Proof of Intrinsic Test Functions at a Maximiser of the Doubled Difference Linked through One-Particle Marginals

Each result cited is universally quantified over the data in its own statement. We write $W$ for $W_{2}$, on $\mathcal{P}_{2}(\mathbb{R}^{d})$ and on $\mathcal{P}_{2}(\mathbb{R}^{dN})$, and $1/n$ for the multiplicative inverse of the positive real attached to $n\in\mathbb{N}$ (The Real Numbers: Standing Notation and Background §numbers); the real sequence $(1/n)_{n\in\mathbb{N}}$ converges to $0$ by The Archimedean Property of the Real Numbers. The number $N$ of particles is read in $\mathbb{R}$ through the same canonical map, so $1\le N$ and $0<N$, hence $0<N^{-1}$, by claims 2 and 3 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field, and $N^{-1}=N^{-1}\cdot1\le N^{-1}N=1$ by claim 5 of Elementary Arithmetic in an Ordered Field. As in The Squared Wasserstein Distance to a Fixed Measure and Functions of the Mean are Intrinsic Test Functions, a point $a\in\mathbb{R}^{r}$ also stands for the class of the constant map with value $a$ in any $L^{2}(\rho;\mathbb{R}^{r})$. Convergence in $\mathbb{R}^{d}$ is convergence in $(\mathbb{R}^{d},d_{E})$, where $d_{E}(a,a')=\lVert a-a'\rVert$ by claim 2 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$. For $\rho\in\mathcal{P}_{2}(\mathbb{R}^{d})$, $m(\rho)\in\mathbb{R}^{d}$ is its mean (The Mean of a Square-Integrable Probability Measure, Its Lift, Its Centring, and Functions of the Mean and Centred Integrals as Test Functions §mean); for $P\in\mathcal{P}_{2}(\mathbb{R}^{dN})$, $M(P)\in\mathbb{R}^{dN}$ is its mean (the same clause at the configuration level) and $\bar{m}(P)=m(P^{[1]})$, defined because $P^{[1]}\in\mathcal{P}_{2}(\mathbb{R}^{d})$ by Tensor Powers and One-Particle Marginals: Particle Laws, Product Integrals, Push-Forwards, Moments, Product Maps and Diagonal Shifts §moments. For $x\in\mathbb{R}^{dN}$ we put $\bar{x}=N^{-1}\sum_{k=1}^{N}\mathfrak{p}_{k}(x)\in\mathbb{R}^{d}$; the centred measures of The Mean of a Square-Integrable Probability Measure, Its Lift, Its Centring, and Functions of the Mean and Centred Integrals as Test Functions §centring do not occur in this proof, so the bar is free. Indices $b(k,i)=(k-1)d+i$ are those of Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points with $q=d$, and $e_{i}$ is the $i$th standard basis vector of $\mathbb{R}^{d}$ (Real Matrices, Symmetric Matrices and the Semidefinite Ordering: Standing Notation §basis).

Step 1 (Particle averages and the matrices $B^{\boxplus}$). Let $x\in\mathbb{R}^{dN}$. The $i$th coordinate of $\mathfrak{p}_{k}(x)$ is $x_{b(k,i)}$ (Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points §blocks), and sums and scalar multiples of points are formed coordinatewise, so $\bar{x}_{i}=N^{-1}\sum_{k=1}^{N}x_{b(k,i)}$ for $i\in[d]$. For $a\in\mathbb{R}^{d}$, $\mathfrak{p}_{k}(a^{\oplus})=a$ for every $k$ (Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points §configuration and Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points §diagonal), hence

$$\overline{a^{\oplus}}=N^{-1}(Na)=a .\qquad(1\mathrm{a})$$

By 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 §diagonal, $\mathfrak{p}_{k}(x-\bar{x}^{\oplus})=\mathfrak{p}_{k}(x)-\bar{x}$, so by Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §inner-product, claim 1 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, Bilinearity and Symmetry of the Dot Product on $\mathbb{R}^n$ and claims 2 and 3 of Properties of Finite Sums,

$$\lVert x-\bar{x}^{\oplus}\rVert^{2}=\sum_{k=1}^{N}\bigl(\lVert\mathfrak{p}_{k}(x)\rVert^{2}-2\,\mathfrak{p}_{k}(x)\cdot\bar{x}+\lVert\bar{x}\rVert^{2}\bigr)=\lVert x\rVert^{2}-2\,(N\bar{x})\cdot\bar{x}+N\lVert\bar{x}\rVert^{2}=\lVert x\rVert^{2}-N\lVert\bar{x}\rVert^{2}.\qquad(1\mathrm{b})$$

As the left side is nonnegative, $N\lVert\bar{x}\rVert^{2}\le\lVert x\rVert^{2}$ (claim 3 of Elementary Arithmetic in an Ordered Field), and multiplying by $N^{-1}\ge0$ (claim 5 there), $\lVert\bar{x}\rVert^{2}\le N^{-1}\lVert x\rVert^{2}$.

For $B\in\mathcal{S}(d)$ let $B^{\boxplus}$ be the real $dN\times dN$ matrix whose entry in row $b(k,i)$ and column $b(l,j)$ is $N^{-2}B_{ij}$, for $k,l\in[N]$ and $i,j\in[d]$; it is well defined because every index in $[dN]$ is $b(k,i)$ for exactly one pair $(k,i)$ (Block Indices: Enumerating an Initial Segment of Length qN by Blocks and Positions §bijection), and it is symmetric, its entry in row $b(l,j)$ and column $b(k,i)$ being $N^{-2}B_{ji}=N^{-2}B_{ij}$ (Real Matrices, Symmetric Matrices and the Semidefinite Ordering: Standing Notation §symmetric). By claim 4 of Linearity of the Matrix-Vector Product and the Quadratic Form as a Double Sum in dimension $dN$, Block Indices: Enumerating an Initial Segment of Length qN by Blocks and Positions §sums applied to both sums, claims 2 and 3 of Properties of Finite Sums, and claim 4 of Linearity of the Matrix-Vector Product and the Quadratic Form as a Double Sum in dimension $d$,

$$x\cdot(B^{\boxplus}x)=\sum_{k,i}\sum_{l,j}N^{-2}B_{ij}\,x_{b(k,i)}\,x_{b(l,j)}=\sum_{i=1}^{d}\sum_{j=1}^{d}B_{ij}\Bigl(N^{-1}\sum_{k=1}^{N}x_{b(k,i)}\Bigr)\Bigl(N^{-1}\sum_{l=1}^{N}x_{b(l,j)}\Bigr)=\bar{x}\cdot(B\bar{x}).\qquad(1\mathrm{c})$$

