes})-\mathcal{J}\bigl(N\Pi_{P}(D),\pi\bigr)\bigr|\le\lVert D-\nabla\psi\rVert_{P}\sqrt{N}\sqrt{I(\pi)}+\sqrt{N}\lVert D-\nabla\psi\rVert_{P}\sqrt{I(\pi)}\le2\varepsilon\sqrt{N}\sqrt{I(\pi)} .$$

A nonnegative real number bounded by $2\varepsilon\sqrt{N}\sqrt{I(\pi)}$ for every $\varepsilon>0$ is $0$ by Comparison of Real Numbers with Arbitrary Positive Slack §vanishing (if $I(\pi)>0$, apply it with $\varepsilon$ replaced by $\varepsilon'(2\sqrt{N}\sqrt{I(\pi)})^{-1}$ for arbitrary positive $\varepsilon'$; if $I(\pi)=0$ the bound is $0$ itself), which proves (7.1).

Step 8 (Property (b) and the gradient). Let $\mu\in\mathcal{Q}$ and $P=\mu^{\otimes N}\in\mathcal{Q}_{N}$. By property (b) of $\Phi$, $\Phi$ is differentiable along couplings at $P$ and $D=\nabla\Phi(P)\in T_{P}$. Put $\eta=N\,\Pi_{P}(D)$; then $\eta\in T_{\mu}$, since $\Pi_{P}(D)\in T_{\mu}$ (Step 7) and $T_{\mu}$ is a linear subspace (Basic Properties of the Tangent Space: Closed Subspace, the Identity Map Belongs to It, Second-Moment Limits, and Representation of Bounded Functionals on Gradients §closed). Let $\varepsilon>0$. By Differentiability of a Function on the Wasserstein Space Along Couplings, and Its Gradient §differentiable at the configuration level, applied with the positive number $\varepsilon/\sqrt{N}$, there is $\theta>0$ with $|\Phi(P')-\Phi(P)-\mathcal{J}(D,\gamma)|\le(\varepsilon/\sqrt{N})\sqrt{I(\gamma)}$ for all $P'\in\mathcal{P}_{2}(\mathbb{R}^{dN})$ and $\gamma\in\Pi(P,P')$ with $I(\gamma)<\theta^{2}$. Let $\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and $\pi\in\Pi(\mu,\nu)$ with $I(\pi)<(\theta/\sqrt{N})^{2}=\theta^{2}/N$. Then $\pi^{\boxtimes}\in\Pi(P,\nu^{\otimes N})$ by (T1) and $I(\pi^{\boxtimes})=N\,I(\pi)<\theta^{2}$ by (T2) and claim 10 of Elementary Order Arithmetic in an Ordered Field, so by (7.1) and $\sqrt{I(\pi^{\boxtimes})}=\sqrt{N}\sqrt{I(\pi)}$,

$$\bigl|\varphi(\nu)-\varphi(\mu)-\mathcal{J}(\eta,\pi)\bigr|=\bigl|\Phi(\nu^{\otimes N})-\Phi(P)-\mathcal{J}(D,\pi^{\boxtimes})\bigr|\le\frac{\varepsilon}{\sqrt{N}}\sqrt{N}\sqrt{I(\pi)}=\varepsilon\sqrt{I(\pi)} .$$

As $\theta/\sqrt{N}>0$, $\varphi$ is differentiable along couplings at $\mu$ with gradient $\eta$ in the sense of Differentiability of a Function on the Wasserstein Space Along Couplings, and Its Gradient §differentiable, and by the uniqueness in Differentiability of a Function on the Wasserstein Space Along Couplings, and Its Gradient §gradient,

$$\nabla\varphi(\mu)=N\,\Pi_{\mu^{\otimes N}}\bigl(\nabla\Phi(\mu^{\otimes N})\bigr)\in T_{\mu}.$$

This is property (b) for $\varphi$ and the gradient formula of clause 1.

