Proof of Scores on the Configuration Space: the Score of a Tensor Power is the Product Field of the Score, and the Score of the One-Particle Marginal is the Projection of the Score

Each result cited is universally quantified over the data in its own statement. Block maps $\mathfrak{p}_{k}:\mathbb{R}^{dN}\to\mathbb{R}^{d}$, configurations and product maps are those of Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points with $q=p=d$, 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 with $q=d$. For $m\in\{d,dN\}$, test functions, gradient maps $\nabla$ and Laplacians $\Delta$ are those of Test Functions on Euclidean Space, Their Gradient Maps and Laplacians with $q=m$, and Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §l2mu, Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §test, The Gradient of a Test Function is Bounded and Square-Integrable, and Its Laplacian Bounded and Integrable, Against Every Probability Measure, Basic Properties of the Tangent Space: Closed Subspace, the Identity Map Belongs to It, Second-Moment Limits, and Representation of Bounded Functionals on Gradients and 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 are read with $m$ in place of $d$, exactly as the statement reads the definitions. $\mathbb{R}^{m}$ is open by claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous, $e_{j}$ is the standard basis vector of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation §matrices, so that $y+he_{j}$ is the point obtained from $y$ by adding $h$ to its $j$th coordinate, as in Partial Derivative on a Euclidean Open Set; the gradient $Df$ is that of Gradient of a Real-Valued Function on a Euclidean Open Set. Composites of Borel maps are Borel and push-forwards obey change of variables, by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps and Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward.

Step 1 (Increments along blocks). Let $x\in\mathbb{R}^{dN}$, $k,l\in[N]$, $j\in[d]$ and $h\in\mathbb{R}$. The $i$th coordinate of $\mathfrak{p}_{k}(x+he_{b(l,j)})$ is $x_{b(k,i)}+h$ if $b(k,i)=b(l,j)$, that is (Block Indices: Enumerating an Initial Segment of Length qN by Blocks and Positions §bijection) if $(k,i)=(l,j)$, and $x_{b(k,i)}$ otherwise (Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points §blocks). Hence $\mathfrak{p}_{k}(x+he_{b(l,j)})=\mathfrak{p}_{k}(x)$ for $k\ne l$ and $\mathfrak{p}_{l}(x+he_{b(l,j)})=\mathfrak{p}_{l}(x)+he_{j}$. Comparing difference quotients in Partial Derivative on a Euclidean Open Set, for $f:\mathbb{R}^{d}\to\mathbb{R}$:

(a) if $k\ne l$, then $\partial_{b(l,j)}(f\circ\mathfrak{p}_{k})(x)$ exists and is $0$; if $\partial_{j}f(\mathfrak{p}_{l}(x))$ exists, then $\partial_{b(l,j)}(f\circ\mathfrak{p}_{l})(x)$ exists and equals $\partial_{j}f(\mathfrak{p}_{l}(x))$;

(b) if $\partial_{j}f$ exists everywhere and $S_{f}(y)=\sum_{k=1}^{N}f(\mathfrak{p}_{k}(y))$, then $\partial_{b(l,j)}S_{f}(x)$ exists and equals $\partial_{j}f(\mathfrak{p}_{l}(x))$: by claims 2, 3 and 7 of Properties of Finite Sums, $S_{f}(x+he_{b(l,j)})-S_{f}(x)=\sum_{k=1}^{N}\bigl(f(\mathfrak{p}_{k}(x+he_{b(l,j)}))-f(\mathfrak{p}_{k}(x))\bigr)=f(\mathfrak{p}_{l}(x)+he_{j})-f(\mathfrak{p}_{l}(x))$.

Step 2 (Sums over the particles of a test function). Let $\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$ and $\Psi=S_{\psi}$, $\Psi(x)=\sum_{k=1}^{N}\psi(\mathfrak{p}_{k}(x))$. The $i$th coordinate of $\mathfrak{p}_{k}$ is the coordinate function $x\mapsto x_{b(k,i)}$, smooth by claim 2 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set, so $\mathfrak{p}_{k}$ is smooth by claim 1 of Coordinate Functions, the $C^k$ Hierarchy, and Partial Derivatives of a Smooth Map, $\psi\circ\mathfrak{p}_{k}$ is smooth by claim 3 of A Composition of $C^k$ Maps Between Euclidean Open Sets is of Class $C^k$, and $\Psi$ is smooth by claim 3 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set along the recursion of claim 1 of Properties of Finite Sums; in particular $\Psi$ is of class $C^{1}$ and $C^{2}$ (Smooth Map on a Euclidean Open Set). Each $\partial_{j}\psi$ is smooth (claim 3 of Coordinate Functions, the $C^k$ Hierarchy, and Partial Derivatives of a Smooth Map) and compactly supported and bounded (The Gradient of a Test Function is Bounded and Square-Integrable, and Its Laplacian Bounded and Integrable, Against Every Probability Measure §gradient), so $\partial_{j}\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$ and each $\partial_{j'}\partial_{j}\psi$ exists everywhere and is bounded, by the same clause applied to $\partial_{j}\psi$. Step 1(b) with $f=\psi$, and then Step 1(a) with $f=\partial_{j}\psi$, give for $x\in\mathbb{R}^{dN}$, $l,l'\in[N]$, $j,j'\in[d]$:

Every index in $[dN]$ is $b(l,j)$ for exactly one pair (Block Indices: Enumerating an Initial Segment of Length qN by Blocks and Positions §bijection); so, with $M\ge0$ a common bound of the finitely many bounded functions $|\partial_{j}\psi|$ and $|\partial_{j'}\partial_{j}\psi|$ (for instance the sum of bounds for each, by claim 6 of Properties of Finite Sums), all first and second partial derivatives of $\Psi$ are bounded by $M$. The coordinate of $D\Psi(x)$ with index $b(l,j)$ is $\partial_{j}\psi(\mathfrak{p}_{l}(x))$, the $j$th coordinate of $\nabla\psi(\mathfrak{p}_{l}(x))$, which is the coordinate with index $b(l,j)$ of $(\nabla\psi)^{\oplus}(x)$ (Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points §product-map, Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points §configuration); thus $D\Psi=(\nabla\psi)^{\oplus}$ as maps. By The Laplacian of a Twice Continuously Differentiable Function §laplacian and Block Indices: Enumerating an Initial Segment of Length qN by Blocks and Positions §sums,

For $P\in\mathcal{P}_{2}(\mathbb{R}^{dN})$, the Borel map $\nabla\psi$ represents its class in $L^{2}(P^{[1]};\mathbb{R}^{d})$, so the class of $D\Psi$ in $L^{2}(P;\mathbb{R}^{dN})$ is the product field $(\nabla\psi)^{\oplus}$ of Product Fields and the Projection onto One-Particle Tangent Fields §product-field.

Step 3 (Claim 2). Let $P\in\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{dN})$ and $\mu=P^{[1]}\in\mathcal{P}_{2}(\mathbb{R}^{d})$. Let $\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$ and $\Psi$ be as in Step 2. 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 §score-identity read with $dN$ in place of $d$ (with $f=\Psi$ and the bound $M$ of Step 2), $\langle\xi_{P},(\nabla\psi)^{\oplus}\rangle_{P}=\langle\xi_{P},D\Psi\rangle_{P}=-\int\Delta\Psi\,dP$. The function $\Delta\psi$ is bounded and Borel (The Gradient of a Test Function is Bounded and Square-Integrable, and Its Laplacian Bounded and Integrable, Against Every Probability Measure §laplacian), hence $\mu$-integrable; since $\mu$ is the measure $A_{P}$ of Existence and Uniqueness of Tensor Powers, and the Average of the Block Marginals §average (The One-Particle Marginal of a Probability Measure on the Configuration Space §marginal), each $\Delta\psi\circ\mathfrak{p}_{l}$ is $P$-integrable and $\sum_{l=1}^{N}\int\Delta\psi\circ\mathfrak{p}_{l}\,dP=N\int\Delta\psi\,d\mu$, so $\int\Delta\Psi\,dP=N\int\Delta\psi\,d\mu$ by Step 2 and Linearity of the Lebesgue Integral over a Finite Sum of Integrable Functions §linear. By Product Fields and the Projection onto One-Particle Tangent Fields §projection,