Step 2 (Functions of the particle average). Let $f:\mathbb{R}^{d}\to\mathbb{R}$ be of class $C^{2}$ on $\mathbb{R}^{d}$ and $f^{\flat}:\mathbb{R}^{dN}\to\mathbb{R}$, $f^{\flat}(x)=f(\bar{x})$. We show that $f^{\flat}$ is of class $C^{2}$ on $\mathbb{R}^{dN}$ and

$$Df^{\flat}(x)=\bigl(N^{-1}Df(\bar{x})\bigr)^{\oplus},\qquad D^{2}f^{\flat}(x)=\bigl(D^{2}f(\bar{x})\bigr)^{\boxplus}\qquad(x\in\mathbb{R}^{dN}).\qquad(2\mathrm{a})$$

Let $L:\mathbb{R}^{dN}\to\mathbb{R}^{d}$, $L(x)=\bar{x}$, and for $i\in[d]$ let $w^{(i)}=(N^{-1}e_{i})^{\oplus}\in\mathbb{R}^{dN}$. By Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §diagonal and claims 4 and 5 of Bilinearity and Symmetry of the Dot Product on $\mathbb{R}^n$, $w^{(i)}\cdot x=(N^{-1}e_{i})\cdot\sum_{k}\mathfrak{p}_{k}(x)=e_{i}\cdot\bar{x}=\bar{x}_{i}$, so the $i$th coordinate function $L_{i}$ of $L$ is $x\mapsto w^{(i)}\cdot x$. By Quadratic and Affine Functions of Class $C^2$, Translation, and Quadratic Perturbation of Semiconvexity §quadratic (in dimension $dN$, with the zero matrix, the vector $w^{(i)}$ and the constant $0$), $L_{i}$ is of class $C^{2}$ on $\mathbb{R}^{dN}$ with gradient $w^{(i)}$ at every point, so by Gradient of a Real-Valued Function on a Euclidean Open Set its partial derivative with respect to the variable $b(l,j)$ is the coordinate of $w^{(i)}$ with index $b(l,j)$, that is, the $j$th coordinate of $\mathfrak{p}_{l}(w^{(i)})=N^{-1}e_{i}$: it is $N^{-1}$ if $j=i$ and $0$ otherwise. The conditions of clauses 1 and 2 of C^k Maps on a Euclidean Open Set are imposed coordinate function by coordinate function, so $L$ is of class $C^{2}$ on $\mathbb{R}^{dN}$, and $f^{\flat}=f\circ L$ is of class $C^{2}$ on $\mathbb{R}^{dN}$ by claim 2 of A Composition of $C^k$ Maps Between Euclidean Open Sets is of Class $C^k$ ($\mathbb{R}^{dN}$ and $\mathbb{R}^{d}$ are open by claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous). By claim 1 of A Composition of $C^k$ Maps Between Euclidean Open Sets is of Class $C^k$ and claim 7 of Properties of Finite Sums,

$$\partial_{b(l,j)}f^{\flat}(x)=\sum_{i=1}^{d}\partial_{i}f(\bar{x})\,\partial_{b(l,j)}L_{i}(x)=N^{-1}\partial_{j}f(\bar{x})\qquad(l\in[N],\ j\in[d]).$$

So the coordinate of $Df^{\flat}(x)$ with index $b(l,j)$ is the $j$th coordinate of $\mathfrak{p}_{l}\bigl((N^{-1}Df(\bar{x}))^{\oplus}\bigr)=N^{-1}Df(\bar{x})$; every index being of this form, the first identity of (2a) holds. Since $f$ is of class $C^{2}$, $\partial_{j}f$ is of class $C^{1}$ on $\mathbb{R}^{d}$ (clause 2 of C^k Maps on a Euclidean Open Set), and the same chain rule gives $\partial_{b(k,i)}(\partial_{j}f\circ L)(x)=N^{-1}\partial_{i}\partial_{j}f(\bar{x})$. The difference quotients of $\partial_{b(l,j)}f^{\flat}=N^{-1}(\partial_{j}f\circ L)$ in Partial Derivative on a Euclidean Open Set are $N^{-1}$ times those of $\partial_{j}f\circ L$, and $|N^{-1}s|=N^{-1}|s|\le|s|$ for real $s$ (claim 4 of Properties of the Absolute Value in an Ordered Field, claim 5 of Elementary Arithmetic in an Ordered Field); hence $\partial_{b(k,i)}\partial_{b(l,j)}f^{\flat}(x)=N^{-2}\partial_{i}\partial_{j}f(\bar{x})$, which by Hessian Matrix of a C^2 Function is the entry of $D^{2}f^{\flat}(x)$ in row $b(k,i)$ and column $b(l,j)$, and equals the corresponding entry of $(D^{2}f(\bar{x}))^{\boxplus}$. This is the second identity of (2a).

Step 3 (The marginal mean, the marginal map and product fields). (3a) $\bar{m}(P)=\overline{M(P)}$ for $P\in\mathcal{P}_{2}(\mathbb{R}^{dN})$. Let $i\in[d]$. The coordinate map $y\mapsto y_{i}$ is Borel and integrable with respect to $P^{[1]}$, and each $x\mapsto x_{b(k,i)}$, which is $x\mapsto\mathfrak{p}_{k}(x)_{i}$, is integrable with respect to $P$ (The Mean of a Square-Integrable Probability Measure, Its Lift, Its Centring, and Functions of the Mean and Centred Integrals as Test Functions §mean at both levels). By the integration identity of The One-Particle Marginal of a Probability Measure on the Configuration Space §marginal, in the integrable form of Existence and Uniqueness of Tensor Powers, and the Average of the Block Marginals §average,

$$m(P^{[1]})_{i}=\frac{1}{N}\sum_{k=1}^{N}\int x_{b(k,i)}\,P(dx)=\frac{1}{N}\sum_{k=1}^{N}M(P)_{b(k,i)}=\overline{M(P)}_{i}$$