Step 9 (Discrepancy of projections). Let $\mu',\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, $\gamma\in\Pi(\mu',\mu)$, $P'=\mu'^{\otimes N}$ and $P=\mu^{\otimes N}$, so that $\gamma^{\boxtimes}\in\Pi(P',P)$ by (T1). We show that for all $D'\in T_{P'}$ and $D\in T_{P}$

$$\Delta_{\gamma}\bigl(\Pi_{P'}(D'),\Pi_{P}(D)\bigr)\le\frac{9}{N}\,\Delta_{\gamma^{\boxtimes}}(D',D).\tag{9.1}$$

(a) Perturbation. For $m\in\{d,dN\}$, $\nu,\nu'\in\mathcal{P}_{2}(\mathbb{R}^{m})$, $\beta\in\Pi(\nu,\nu')$, $q,\tilde q\in L^{2}(\nu;\mathbb{R}^{m})$ and $\eta,\tilde\eta\in L^{2}(\nu';\mathbb{R}^{m})$,

$$\Delta_{\beta}(q,\eta)\le3\bigl(\lVert q-\tilde q\rVert_{\nu}^{2}+\Delta_{\beta}(\tilde q,\tilde\eta)+\lVert\tilde\eta-\eta\rVert_{\nu'}^{2}\bigr).\tag{9.2}$$

Indeed $q(x)-\eta(y)=(q(x)-\tilde q(x))+(\tilde q(x)-\tilde\eta(y))+(\tilde\eta(y)-\eta(y))$, so (V3) with $n=3$ bounds $\lVert q(x)-\eta(y)\rVert^{2}$ pointwise by three times the sum of the squared norms of the three terms; integrating against $\beta$ with claim 1 of Linearity and Monotonicity of the Lebesgue Integral, and using change of variables for $\mathrm{pr}_{1}$ and $\mathrm{pr}_{2}$, whose push-forwards of $\beta$ are $\nu$ and $\nu'$ (Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost §coupling), gives (9.2).

(b) Gradients. Let $D'=\nabla\psi$ and $D=\nabla\chi$ with $\psi,\chi\in C_{c}^{\infty}(\mathbb{R}^{dN})$. For $k\in[N]$ let $F_{k}(w)=\mathfrak{p}_{k}(\nabla\psi(\mathrm{pr}_{1}^{\oplus}(w)))-\mathfrak{p}_{k}(\nabla\chi(\mathrm{pr}_{2}^{\oplus}(w)))$ on $\mathbb{R}^{(d+d)N}$, a bounded Borel map. For $z\in\mathbb{R}^{d+d}$, (5.1) for $(\mu',\psi)$ and for $(\mu,\chi)$, change of variables for $\mathrm{pr}_{1}^{\oplus}$ and $\mathrm{pr}_{2}^{\oplus}$ with (T1) (here $(\mathrm{pr}_{1}^{\oplus})_{\#}\gamma^{\otimes N}=P'$ and $(\mathrm{pr}_{2}^{\oplus})_{\#}\gamma^{\otimes N}=P$), (T3) and (V1) give

$$a^{\mu',\psi}_{k}(x)-a^{\mu,\chi}_{k}(y)=\int_{\mathbb{R}^{(d+d)N}}F_{k}\bigl(\hat r_{k}(z,w')\bigr)\,\gamma^{\otimes N}(dw').$$

Summing over $k$ and applying (V3) with $n=N$ and then (V2),

$$\bigl\lVert V^{\mu',\psi}(x)-V^{\mu,\chi}(y)\bigr\rVert^{2}\le N\sum_{k=1}^{N}\int_{\mathbb{R}^{(d+d)N}}\bigl\lVert F_{k}\bigl(\hat r_{k}(z,w')\bigr)\bigr\rVert^{2}\,\gamma^{\otimes N}(dw').$$

The right side is a Borel function of $z$ by (R3)(i); integrating against $\gamma$ (claim 1 of Linearity and Monotonicity of the Lebesgue Integral), then using (R3)(i) with $q=d+d$, $\rho=\gamma$ and $F=\lVert F_{k}\rVert^{2}$, Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §inner-product with the linearity of $\mathfrak{p}_{k}$, and change of variables for $\Theta$,

$$\int_{\mathbb{R}^{d+d}}\bigl\lVert V^{\mu',\psi}(x)-V^{\mu,\chi}(y)\bigr\rVert^{2}\gamma(dz)\le N\int_{\mathbb{R}^{(d+d)N}}\bigl\lVert\nabla\psi(\mathrm{pr}_{1}^{\oplus}(w))-\nabla\chi(\mathrm{pr}_{2}^{\oplus}(w))\bigr\rVert^{2}\gamma^{\otimes N}(dw)=N\,\Delta_{\gamma^{\boxtimes}}(\nabla\psi,\nabla\chi).$$