By The Cauchy-Schwarz Inequality in a Real Inner Product Space, $|\int\Delta\psi\,d\mu|\le\lVert\Pi_{P}(\xi_{P})\rVert_{\mu}\lVert\nabla\psi\rVert_{\mu}$, so $\mu\in\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{d})$ (Finite Fisher Information, the Score and the Fisher Information of a Probability Measure §finite). As $\Pi_{P}(\xi_{P})\in T_{\mu}$ satisfies the identity of Finite Fisher Information, the Score and the Fisher Information of a Probability Measure §score, uniqueness there gives $\xi_{\mu}=\Pi_{P}(\xi_{P})$. Then The Projection onto One-Particle Tangent Fields: Pairing with Product Fields, Contraction, and Product Fields of Tangent Fields §contraction and Finite Fisher Information, the Score and the Fisher Information of a Probability Measure §information give $N\,\mathcal{I}(\mu)=N\lVert\Pi_{P}(\xi_{P})\rVert_{\mu}^{2}\le\lVert\xi_{P}\rVert_{P}^{2}=\mathcal{I}(P)$, and for $g\in T_{\mu}$, The Projection onto One-Particle Tangent Fields: Pairing with Product Fields, Contraction, and Product Fields of Tangent Fields §pairing with $D=\xi_{P}$ gives $\langle\xi_{P},g^{\oplus}\rangle_{P}=N\langle\Pi_{P}(\xi_{P}),g\rangle_{\mu}=N\langle\xi_{\mu},g\rangle_{\mu}$. This is claim 2.

Step 4 (Setting for claim 1). Let $\rho\in\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{d})$ and $P=\rho^{\otimes N}$. Then $P\in\mathcal{P}_{2}(\mathbb{R}^{dN})$ by Tensor Powers and One-Particle Marginals: Particle Laws, Product Integrals, Push-Forwards, Moments, Product Maps and Diagonal Shifts §moments and $P^{[1]}=\rho$ by Tensor Powers and One-Particle Marginals: Particle Laws, Product Integrals, Push-Forwards, Moments, Product Maps and Diagonal Shifts §marginal-of-tensor. Fix a Borel representative $\tilde\xi$ of $\xi_{\rho}$; by Product Fields and the Projection onto One-Particle Tangent Fields §product-field, $(\xi_{\rho})^{\oplus}\in L^{2}(P;\mathbb{R}^{dN})$ is the class of $\tilde\xi^{\oplus}$ and $\lVert(\xi_{\rho})^{\oplus}\rVert_{P}^{2}=N\lVert\xi_{\rho}\rVert_{\rho}^{2}$. For $g\in L^{2}(\rho;\mathbb{R}^{d})$ with Borel representative $\tilde g$, $\tilde\xi+(-1)\tilde g$ represents $\xi_{\rho}-g$ by The Space of Square-Integrable Random Vectors §classes, and $(\tilde\xi+(-1)\tilde g)^{\oplus}=\tilde\xi^{\oplus}+(-1)\tilde g^{\oplus}$ pointwise by Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §linear; hence $(\xi_{\rho})^{\oplus}-g^{\oplus}=(\xi_{\rho}-g)^{\oplus}$ and $\lVert(\xi_{\rho})^{\oplus}-g^{\oplus}\rVert_{P}^{2}=N\lVert\xi_{\rho}-g\rVert_{\rho}^{2}$.

Step 5 (Tangency). Let $\varepsilon>0$ and $\delta=\varepsilon N^{-1}>0$. Since $\xi_{\rho}\in T_{\rho}$, the closure of the gradients of test functions (The Tangent Space of the Wasserstein Space at a Probability Measure §tangent), Characterization of the Closure in a Metric Space by Open Balls (claim 1 implies claim 3) gives $\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$ with $\lVert\xi_{\rho}-\nabla\psi\rVert_{\rho}<\delta$. With $\Psi$ as in Step 2, $D\Psi\in T_{P}$ 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 read with $dN$ in place of $d$, its class being $(\nabla\psi)^{\oplus}$, and by Step 4, $\lVert(\xi_{\rho})^{\oplus}-(\nabla\psi)^{\oplus}\rVert_{P}^{2}=N\lVert\xi_{\rho}-\nabla\psi\rVert_{\rho}^{2}<N\delta^{2}=\varepsilon^{2}N^{-1}\le\varepsilon^{2}$, so $\lVert(\xi_{\rho})^{\oplus}-(\nabla\psi)^{\oplus}\rVert_{P}<\varepsilon$. By Characterization of the Closure in a Metric Space by Open Balls (claim 3 implies claim 1), $(\xi_{\rho})^{\oplus}$ lies in the closure of $T_{P}$, which is $T_{P}$ by claim 4 of The Closure is the Smallest Closed Superset, $T_{P}$ being closed 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 read with $dN$ in place of $d$. So $(\xi_{\rho})^{\oplus}\in T_{P}$.