by Step 1. (3b) For $P,P'\in\mathcal{P}_{2}(\mathbb{R}^{dN})$, $W(P^{[1]},P'^{[1]})\le W(P,P')$ and $\lVert\bar{m}(P)-\bar{m}(P')\rVert\le W(P,P')$. 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} §marginal, $N\,W(P^{[1]},P'^{[1]})^{2}\le W(P,P')^{2}$, and $W(P^{[1]},P'^{[1]})^{2}\le N\,W(P^{[1]},P'^{[1]})^{2}$ as $1\le N$ (claim 5 of Elementary Arithmetic in an Ordered Field); both distances being nonnegative (The Quadratic Wasserstein Distance is a Metric on the Wasserstein Space), claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives the first inequality, and The Mean of a Square-Integrable Probability Measure, Its Lift, Its Centring, and Functions of the Mean and Centred Integrals as Test Functions §mean gives the second. In particular, if $P_{n}\to P$ in $(\mathcal{P}_{2}(\mathbb{R}^{dN}),W)$, then $P_{n}^{[1]}\to P^{[1]}$ and $\bar{m}(P_{n})\to\bar{m}(P)$. (3c) For $a\in\mathbb{R}^{d}$ and $P\in\mathcal{P}_{2}(\mathbb{R}^{dN})$, $\bar{m}((\tau_{a^{\oplus}})_{\#}P)=\bar{m}(P)+a$, by Tensor Powers and One-Particle Marginals: Particle Laws, Product Integrals, Push-Forwards, Moments, Product Maps and Diagonal Shifts §diagonal and The Mean of a Square-Integrable Probability Measure, Its Lift, Its Centring, and Functions of the Mean and Centred Integrals as Test Functions §mean. (3d) Product fields. For $g,h\in L^{2}(P^{[1]};\mathbb{R}^{d})$, $t\in\mathbb{R}$ and $a\in\mathbb{R}^{d}$: $(g+th)^{\oplus}=g^{\oplus}+t\,h^{\oplus}$ in $L^{2}(P;\mathbb{R}^{dN})$, and the product field of the constant $a$ is the constant $a^{\oplus}$. Indeed, for Borel representatives $\tilde{g},\tilde{h}$, the map $\tilde{g}+t\tilde{h}$ represents $g+th$, and its product map at $x$ is $[\tilde{g}(\mathfrak{p}_{1}(x))+t\tilde{h}(\mathfrak{p}_{1}(x)),\dots]=\tilde{g}^{\oplus}(x)+t\,\tilde{h}^{\oplus}(x)$ by Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §linear; the product map of the constant map $a$ is the constant map $a^{\oplus}$ (Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points §product-map); and Product Fields and the Projection onto One-Particle Tangent Fields §product-field takes classes.

Step 4 (Averaged couplings). Let $P,P'\in\mathcal{P}_{2}(\mathbb{R}^{dN})$ and $\gamma\in\Pi(P,P')$. We construct $\bar{\gamma}\in\Pi(P^{[1]},P'^{[1]})$ with $I(\bar{\gamma})=N^{-1}I(\gamma)$ and such that for all $g\in L^{2}(P^{[1]};\mathbb{R}^{d})$ and $h\in L^{2}(P'^{[1]};\mathbb{R}^{d})$

$$\int_{\mathbb{R}^{dN+dN}}\bigl\lVert g^{\oplus}(x)-h^{\oplus}(y)\bigr\rVert^{2}\,\gamma(dz)=N\int_{\mathbb{R}^{d+d}}\bigl\lVert g(x')-h(y')\bigr\rVert^{2}\,\bar{\gamma}(dz').\qquad(4\mathrm{a})$$

The construction is that of the proof of Tensor Powers Scale the Wasserstein Distance by the Square Root of N, and the One-Particle Marginal is Lipschitz with Constant N^{-1/2} §marginal, which we repeat. Write $\mathrm{pr}_{1},\mathrm{pr}_{2}$ for the coordinate projections of $\mathbb{R}^{dN+dN}$ and of $\mathbb{R}^{d+d}$ (Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections), $\hat{\mathfrak{p}}_{k}$ for the block maps of $\mathbb{R}^{(d+d)N}$, and apply Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points, The One-Particle Marginal of a Probability Measure on the Configuration Space and Existence and Uniqueness of Tensor Powers, and the Average of the Block Marginals with $d+d$ in place of $q$ where these maps occur. For $k\in[N]$ let $S_{k}=(\mathfrak{p}_{k}\circ\mathrm{pr}_{1},\mathfrak{p}_{k}\circ\mathrm{pr}_{2}):\mathbb{R}^{dN+dN}\to\mathbb{R}^{d+d}$, 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 §pairing, with $\mathrm{pr}_{i}\circ S_{k}=\mathfrak{p}_{k}\circ\mathrm{pr}_{i}$ (Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections). Let $J(z)=[S_{1}(z),\dots,S_{N}(z)]\in\mathbb{R}^{(d+d)N}$ (Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points §configuration), so that $\hat{\mathfrak{p}}_{k}\circ J=S_{k}$; each component of $J$ is a component of some $S_{k}$ (Block Indices: Enumerating an Initial Segment of Length qN by Blocks and Positions §bijection), so $J$ 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. Put $\bar{\gamma}=(J_{\#}\gamma)^{[1]}$. By Existence and Uniqueness of Tensor Powers, and the Average of the Block Marginals §average for $J_{\#}\gamma$ and change of variables for $J$ (Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward), for every Borel $f:\mathbb{R}^{d+d}\to[0,\infty]$,

$$\int f\,d\bar{\gamma}=\frac{1}{N}\sum_{k=1}^{N}\int f\circ S_{k}\,d\gamma .\qquad(4\mathrm{b})$$