By (6.1), $\Pi_{P'}(\nabla\psi)=N^{-1}V^{\mu',\psi}$ and $\Pi_{P}(\nabla\chi)=N^{-1}V^{\mu,\chi}$, so, the factor $N^{-2}$ being taken out by claim 1 of Linearity and Monotonicity of the Lebesgue Integral,

$$\Delta_{\gamma}\bigl(\Pi_{P'}(\nabla\psi),\Pi_{P}(\nabla\chi)\bigr)\le\frac{1}{N}\,\Delta_{\gamma^{\boxtimes}}(\nabla\psi,\nabla\chi).\tag{9.3}$$

(c) General fields. Let $D'\in T_{P'}$, $D\in T_{P}$ and $\varepsilon>0$. By density (Basic Properties of the Tangent Space: Closed Subspace, the Identity Map Belongs to It, Second-Moment Limits, and Representation of Bounded Functionals on Gradients §closed at the configuration level) and claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field there are $\psi,\chi\in C_{c}^{\infty}(\mathbb{R}^{dN})$ with $\lVert D'-\nabla\psi\rVert_{P'}^{2}<\varepsilon$ and $\lVert D-\nabla\chi\rVert_{P}^{2}<\varepsilon$. The projections are linear with $\lVert\Pi_{P}(E)\rVert_{\mu}^{2}\le N^{-1}\lVert E\rVert_{P}^{2}$, and likewise for $P'$ (The Projection onto One-Particle Tangent Fields: Pairing with Product Fields, Contraction, and Product Fields of Tangent Fields §contraction). Hence (9.2) at the particle dimension, (9.3), and (9.2) at the configuration level give

$$\Delta_{\gamma}\bigl(\Pi_{P'}(D'),\Pi_{P}(D)\bigr)\le3\Bigl(\frac{\varepsilon}{N}+\frac{1}{N}\Delta_{\gamma^{\boxtimes}}(\nabla\psi,\nabla\chi)+\frac{\varepsilon}{N}\Bigr),\qquad\Delta_{\gamma^{\boxtimes}}(\nabla\psi,\nabla\chi)\le3\bigl(2\varepsilon+\Delta_{\gamma^{\boxtimes}}(D',D)\bigr),$$

so that $\Delta_{\gamma}(\Pi_{P'}(D'),\Pi_{P}(D))\le\frac{9}{N}\Delta_{\gamma^{\boxtimes}}(D',D)+\frac{24}{N}\varepsilon$. As $\varepsilon>0$ is arbitrary, (9.1) follows by Comparison of Real Numbers with Arbitrary Positive Slack §slack-above, applied to the error term with $\varepsilon$ replaced by $N\varepsilon'/24$ for arbitrary positive $\varepsilon'$.

Step 10 (Property (c)). Let $\mu\in\mathcal{Q}$, let $(\mu_{n})_{n\in\mathbb{N}}$ be a sequence in $\mathcal{Q}$, and let $\pi_{n}\in\Pi(\mu_{n},\mu)$ with $I(\pi_{n})\to0$. Put $P_{n}=\mu_{n}^{\otimes N}$ and $P=\mu^{\otimes N}$, which lie in $\mathcal{Q}_{N}$. By (T1) and (T2) (with $\mu_{n},\mu,\pi_{n}$ in place of $\mu,\nu,\pi$), $\pi_{n}^{\boxtimes}\in\Pi(P_{n},P)$ and $I(\pi_{n}^{\boxtimes})=N\,I(\pi_{n})$, which converges to $0$ (Limit of a Sequence of Real Numbers: given $\varepsilon>0$, eventually $I(\pi_{n})<\varepsilon/N$, hence $N\,I(\pi_{n})<\varepsilon$ by claim 10 of Elementary Order Arithmetic in an Ordered Field). By property (c) of $\Phi$ at the configuration level, $\Delta_{\pi_{n}^{\boxtimes}}(\nabla\Phi(P_{n}),\nabla\Phi(P))\to0$. By Step 8, $\nabla\varphi(\mu_{n})=N\,\Pi_{P_{n}}(\nabla\Phi(P_{n}))$ and $\nabla\varphi(\mu)=N\,\Pi_{P}(\nabla\Phi(P))$, and $\nabla\Phi(P_{n})\in T_{P_{n}}$, $\nabla\Phi(P)\in T_{P}$ by property (b) of $\Phi$. Since $\lVert Na-Nb\rVert^{2}=N^{2}\lVert a-b\rVert^{2}$, claim 1 of Linearity and Monotonicity of the Lebesgue Integral and (9.1) (with $\mu_{n}$ and $\pi_{n}$ in place of $\mu'$ and $\gamma$) give