Step 6 (Replacing one particle). Fix $k\in[N]$. Let $\mathrm{pr}_{1}:\mathbb{R}^{dN+d}\to\mathbb{R}^{dN}$, $\mathrm{pr}_{2}:\mathbb{R}^{dN+d}\to\mathbb{R}^{d}$ and $\iota=\iota^{dN,d}$ be as in Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections, and let $R_{k}:\mathbb{R}^{dN+d}\to\mathbb{R}^{dN}$ send $z$ to the configuration whose $l$th particle is $\mathfrak{p}_{l}(\mathrm{pr}_{1}(z))$ for $l\ne k$ and whose $k$th particle is $\mathrm{pr}_{2}(z)$. For $x\in\mathbb{R}^{dN}$ put $E_{x}(v)=R_{k}(\iota(x,v))$ ($v\in\mathbb{R}^{d}$); by Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points §configuration and Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections, $\mathfrak{p}_{l}(E_{x}(v))=\mathfrak{p}_{l}(x)$ for $l\ne k$ and $\mathfrak{p}_{k}(E_{x}(v))=v$. Each component of $R_{k}$ is a component of $\mathfrak{p}_{l}\circ\mathrm{pr}_{1}$ or of $\mathrm{pr}_{2}$ (Block Indices: Enumerating an Initial Segment of Length qN by Blocks and Positions §bijection), which are Borel by Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §linear and Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections; so $R_{k}$ is 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.

Let $B_{1},\dots,B_{N}\in\mathcal{B}(\mathbb{R}^{d})$, put $B'_{l}=B_{l}$ for $l\ne k$ and $B'_{k}=\mathbb{R}^{d}$, and $A=\bigcap_{l=1}^{N}\mathfrak{p}_{l}^{-1}(B'_{l})\in\mathcal{B}(\mathbb{R}^{dN})$. Then $R_{k}^{-1}\bigl(\bigcap_{l}\mathfrak{p}_{l}^{-1}(B_{l})\bigr)=\mathrm{pr}_{1}^{-1}(A)\cap\mathrm{pr}_{2}^{-1}(B_{k})=\iota(A\times B_{k})$, so by 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,

The last equality: for $N=1$ both sides are $\rho(B_{1})$, as $\rho(\mathbb{R}^{d})=1$ and a product over $[1]$ is its factor (Finite Product Notation); for $N=S(n)$ with $n\in\mathbb{N}$, claim 3 of Extraction of a Term from a Finite Sum or Product in a Field for the field $\mathbb{R}$ (whose finite products satisfy the recursion of Finite Product Notation, hence are those products) with $j=k$ gives $\prod_{l}\rho(B'_{l})=\bigl(\prod_{m=1}^{n}\rho(B_{g_{k}(m)})\bigr)\cdot1$ and $\prod_{l}\rho(B_{l})=\bigl(\prod_{m=1}^{n}\rho(B_{g_{k}(m)})\bigr)\rho(B_{k})$, since $g_{k}(m)\ne k$ by claim 1 there. As $(R_{k})_{\#}(P\boxtimes\rho)\in\mathcal{P}(\mathbb{R}^{dN})$, the uniqueness in Existence and Uniqueness of Tensor Powers, and the Average of the Block Marginals §tensor gives $(R_{k})_{\#}(P\boxtimes\rho)=P$.

Consequently, if $H:\mathbb{R}^{dN}\to\mathbb{R}$ is Borel and $P$-integrable, then $H\circ R_{k}$ is $P\boxtimes\rho$-integrable with the same integral (change of variables), and since $P\boxtimes\rho$ is the image of $P\otimes\rho$ under the $\mathcal{B}(\mathbb{R}^{dN})\otimes\mathcal{B}(\mathbb{R}^{d})$-measurable map $\iota$ (Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §product), claim 2 of Image Measures, Measures with Densities, and Change of Variables shows that $\hat H(x,v)=H(E_{x}(v))$ is $P\otimes\rho$-integrable with $\int\hat H\,d(P\otimes\rho)=\int H\,dP$.