For $B\in\mathcal{B}(\mathbb{R}^{d})$, (4b) with the indicator of $\mathrm{pr}_{1}^{-1}(B)$, $S_{k}^{-1}(\mathrm{pr}_{1}^{-1}(B))=\mathrm{pr}_{1}^{-1}(\mathfrak{p}_{k}^{-1}(B))$ and $\gamma\in\Pi(P,P')$ give $\bar{\gamma}(\mathrm{pr}_{1}^{-1}(B))=\frac1N\sum_{k}P(\mathfrak{p}_{k}^{-1}(B))=P^{[1]}(B)$, and likewise $\bar{\gamma}(\mathrm{pr}_{2}^{-1}(B))=P'^{[1]}(B)$; so $\bar{\gamma}\in\Pi(P^{[1]},P'^{[1]})$ (Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost §coupling). Now let $\tilde{g},\tilde{h}$ be Borel representatives of $g,h$, put $f(z')=\lVert\tilde{g}(\mathrm{pr}_{1}(z'))-\tilde{h}(\mathrm{pr}_{2}(z'))\rVert^{2}$ and $F(z)=\lVert\tilde{g}^{\oplus}(\mathrm{pr}_{1}(z))-\tilde{h}^{\oplus}(\mathrm{pr}_{2}(z))\rVert^{2}$, nonnegative and Borel by The Discrepancy of Two Square-Integrable Vector Fields Along a Coupling of Their Base Measures §well-defined (at both levels), with $\int F\,d\gamma<\infty$ by the same clause, $g^{\oplus}$ and $h^{\oplus}$ being square-integrable against $P$ and $P'$ (Product Fields and the Projection onto One-Particle Tangent Fields §product-field). By Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §product-map and Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §linear, $f(S_{k}(z))=\lVert\mathfrak{p}_{k}\bigl(\tilde{g}^{\oplus}(\mathrm{pr}_{1}(z))-\tilde{h}^{\oplus}(\mathrm{pr}_{2}(z))\bigr)\rVert^{2}$, so $\sum_{k=1}^{N}f\circ S_{k}=F$ by Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §inner-product; each $f\circ S_{k}\le F$ (claim 6 of Properties of Finite Sums) is integrable against $\gamma$ (claim 1 of Linearity and Monotonicity of the Lebesgue Integral), and Linearity of the Lebesgue Integral over a Finite Sum of Integrable Functions §linear with (4b) gives $\int F\,d\gamma=\sum_{k}\int f\circ S_{k}\,d\gamma=N\int f\,d\bar{\gamma}$, which is (4a) (both sides being the discrepancies of The Discrepancy of Two Square-Integrable Vector Fields Along a Coupling of Their Base Measures §well-defined, independent of representatives). The same computation with $c_{m}(z)=\lVert\mathrm{pr}_{1}(z)-\mathrm{pr}_{2}(z)\rVert^{2}$ on $\mathbb{R}^{m+m}$ (Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §functions) in place of $f$ and $F$, where $\sum_{k}c_{d}\circ S_{k}=c_{dN}$ by 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 §inner-product and $I(\gamma)=\int c_{dN}\,d\gamma<\infty$ (Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost §cost, The Intrinsic Calculus on the Wasserstein Space: Standing Notation §couplings), gives $I(\gamma)=N\,I(\bar{\gamma})$.

Step 5 (The midpoint split). Put $\hat{\nu}=\hat{P}^{[1]}$, which lies in $\mathcal{D}$ by hypothesis. By Existence of an Optimal Coupling of Two Probability Measures with Finite Second Moment fix an optimal coupling $\hat{\pi}\in\Pi(\hat{\mu},\hat{\nu})$, and let $\hat{m}$, $c$ and $A$ be as in The Displacement Midpoint and the Midpoint Split of the Squared Wasserstein Distance for $\mu=\hat{\mu}$, $\nu=\hat{\nu}$ and $\hat{\pi}$. Put $\hat{\zeta}=m(\hat{\mu})$, $\hat{\omega}=m(\hat{\nu})=\bar{m}(\hat{P})$ and $M_{0}=\Psi(\hat{\mu},\hat{P})$, so that $c=\tfrac12(\hat{\zeta}+\hat{\omega})$, $\hat{m}\in\mathcal{P}_{2}(\mathbb{R}^{d})$ by The Displacement Midpoint and the Midpoint Split of the Squared Wasserstein Distance §midpoint, and $0\le A(\rho)$ for every $\rho$ by The Displacement Midpoint and the Midpoint Split of the Squared Wasserstein Distance §nonnegative. By Basic Properties of a Wasserstein-Coercive Penalty Pair §bounded-below, at the particle level and at the configuration level, fix $e_{0},e'_{0}\in\mathbb{R}$ with $e_{0}\le\mathcal{E}(\rho)$ for $\rho\in\mathcal{D}$ and $e'_{0}\le\mathcal{E}_{N}(P)$ for $P\in\mathcal{D}_{N}$. For $\rho\in\mathcal{D}$ and $P\in\mathcal{D}_{N}$ put

$$\Theta(\rho)=N\,u^{-}_{\delta}(\rho)-N\alpha A(\rho),\qquad\Xi(P)=U^{+}_{\delta}(P)+N\alpha A(P^{[1]}).$$

By The Delta-Envelopes of Bounded Functions and Their Monotonicity in the Weight, for a Wasserstein-Coercive Penalty Pair §bounded for $u$ (particle level) and for $U$ (configuration level), $u^{-}_{\delta}(\rho)\le b-\delta\mathcal{E}(\rho)$ and $b'+\delta\mathcal{E}_{N}(P)\le U^{+}_{\delta}(P)$; as $0\le N\alpha A(\cdot)$, $\delta e_{0}\le\delta\mathcal{E}(\rho)$ and $\delta e'_{0}\le\delta\mathcal{E}_{N}(P)$ (claims 2 and 5 of Elementary Arithmetic in an Ordered Field, $N\alpha$ being positive by claim 5 of Elementary Order Arithmetic in an Ordered Field),

$$\Theta(\rho)\le N\bigl(b-\delta\,\mathcal{E}(\rho)\bigr)\le N(b-\delta e_{0}),\qquad b'+\delta e'_{0}\le b'+\delta\,\mathcal{E}_{N}(P)\le\Xi(P).\qquad(5\mathrm{a})$$