$$0\le\Delta_{\pi_{n}}\bigl(\nabla\varphi(\mu_{n}),\nabla\varphi(\mu)\bigr)=N^{2}\,\Delta_{\pi_{n}}\bigl(\Pi_{P_{n}}(\nabla\Phi(P_{n})),\Pi_{P}(\nabla\Phi(P))\bigr)\le9N\,\Delta_{\pi_{n}^{\boxtimes}}\bigl(\nabla\Phi(P_{n}),\nabla\Phi(P)\bigr).$$

Given $\varepsilon>0$, the right side is eventually below $\varepsilon$, hence so is the middle term; thus the discrepancies of $\nabla\varphi(\mu_{n})$ and $\nabla\varphi(\mu)$ along $\pi_{n}$ converge to $0$, which is property (c) for $\varphi$.

Step 11 (Property (d) and the quadratic-form identity). Let $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and $P=\mu^{\otimes N}\in\mathcal{P}_{2}(\mathbb{R}^{dN})$, and let $\Phi_{P}(v)=\Phi((\tau_{v})_{\#}P)$ for $v\in\mathbb{R}^{dN}$, which is of class $C^{2}$ on $\mathbb{R}^{dN}$ by property (d) of $\Phi$ at the configuration level. For $a\in\mathbb{R}^{d}$, Tensor Powers and One-Particle Marginals: Particle Laws, Product Integrals, Push-Forwards, Moments, Product Maps and Diagonal Shifts §diagonal gives $((\tau_{a})_{\#}\mu)^{\otimes N}=(\tau_{a^{\oplus}})_{\#}P$, hence

$$\phi_{\mu}(a)=\varphi\bigl((\tau_{a})_{\#}\mu\bigr)=\Phi\bigl(((\tau_{a})_{\#}\mu)^{\otimes N}\bigr)=\Phi\bigl((\tau_{a^{\oplus}})_{\#}P\bigr)=\Phi_{P}(a^{\oplus}).\tag{11.1}$$

The map $L:\mathbb{R}^{d}\to\mathbb{R}^{dN}$, $L(a)=a^{\oplus}$, has as coordinate with index $b(k,i)$ the coordinate function $a\mapsto a_{i}$ (Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points §diagonal, Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points §configuration, Block Indices: Enumerating an Initial Segment of Length qN by Blocks and Positions §bijection), so it is smooth by claim 2 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set and claim 1 of Coordinate Functions, the $C^k$ Hierarchy, and Partial Derivatives of a Smooth Map, in particular of class $C^{2}$ (Smooth Map on a Euclidean Open Set). By claim 2 of A Composition of $C^k$ Maps Between Euclidean Open Sets is of Class $C^k$, $\phi_{\mu}=\Phi_{P}\circ L$ is of class $C^{2}$ on $\mathbb{R}^{d}$. This is property (d) for $\varphi$.

We now show that for every $h\in\mathbb{R}^{d}$

$$h\cdot\bigl(D^{2}\phi_{\mu}(0_{\mathbb{R}^{d}})\,h\bigr)=h^{\oplus}\cdot\bigl(D^{2}\Phi_{P}(0_{\mathbb{R}^{dN}})\,h^{\oplus}\bigr).\tag{11.2}$$