Step 7 (Sections of test functions). Let $\Phi\in C_{c}^{\infty}(\mathbb{R}^{dN})$, $x\in\mathbb{R}^{dN}$, $\phi=\Phi\circ E_{x}$, and $G_{k}=\sum_{i=1}^{d}\partial_{b(k,i)}\partial_{b(k,i)}\Phi$. The coordinate of $E_{x}(v)$ with index $b(l,i)$ is the constant $x_{b(l,i)}$ for $l\ne k$ and $v_{i}$ for $l=k$, so $E_{x}$ is smooth (claim 2 of Constants, Coordinate Functions, Sums and Products of $C^k$ Functions on a Euclidean Open Set, claim 1 of Coordinate Functions, the $C^k$ Hierarchy, and Partial Derivatives of a Smooth Map) and $\phi$ is smooth (claim 3 of A Composition of $C^k$ Maps Between Euclidean Open Sets is of Class $C^k$); moreover $E_{x}(v+he_{i})=E_{x}(v)+he_{b(k,i)}$ (Block Indices: Enumerating an Initial Segment of Length qN by Blocks and Positions §bijection). Comparing difference quotients in Partial Derivative on a Euclidean Open Set, for every $\Theta:\mathbb{R}^{dN}\to\mathbb{R}$ with $\partial_{b(k,i)}\Theta$ existing everywhere, $\partial_{i}(\Theta\circ E_{x})=(\partial_{b(k,i)}\Theta)\circ E_{x}$. Applied to $\Theta=\Phi$ and to the smooth $\Theta=\partial_{b(k,i)}\Phi$ (claim 3 of Coordinate Functions, the $C^k$ Hierarchy, and Partial Derivatives of a Smooth Map), this gives $\partial_{i}\phi=(\partial_{b(k,i)}\Phi)\circ E_{x}$ and $\partial_{i}\partial_{i}\phi=(\partial_{b(k,i)}\partial_{b(k,i)}\Phi)\circ E_{x}$; since the $i$th coordinate of $\mathfrak{p}_{k}(\nabla\Phi(y))$ is $\partial_{b(k,i)}\Phi(y)$, $\nabla\phi=\mathfrak{p}_{k}\circ\nabla\Phi\circ E_{x}$ and $\Delta\phi=G_{k}\circ E_{x}$. By claim 2 of Compact Support on $\mathbb{R}^n$ Means Vanishing Outside a Bounded Set (its topology being that of Test Functions on Euclidean Space, Their Gradient Maps and Laplacians §space by Euclidean Openness Agrees with Metric Openness on $\mathbb{R}^n$) there is $R>0$ with $\Phi(y)=0$ for $\lVert y\rVert>R$; as $\lVert v\rVert=\lVert\mathfrak{p}_{k}(E_{x}(v))\rVert\le\lVert E_{x}(v)\rVert$ (Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §inner-product), $\phi(v)=0$ for $\lVert v\rVert>R$, and the same criterion shows $\phi$ compactly supported. So $\phi\in C_{c}^{\infty}(\mathbb{R}^{d})$.

Step 8 (Integration by parts for the product field). Let $\Phi\in C_{c}^{\infty}(\mathbb{R}^{dN})$, let $K\ge0$ be the bound of $\lVert\nabla\Phi\rVert$ given by The Gradient of a Test Function is Bounded and Square-Integrable, and Its Laplacian Bounded and Integrable, Against Every Probability Measure §gradient, and, for $k\in[N]$ (Steps 6 and 7 being applied with this $k$), let $F_{k}(y)=\tilde\xi(\mathfrak{p}_{k}(y))\cdot\mathfrak{p}_{k}(\nabla\Phi(y))$ and $G_{k}$ be as in Step 7. The maps $\tilde\xi\circ\mathfrak{p}_{k}$ and $\mathfrak{p}_{k}\circ\nabla\Phi$ are Borel (The Gradient of a Test Function is Bounded and Square-Integrable, and Its Laplacian Bounded and Integrable, Against Every Probability Measure §gradient); $\int\lVert\tilde\xi\circ\mathfrak{p}_{k}\rVert^{2}\,dP=\int\lVert\tilde\xi\rVert^{2}\,d\rho<\infty$ by Tensor Powers and One-Particle Marginals: Particle Laws, Product Integrals, Push-Forwards, Moments, Product Maps and Diagonal Shifts §particle-laws and change of variables, and $\lVert\mathfrak{p}_{k}(\nabla\Phi(y))\rVert\le\lVert\nabla\Phi(y)\rVert\le K$ by Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §inner-product, so $\int\lVert\mathfrak{p}_{k}\circ\nabla\Phi\rVert^{2}\,dP\le K^{2}$ (claim 1 of Linearity and Monotonicity of the Lebesgue Integral). So $F_{k}$ is $P$-integrable by Square-Integrable Random Vectors: Coordinates, Operations, Almost Sure Equality and the Mean-Square Form §operations on $(\mathbb{R}^{dN},\mathcal{B}(\mathbb{R}^{dN}),P)$. Each $\partial_{b(k,i)}\Phi$ is smooth (claim 3 of Coordinate Functions, the $C^k$ Hierarchy, and Partial Derivatives of a Smooth Map) and compactly supported (The Gradient of a Test Function is Bounded and Square-Integrable, and Its Laplacian Bounded and Integrable, Against Every Probability Measure §gradient), so $\partial_{b(k,i)}\partial_{b(k,i)}\Phi$ is Borel and bounded by the same clause, hence $P$-integrable (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures), and $G_{k}$ is $P$-integrable by Linearity of the Lebesgue Integral over a Finite Sum of Integrable Functions §integrable. Thus $H_{k}=F_{k}+G_{k}$ is $P$-integrable (claim 2 of Linearity and Monotonicity of the Lebesgue Integral), and by Step 6, $\hat H_{k}(x,v)=H_{k}(E_{x}(v))$ is $P\otimes\rho$-integrable with $\int\hat H_{k}\,d(P\otimes\rho)=\int H_{k}\,dP$.