For $\mu\in\mathcal{D}$ and $P\in\mathcal{D}_{N}$, since $m(P^{[1]})=\bar{m}(P)$,

$$\Psi(\mu,P)-\Bigl(\Theta(\mu)-\Xi(P)-\tfrac{N\alpha}{2}\lVert m(\mu)-\bar{m}(P)\rVert^{2}\Bigr)=\tfrac{N\alpha}{2}\Bigl(2A(\mu)+2A(P^{[1]})+\lVert m(\mu)-m(P^{[1]})\rVert^{2}-W(\mu,P^{[1]})^{2}\Bigr),$$

which is nonnegative by The Displacement Midpoint and the Midpoint Split of the Squared Wasserstein Distance §split and vanishes exactly when equality holds there for $(\mu,P^{[1]})$, $\tfrac{N\alpha}{2}$ being positive (claims 5 and 8 of Elementary Order Arithmetic in an Ordered Field, claim 3 of Zero Products and Elementary Identities in a Field). With the maximality of $(\hat{\mu},\hat{P})$,

$$\Theta(\mu)-\Xi(P)-\tfrac{N\alpha}{2}\lVert m(\mu)-\bar{m}(P)\rVert^{2}\le\Psi(\mu,P)\le M_{0}\qquad(\mu\in\mathcal{D},\ P\in\mathcal{D}_{N}),\qquad(5\mathrm{b})$$

with equality in the first inequality exactly when equality holds in The Displacement Midpoint and the Midpoint Split of the Squared Wasserstein Distance §split for $(\mu,P^{[1]})$. By The Displacement Midpoint and the Midpoint Split of the Squared Wasserstein Distance §endpoints this is so for $(\hat{\mu},\hat{P})$:

$$\Theta(\hat{\mu})-\Xi(\hat{P})-\tfrac{N\alpha}{2}\lVert\hat{\zeta}-\hat{\omega}\rVert^{2}=M_{0}.\qquad(5\mathrm{c})$$

Step 6 ($A$ is an intrinsic test function). Let $q_{c}:\mathbb{R}^{d}\to\mathbb{R}$, $q_{c}(a)=\lVert a-c\rVert^{2}=d_{E}(a,c)^{2}$. By A Scaled Squared Distance to a Point is of Class $C^2$, with Gradient and Hessian with the point $c$ and the scalar $1$, $q_{c}$ is of class $C^{2}$ on $\mathbb{R}^{d}$ with $Dq_{c}(a)=2(a-c)$ and $D^{2}q_{c}(a)=2I_{d}$. Since $A(\rho)=W(\rho,\hat{m})^{2}-q_{c}(m(\rho))$ and $\mathcal{D}$ has the map property, The Squared Wasserstein Distance to a Fixed Measure and Functions of the Mean are Intrinsic Test Functions §distance (with $\nu_{0}=\hat{m}$), The Squared Wasserstein Distance to a Fixed Measure and Functions of the Mean are Intrinsic Test Functions §mean (with $\phi=q_{c}$) and Restrictions, Sums, Real Multiples and Differences of Intrinsic Test Functions on the Wasserstein Space §difference show that $A$ is an intrinsic test function on $\mathcal{D}$, with

$$\nabla A(\rho)=2(\mathrm{id}-G_{\rho})-2\bigl(m(\rho)-c\bigr)\quad(\rho\in\mathcal{D}),\qquad H_{A}(\rho)=2I_{d}-2I_{d}=0_{d}\quad\bigl(\rho\in\mathcal{P}_{2}(\mathbb{R}^{d})\bigr),\qquad(6\mathrm{a})$$

where $G_{\rho}$ is any optimal map from $\rho$ to $\hat{m}$. By property (a) of Intrinsic Test Functions on the Wasserstein Space and Their Translation Hessians §test, $A$ is continuous on $\mathcal{P}_{2}(\mathbb{R}^{d})$; with (3b), $P\mapsto A(P^{[1]})$ is continuous on $\mathcal{P}_{2}(\mathbb{R}^{dN})$ (Continuous Map Between Metric Spaces).

Step 7 (Compactness). We show:

(K-u) Let $(\rho_{n})_{n\in\mathbb{N}}$ be a sequence in $\mathcal{D}$, $\zeta\in\mathbb{R}^{d}$ and $\ell\in\mathbb{R}$ with $(m(\rho_{n}))_{n}$ converging to $\zeta$ and $\ell\le\Theta(\rho_{n})$ for every $n$. Then there are a strictly increasing sequence $(n_{j})_{j\in\mathbb{N}}$ in $\mathbb{N}$ and $\rho\in\mathcal{D}$ with $m(\rho)=\zeta$ such that $\rho_{n_{j}}\to\rho$ in $(\mathcal{P}_{2}(\mathbb{R}^{d}),W)$ and, for every positive $\varepsilon\in\mathbb{R}$, $\Theta(\rho_{n_{j}})<\Theta(\rho)+\varepsilon$ for all sufficiently large $j$.

(K-v) Let $(P_{n})_{n\in\mathbb{N}}$ be a sequence in $\mathcal{D}_{N}$, $\omega\in\mathbb{R}^{d}$ and $\ell'\in\mathbb{R}$ with $(\bar{m}(P_{n}))_{n}$ converging to $\omega$ and $\Xi(P_{n})\le\ell'$ for every $n$. Then there are a strictly increasing $(n_{j})_{j}$ and $P\in\mathcal{D}_{N}$ with $\bar{m}(P)=\omega$ such that $P_{n_{j}}\to P$ in $(\mathcal{P}_{2}(\mathbb{R}^{dN}),W)$ and, for every positive $\varepsilon$, $\Xi(P)-\varepsilon<\Xi(P_{n_{j}})$ for all sufficiently large $j$.