Write $H=D^{2}\phi_{\mu}(0_{\mathbb{R}^{d}})$, $p=D\phi_{\mu}(0_{\mathbb{R}^{d}})$, $H'=D^{2}\Phi_{P}(0_{\mathbb{R}^{dN}})$ and $p'=D\Phi_{P}(0_{\mathbb{R}^{dN}})$. By Basic Properties of Twice Differentiability at a Point §c2, $\phi_{\mu}$ and $\Phi_{P}$ are twice differentiable at the origin with first-order coefficients $p$, $p'$ and Hessians $H$, $H'$. Fix $h\in\mathbb{R}^{d}$ and $\varepsilon>0$, and let $\delta_{1},\delta_{2}>0$ be as in that definition for $\phi_{\mu}$ and for $\Phi_{P}$, with this $\varepsilon$. Choose a real $t>0$ with $t\lVert h\rVert<\delta_{1}$ and $t\lVert h^{\oplus}\rVert<\delta_{2}$, for instance $t=\min\{\delta_{1},\delta_{2}\}\,(1+\lVert h\rVert+\lVert h^{\oplus}\rVert)^{-1}$ (claims 5, 7 and 9 of Elementary Order Arithmetic in an Ordered Field). For $s\in\{t,-t\}$ we have $(sh)^{\oplus}=s\,h^{\oplus}$ and $0_{\mathbb{R}^{d}}^{\oplus}=0_{\mathbb{R}^{dN}}$ (Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §linear), so (11.1) gives $\phi_{\mu}(sh)=\Phi_{P}(s\,h^{\oplus})$ and $\phi_{\mu}(0_{\mathbb{R}^{d}})=\Phi_{P}(0_{\mathbb{R}^{dN}})$. Let

$$A(s)=\phi_{\mu}(sh)-\phi_{\mu}(0_{\mathbb{R}^{d}})-s\,p\cdot h-\tfrac{1}{2}s^{2}\,h\cdot(Hh),\qquad B(s)=\Phi_{P}(s\,h^{\oplus})-\Phi_{P}(0_{\mathbb{R}^{dN}})-s\,p'\cdot h^{\oplus}-\tfrac{1}{2}s^{2}\,h^{\oplus}\cdot(H'h^{\oplus}),$$

where the quadratic terms are those of the definition at $sh$ and $sh^{\oplus}$ because the matrix-vector product is linear in the vector (claim 1 of Linearity, Compatibility with the Matrix Product, and a Norm Bound for the Matrix-Vector Product). Then $|A(s)|\le\varepsilon t^{2}\lVert h\rVert^{2}$ and $|B(s)|\le\varepsilon t^{2}\lVert h^{\oplus}\rVert^{2}=\varepsilon Nt^{2}\lVert h\rVert^{2}$ (Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §diagonal), and, since $p'\cdot h^{\oplus}=\bigl(\sum_{k}\mathfrak{p}_{k}(p')\bigr)\cdot h$ by that clause,

$$A(s)-B(s)=s\,\ell\cdot h+s^{2}\,Q(h),\qquad\ell=\sum_{k=1}^{N}\mathfrak{p}_{k}(p')-p,\qquad Q(h)=\tfrac{1}{2}\bigl(h^{\oplus}\cdot(H'h^{\oplus})-h\cdot(Hh)\bigr).$$

Adding the cases $s=t$ and $s=-t$ gives $2t^{2}Q(h)=A(t)-B(t)+A(-t)-B(-t)$, so the triangle inequality (claim 5 of Properties of the Absolute Value in an Ordered Field, applied three times) gives $2t^{2}|Q(h)|\le2\varepsilon(1+N)t^{2}\lVert h\rVert^{2}$, and multiplying by $(2t^{2})^{-1}>0$ gives $|Q(h)|\le\varepsilon(1+N)\lVert h\rVert^{2}$. As $\varepsilon>0$ is arbitrary, Comparison of Real Numbers with Arbitrary Positive Slack §vanishing, applied to the nonnegative number $|Q(h)|$ with $\varepsilon$ replaced by $\varepsilon'((1+N)\lVert h\rVert^{2})^{-1}$ for arbitrary positive $\varepsilon'$ when $h$ is not the origin (for $h$ the origin the bound is $0$ itself), gives $|Q(h)|=0$, hence $Q(h)=0$ by claim 1 of Properties of the Absolute Value in an Ordered Field, which is (11.2).