For every $x\in\mathbb{R}^{dN}$, with $\phi_{x}=\Phi\circ E_{x}\in C_{c}^{\infty}(\mathbb{R}^{d})$ (Step 7) and $\mathfrak{p}_{k}(E_{x}(v))=v$, the section is $\hat H_{k}(x,v)=\tilde\xi(v)\cdot\nabla\phi_{x}(v)+\Delta\phi_{x}(v)$. Here $\tilde\xi\cdot\nabla\phi_{x}$ is $\rho$-integrable with integral $\langle\xi_{\rho},\nabla\phi_{x}\rangle_{\rho}$ (Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §l2mu, Square-Integrable Random Vectors: Coordinates, Operations, Almost Sure Equality and the Mean-Square Form §operations) and $\Delta\phi_{x}$ is $\rho$-integrable (Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §test), so the section is $\rho$-integrable with integral $\langle\xi_{\rho},\nabla\phi_{x}\rangle_{\rho}+\int\Delta\phi_{x}\,d\rho=0$ by Finite Fisher Information, the Score and the Fisher Information of a Probability Measure §score. By the Fubini part of Tonelli and Fubini Theorems ($P$ and $\rho$ being probability measures, hence $\sigma$-finite), the function equal to the section integral off a $P$-null set and to $0$ on it, which is identically $0$, has $P$-integral $\int\hat H_{k}\,d(P\otimes\rho)$; hence $\int H_{k}\,dP=0$, that is, $\int F_{k}\,dP=-\int G_{k}\,dP$.

For $y\in\mathbb{R}^{dN}$, Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §inner-product and Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §product-map give $\tilde\xi^{\oplus}(y)\cdot\nabla\Phi(y)=\sum_{k=1}^{N}F_{k}(y)$, and The Laplacian of a Twice Continuously Differentiable Function §laplacian with Block Indices: Enumerating an Initial Segment of Length qN by Blocks and Positions §sums gives $\Delta\Phi(y)=\sum_{k=1}^{N}G_{k}(y)$. Therefore, by Linearity of the Lebesgue Integral over a Finite Sum of Integrable Functions §linear and claim 3 of Properties of Finite Sums,

Step 9 (Claim 1). By Step 8 and The Cauchy-Schwarz Inequality in a Real Inner Product Space, $|\int\Delta\Phi\,dP|\le\lVert(\xi_{\rho})^{\oplus}\rVert_{P}\lVert\nabla\Phi\rVert_{P}$ for every $\Phi\in C_{c}^{\infty}(\mathbb{R}^{dN})$, so $P=\rho^{\otimes N}\in\mathcal{P}_{2}^{\mathcal{I}}(\mathbb{R}^{dN})$ (Finite Fisher Information, the Score and the Fisher Information of a Probability Measure §finite read with $dN$). By Steps 5 and 8, $(\xi_{\rho})^{\oplus}\in T_{P}$ satisfies the identity of Finite Fisher Information, the Score and the Fisher Information of a Probability Measure §score, so $\xi_{\rho^{\otimes N}}=(\xi_{\rho})^{\oplus}$ by uniqueness there. Finally, by Finite Fisher Information, the Score and the Fisher Information of a Probability Measure §information and Step 4, $\mathcal{I}(\rho^{\otimes N})=\lVert(\xi_{\rho})^{\oplus}\rVert_{P}^{2}=N\lVert\xi_{\rho}\rVert_{\rho}^{2}=N\,\mathcal{I}(\rho)$.