For (K-u): by (5a), $N^{-1}\ell\le b-\delta\mathcal{E}(\rho_{n})$, so $\mathcal{E}(\rho_{n})\le c_{1}$ with $c_{1}=\delta^{-1}(b-N^{-1}\ell)$ (claims 3 and 5 of Elementary Arithmetic in an Ordered Field, $\delta^{-1}$ being positive by claim 7 of Elementary Order Arithmetic in an Ordered Field). The set $\{\rho'\in\mathcal{D}:\mathcal{E}(\rho')\le c_{1}\}$ is sequentially compact (Wasserstein-Coercive Penalty Pairs §coercive), so there are a strictly increasing $(n_{j})_{j}$ and a point $\rho$ of that set, hence of $\mathcal{D}$, with $\rho_{n_{j}}\to\rho$ (Sequentially Compact Subset of a Metric Space). 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, $\lVert m(\rho_{n_{j}})-m(\rho)\rVert\le W(\rho_{n_{j}},\rho)$, so $m(\rho_{n_{j}})\to m(\rho)$; also $m(\rho_{n_{j}})\to\zeta$ by A Subsequence of a Convergent Sequence Has the Same Limit, so $m(\rho)=\zeta$ by Uniqueness of Limits in a Metric Space. Let $\varepsilon>0$. Since $u$ has penalty-subordinate growth from above (The Delta-Envelopes of Bounded Functions and Their Monotonicity in the Weight, for a Wasserstein-Coercive Penalty Pair §growth), $u^{-}_{\delta}$ is upper semicontinuous on $\mathcal{D}$ relative to $\mathcal{D}$ by Basic Properties of the Delta-Envelopes on the Wasserstein Space §semicontinuity; with the continuity of $A$ there is a positive $r$ such that every $\rho'\in\mathcal{D}$ with $W(\rho',\rho)<r$ satisfies $u^{-}_{\delta}(\rho')<u^{-}_{\delta}(\rho)+\tfrac{\varepsilon}{2}N^{-1}$ and $|A(\rho')-A(\rho)|<\tfrac{\varepsilon}{2}(N\alpha)^{-1}$ (the least of two radii, claim 9 of Elementary Order Arithmetic in an Ordered Field), hence $\Theta(\rho')<\Theta(\rho)+\varepsilon$ by claims 3 and 10 of Elementary Order Arithmetic in an Ordered Field and claim 3 of Properties of the Absolute Value in an Ordered Field. As $W(\rho_{n_{j}},\rho)<r$ for all large $j$, (K-u) follows. (K-v) is proved in the same way at the configuration level: (5a) gives $\mathcal{E}_{N}(P_{n})\le\delta^{-1}(\ell'-b')$; sublevel sets of $\mathcal{E}_{N}$ are sequentially compact in $(\mathcal{P}_{2}(\mathbb{R}^{dN}),W)$ (Wasserstein-Coercive Penalty Pairs §coercive at the configuration level); $\bar{m}(P_{n_{j}})\to\bar{m}(P)$ by (3b), so $\bar{m}(P)=\omega$; $U^{+}_{\delta}$ is lower semicontinuous on $\mathcal{D}_{N}$ relative to $\mathcal{D}_{N}$ by the second part of Basic Properties of the Delta-Envelopes on the Wasserstein Space §semicontinuity at the configuration level, $U$ having penalty-subordinate growth from below by The Delta-Envelopes of Bounded Functions and Their Monotonicity in the Weight, for a Wasserstein-Coercive Penalty Pair §growth; and $P'\mapsto N\alpha A(P'^{[1]})$ is continuous by Step 6.

Step 8 (The fibre functions). For $\zeta,\omega\in\mathbb{R}^{d}$ let $\mathcal{D}_{\zeta}=\{\rho\in\mathcal{D}:m(\rho)=\zeta\}$ and $\mathcal{D}_{N,\omega}=\{P\in\mathcal{D}_{N}:\bar{m}(P)=\omega\}$. Both are nonempty: $(\tau_{\zeta-\hat{\zeta}})_{\#}\hat{\mu}\in\mathcal{D}$ by Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §translation, with mean $\hat{\zeta}+(\zeta-\hat{\zeta})=\zeta$ 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 with $a=\omega-\hat{\omega}$, $(\tau_{a^{\oplus}})_{\#}\hat{P}\in\mathcal{D}_{N}$ by Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §translation at the configuration level ($a^{\oplus}\in\mathbb{R}^{dN}$), with $\bar{m}\bigl((\tau_{a^{\oplus}})_{\#}\hat{P}\bigr)=\hat{\omega}+a=\omega$ by (3c). By (5a) the set $\{\Theta(\rho):\rho\in\mathcal{D}_{\zeta}\}$ is bounded above and $\{\Xi(P):P\in\mathcal{D}_{N,\omega}\}$ bounded below; let $U(\zeta)\in\mathbb{R}$ be the supremum of the first and $V(\omega)\in\mathbb{R}$ the infimum of the second (Approximation Property of the Supremum and the Infimum in $\mathbb{R}$).

(8a) The fibre extrema are attained. Let $\zeta\in\mathbb{R}^{d}$. By claim 3 of Approximation Property of the Supremum and the Infimum in $\mathbb{R}$ choose $\rho_{n}\in\mathcal{D}_{\zeta}$ with $U(\zeta)-1/n<\Theta(\rho_{n})$ for each $n$; then $U(\zeta)-1\le\Theta(\rho_{n})$, as $1/n\le1$. (K-u), with the constant sequence of means $\zeta$ and $\ell=U(\zeta)-1$, gives $(n_{j})_{j}$ and $\rho\in\mathcal{D}_{\zeta}$. For $\varepsilon>0$ take $j$ so large that $\Theta(\rho_{n_{j}})<\Theta(\rho)+\varepsilon$ and $1/n_{j}<\varepsilon$ (the sequence $(1/n_{j})_{j}$ converges to $0$ by A Subsequence of a Convergent Sequence Has the Same Limit); then $U(\zeta)<\Theta(\rho)+2\varepsilon$. By Comparison of Real Numbers with Arbitrary Positive Slack §slack-above, $U(\zeta)\le\Theta(\rho)\le U(\zeta)$, so $\Theta(\rho)=U(\zeta)$. In the same way, with claim 4 of Approximation Property of the Supremum and the Infimum in $\mathbb{R}$, (K-v) and Comparison of Real Numbers with Arbitrary Positive Slack §slack-below, for every $\omega$ there is $P\in\mathcal{D}_{N,\omega}$ with $\Xi(P)=V(\omega)$.

(8b) $U$ is upper and $V$ lower semicontinuous on $\mathbb{R}^{d}$, in the sense of Upper Semicontinuous Function on a Subset of a Metric Space and Lower Semicontinuous Function on a Subset of a Metric Space in $(\mathbb{R}^{d},d_{E})$, which is the reading of Second-Order Equations on Euclidean Open Sets §extrema. Suppose $U$ were not upper semicontinuous at $\zeta$. Then there is $\varepsilon>0$ such that for every $n$ some $\zeta_{n}$ has $d_{E}(\zeta_{n},\zeta)<1/n$ and $U(\zeta)+\varepsilon\le U(\zeta_{n})$; so $\zeta_{n}\to\zeta$. By (8a) choose $\rho_{n}\in\mathcal{D}_{\zeta_{n}}$ with $\Theta(\rho_{n})=U(\zeta_{n})$. (K-u) with $\ell=U(\zeta)+\varepsilon$ gives $\rho\in\mathcal{D}_{\zeta}$ and, for large $j$, $U(\zeta)+\varepsilon\le\Theta(\rho_{n_{j}})<\Theta(\rho)+\tfrac{\varepsilon}{2}\le U(\zeta)+\tfrac{\varepsilon}{2}$, which is impossible as $\tfrac{\varepsilon}{2}<\varepsilon$ (claim 8 of Elementary Order Arithmetic in an Ordered Field). The lower semicontinuity of $V$ follows in the same way from (8a) and (K-v).