Step 12 (Property (e)). Let $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, $P=\mu^{\otimes N}$ and $\varepsilon>0$; then $\varepsilon/(2N)>0$ (claims 7 and 8 of Elementary Order Arithmetic in an Ordered Field). By property (e) of $\Phi$ at the configuration level there is $\theta>0$ with $d_{\mathcal{S}(dN)}(D^{2}\Phi_{P'}(0_{\mathbb{R}^{dN}}),D^{2}\Phi_{P}(0_{\mathbb{R}^{dN}}))<\varepsilon/(2N)$ for every $P'\in\mathcal{P}_{2}(\mathbb{R}^{dN})$ with $W_{2}(P',P)<\theta$. Let $\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ with $W_{2}(\nu,\mu)<\theta/\sqrt{N}$. By (W), $W_{2}(\nu^{\otimes N},P)<\theta$, so $M=D^{2}\Phi_{\nu^{\otimes N}}(0_{\mathbb{R}^{dN}})-D^{2}\Phi_{P}(0_{\mathbb{R}^{dN}})\in\mathcal{S}(dN)$ has $\lVert M\rVert<\varepsilon/(2N)$ (Real Matrices, Symmetric Matrices and the Semidefinite Ordering: Standing Notation §norm). Let $\xi\in\mathbb{R}^{d}$ with $\lVert\xi\rVert\le1$. By (11.2) at $\nu$ and at $\mu$, and the linearity of the matrix-vector product in the matrix (each coordinate of $Av$ is, by the formula of Matrix-Vector Product, a finite sum of the entries of $A$ times fixed reals, so it is additive in $A$ and commutes with scalar multiples of $A$ by claims 2 and 3 of Properties of Finite Sums),

$$\xi\cdot\Bigl(\bigl(D^{2}\phi_{\nu}(0_{\mathbb{R}^{d}})-D^{2}\phi_{\mu}(0_{\mathbb{R}^{d}})\bigr)\xi\Bigr)=\xi^{\oplus}\cdot(M\xi^{\oplus}).$$

The point $\zeta=N^{-1/2}\xi^{\oplus}$ satisfies $\lVert\zeta\rVert^{2}=N^{-1}\lVert\xi^{\oplus}\rVert^{2}=\lVert\xi\rVert^{2}\le1$ (Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §diagonal), hence $\lVert\zeta\rVert\le1$ (claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field), and $\xi^{\oplus}\cdot(M\xi^{\oplus})=N\,\zeta\cdot(M\zeta)$. By Norm of a Symmetric Real Matrix, $|\zeta\cdot(M\zeta)|\le\lVert M\rVert$, so

$$\Bigl|\xi\cdot\Bigl(\bigl(D^{2}\phi_{\nu}(0_{\mathbb{R}^{d}})-D^{2}\phi_{\mu}(0_{\mathbb{R}^{d}})\bigr)\xi\Bigr)\Bigr|\le N\lVert M\rVert<\frac{\varepsilon}{2}.$$

The difference of the two Hessians lies in $\mathcal{S}(d)$ (Real Matrices, Symmetric Matrices and the Semidefinite Ordering: Standing Notation §symmetric), and $\varepsilon/2$ is an upper bound of the set whose least upper bound is its norm, so $d_{\mathcal{S}(d)}(D^{2}\phi_{\nu}(0_{\mathbb{R}^{d}}),D^{2}\phi_{\mu}(0_{\mathbb{R}^{d}}))\le\varepsilon/2<\varepsilon$ by Real Matrices, Symmetric Matrices and the Semidefinite Ordering: Standing Notation §norm and claim 8 of Elementary Order Arithmetic in an Ordered Field. Since $\theta/\sqrt{N}>0$, this is property (e) for $\varphi$.

Step 13 (Conclusion). Steps 2, 8, 10, 11 and 12 establish properties (a) to (e) of Intrinsic Test Functions on the Wasserstein Space and Their Translation Hessians §test for $\varphi$ on $\mathcal{Q}$, so $\varphi$ is an intrinsic test function on $\mathcal{Q}$, and Step 8 gives its gradient along couplings; this is clause 1. By Intrinsic Test Functions on the Wasserstein Space and Their Translation Hessians §hessian, $H_{\varphi}(\mu)=D^{2}\phi_{\mu}(0_{\mathbb{R}^{d}})$ for every $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, and $H_{\Phi}(\mu^{\otimes N})=D^{2}\Phi_{\mu^{\otimes N}}(0_{\mathbb{R}^{dN}})$, read at the configuration level; so (11.2) with $h=a$ is clause 2.