Step 9 (Ishii's lemma on the means; the matrices; claim 2). Let $\zeta,\omega\in\mathbb{R}^{d}$ and, by (8a), $\rho\in\mathcal{D}_{\zeta}$ and $P\in\mathcal{D}_{N,\omega}$ with $\Theta(\rho)=U(\zeta)$ and $\Xi(P)=V(\omega)$. By (5b),

$$U(\zeta)-V(\omega)-\tfrac{N\alpha}{2}\lVert\zeta-\omega\rVert^{2}\le M_{0}.\qquad(9\mathrm{a})$$

Since $\hat{\mu}\in\mathcal{D}_{\hat{\zeta}}$ and $\hat{P}\in\mathcal{D}_{N,\hat{\omega}}$, $\Theta(\hat{\mu})\le U(\hat{\zeta})$ and $V(\hat{\omega})\le\Xi(\hat{P})$, so (5c) and (9a) give

$$U(\hat{\zeta})-V(\hat{\omega})-\tfrac{N\alpha}{2}\lVert\hat{\zeta}-\hat{\omega}\rVert^{2}=M_{0}.\qquad(9\mathrm{b})$$

Apply Ishii's Lemma: Test Data and Matrix Bounds at a Maximum of a Quadratically Penalised Difference with $n=d$, the open set $\Omega=\mathbb{R}^{d}$ (claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous), the functions $U$ and $V$ (Step 8), the positive real $N\alpha$ in place of its $\alpha$, $\hat{x}=\hat{\zeta}$, $\hat{y}=\hat{\omega}$ and radius $1$: its hypothesis holds by (9a) and (9b) at all points. Let $X_{I},Y_{I}\in\mathcal{S}(d)$ be the matrices it provides and $p=N\alpha(\hat{\zeta}-\hat{\omega})$. By Ishii's Lemma: Test Data and Matrix Bounds at a Maximum of a Quadratically Penalised Difference §test-data, $(\hat{\zeta},U(\hat{\zeta}),p,X_{I})$ is approximable by test data from above for $U$ and $(\hat{\omega},V(\hat{\omega}),p,Y_{I})$ from below for $V$, with open set $\mathbb{R}^{d}$; and by Ishii's Lemma: Test Data and Matrix Bounds at a Maximum of a Quadratically Penalised Difference §ordering, Ishii's Lemma: Test Data and Matrix Bounds at a Maximum of a Quadratically Penalised Difference §norm-bound and Ishii's Lemma: Test Data and Matrix Bounds at a Maximum of a Quadratically Penalised Difference §quadratic-bound, $X_{I}\preceq Y_{I}$, $\lVert X_{I}\rVert\le6N\alpha$, $\lVert Y_{I}\rVert\le6N\alpha$ and $-3N\alpha(\lVert z\rVert^{2}+\lVert w\rVert^{2})\le z\cdot(X_{I}z)-w\cdot(Y_{I}w)\le3N\alpha\lVert z-w\rVert^{2}$ for $z,w\in\mathbb{R}^{d}$.

Put $\mathbb{X}=N^{-1}X_{I}$ and $\mathbb{Y}=N^{-1}Y_{I}$, members of $\mathcal{S}(d)$ (Real Matrices, Symmetric Matrices and the Semidefinite Ordering: Standing Notation §symmetric). The pair $(\mathbb{X},\mathbb{Y})$ is admitted at $\alpha$ (The Second-Order Structure Condition at Optimally Coupled Pairs on the Lift of the Wasserstein Space §admitted): $\mathbb{X}\preceq\mathbb{Y}$ by claim 4 of The Positive Semidefinite Ordering is a Partial Order Compatible with the Linear Structure; $\lVert\mathbb{X}\rVert=N^{-1}\lVert X_{I}\rVert\le N^{-1}(6N\alpha)=6\alpha$ and likewise for $\mathbb{Y}$, by claim 5 of Properties of the Norm of a Symmetric Real Matrix and claim 5 of Elementary Arithmetic in an Ordered Field; and $z\cdot(\mathbb{X}z)-w\cdot(\mathbb{Y}w)=N^{-1}\bigl(z\cdot(X_{I}z)-w\cdot(Y_{I}w)\bigr)$ by claim 1 of Linearity of the Matrix-Vector Product and the Quadratic Form as a Double Sum and claim 5 of Bilinearity and Symmetry of the Dot Product on $\mathbb{R}^n$, so multiplying the quadratic bounds by $N^{-1}\ge0$ gives $-3\alpha(\lVert z\rVert^{2}+\lVert w\rVert^{2})\le z\cdot(\mathbb{X}z)-w\cdot(\mathbb{Y}w)\le3\alpha\lVert z-w\rVert^{2}$.

Let $\mathbb{Y}_{N}=(-\alpha)(2I_{dN})+(N\alpha)(2I_{d})^{\boxplus}+Y_{I}^{\boxplus}$, a member of $\mathcal{S}(dN)$ by Step 1 and Real Matrices, Symmetric Matrices and the Semidefinite Ordering: Standing Notation §symmetric. For $x\in\mathbb{R}^{dN}$, claim 1 of Linearity of the Matrix-Vector Product and the Quadratic Form as a Double Sum, Bilinearity and Symmetry of the Dot Product on $\mathbb{R}^n$, Vector, Entry and Comparison Bounds for the Norm of a Symmetric Real Matrix §identity, (1c) and (1b) give

$$x\cdot(\mathbb{Y}_{N}x)=-2\alpha\lVert x\rVert^{2}+2N\alpha\lVert\bar{x}\rVert^{2}+\bar{x}\cdot(Y_{I}\bar{x})=-2\alpha\lVert x-\bar{x}^{\oplus}\rVert^{2}+\bar{x}\cdot(Y_{I}\bar{x}).\qquad(9\mathrm{c})$$

For $a\in\mathbb{R}^{d}$ and $x=a^{\oplus}$, (1a) gives $\bar{x}=a$ and $x-\bar{x}^{\oplus}=0_{\mathbb{R}^{dN}}$, so $a^{\oplus}\cdot(\mathbb{Y}_{N}a^{\oplus})=a\cdot(Y_{I}a)=N\,a\cdot(\mathbb{Y}a)$, using $Y_{I}=N\mathbb{Y}$ and claim 1 of Linearity of the Matrix-Vector Product and the Quadratic Form as a Double Sum. This is claim 2.

Step 10 (Test functions at fibre maximisers). Let $n\in\mathbb{N}$. By Quadruple Approximable by Test-Function Data §above with $\varepsilon=1/n$ there are $\zeta_{n}\in\mathbb{R}^{d}$, a function $\chi_{n}$ of class $C^{2}$ on $\mathbb{R}^{d}$ and a positive $r_{n}$ such that $U(\zeta)-\chi_{n}(\zeta)\le U(\zeta_{n})-\chi_{n}(\zeta_{n})$ whenever $d_{E}(\zeta,\zeta_{n})<r_{n}$, and

$$d_{E}(\zeta_{n},\hat{\zeta})<\tfrac1n,\qquad|U(\zeta_{n})-U(\hat{\zeta})|<\tfrac1n,\qquad\lVert D\chi_{n}(\zeta_{n})-p\rVert<\tfrac1n,\qquad\lVert D^{2}\chi_{n}(\zeta_{n})-X_{I}\rVert<\tfrac1n,$$

the last because $d_{\mathcal{S}(d)}(B,B')=\lVert B-B'\rVert$ (Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation §matrices). By (8a) choose $\rho_{n}\in\mathcal{D}_{\zeta_{n}}$ with $\Theta(\rho_{n})=U(\zeta_{n})$, and let $\varphi_{n}=\alpha A+N^{-1}(\chi_{n}\circ m)$. By Step 6, The Squared Wasserstein Distance to a Fixed Measure and Functions of the Mean are Intrinsic Test Functions §mean and Restrictions, Sums, Real Multiples and Differences of Intrinsic Test Functions on the Wasserstein Space §linear, $\varphi_{n}$ is an intrinsic test function on $\mathcal{D}$ with

$$\nabla\varphi_{n}(\rho_{n})=\alpha\nabla A(\rho_{n})+N^{-1}D\chi_{n}(\zeta_{n}),\qquad H_{\varphi_{n}}(\rho_{n})=\alpha0_{d}+N^{-1}D^{2}\chi_{n}(\zeta_{n})=N^{-1}D^{2}\chi_{n}(\zeta_{n}),$$

so that, by claim 5 of Properties of the Norm of a Symmetric Real Matrix, claim 10 of Elementary Order Arithmetic in an Ordered Field and $N^{-1}\le1$,

$$\lVert H_{\varphi_{n}}(\rho_{n})-\mathbb{X}\rVert=N^{-1}\lVert D^{2}\chi_{n}(\zeta_{n})-X_{I}\rVert<N^{-1}\tfrac1n\le\tfrac1n .\qquad(10\mathrm{a})$$

For $\rho'\in\mathcal{D}$ with $W(\rho',\rho_{n})<r_{n}$ we have $d_{E}(m(\rho'),\zeta_{n})\le W(\rho',\rho_{n})<r_{n}$ 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, so, since $u^{-}_{\delta}-\alpha A=N^{-1}\Theta$ on $\mathcal{D}$, by the definition of $U$ with $\rho'\in\mathcal{D}_{m(\rho')}$, and multiplying by $N^{-1}\ge0$,

$$u^{-}_{\delta}(\rho')-\varphi_{n}(\rho')=N^{-1}\bigl(\Theta(\rho')-\chi_{n}(m(\rho'))\bigr)\le N^{-1}\bigl(U(m(\rho'))-\chi_{n}(m(\rho'))\bigr)\le N^{-1}\bigl(U(\zeta_{n})-\chi_{n}(\zeta_{n})\bigr)=u^{-}_{\delta}(\rho_{n})-\varphi_{n}(\rho_{n}).$$

So $u^{-}_{\delta}-\varphi_{n}$ has a local maximum relative to $\mathcal{D}$ at $\rho_{n}$ (Local Maximum of a Function Relative to a Subset of a Metric Space).

Step 11 (Test functions at fibre minimisers, through the majorant). Let $n\in\mathbb{N}$. By Quadruple Approximable by Test-Function Data §below with $\varepsilon=1/n$ there are $\omega_{n}\in\mathbb{R}^{d}$, $\chi'_{n}$ of class $C^{2}$ on $\mathbb{R}^{d}$ and $r'_{n}>0$ with $V(\omega)-\chi'_{n}(\omega)\ge V(\omega_{n})-\chi'_{n}(\omega_{n})$ whenever $d_{E}(\omega,\omega_{n})<r'_{n}$, and

$$d_{E}(\omega_{n},\hat{\omega})<\tfrac1n,\qquad|V(\omega_{n})-V(\hat{\omega})|<\tfrac1n,\qquad\lVert D\chi'_{n}(\omega_{n})-p\rVert<\tfrac1n,\qquad\lVert D^{2}\chi'_{n}(\omega_{n})-Y_{I}\rVert<\tfrac1n .$$

By (8a) choose $\sigma_{n}\in\mathcal{D}_{N,\omega_{n}}$ with $\Xi(\sigma_{n})=V(\omega_{n})$. By hypothesis $\sigma_{n}^{[1]}\in\mathcal{D}$, and $m(\sigma_{n}^{[1]})=\omega_{n}$. As $\mathcal{D}$ has the map property and $\hat{m}\in\mathcal{P}_{2}(\mathbb{R}^{d})$, the pair $(\sigma_{n}^{[1]},\hat{m})$ is uniquely mapped (The Map Property of a Set of Probability Measures §map-property); let $T_{n}$ be an optimal map from $\sigma_{n}^{[1]}$ to $\hat{m}$ and $\sigma_{n,T}=(T_{n}^{\oplus})_{\#}\sigma_{n}$. By A Configuration-Level Majorant of the Distance from a One-Particle Marginal, Touching at a Given Configuration Law §majorant and A Configuration-Level Majorant of the Distance from a One-Particle Marginal, Touching at a Given Configuration Law §touching, $\sigma_{n,T}\in\mathcal{P}_{2}(\mathbb{R}^{dN})$, $N\,W(P^{[1]},\hat{m})^{2}\le W(P,\sigma_{n,T})^{2}$ for every $P\in\mathcal{P}_{2}(\mathbb{R}^{dN})$, with equality at $P=\sigma_{n}$. Put

$$B_{n}(P)=N^{-1}W(P,\sigma_{n,T})^{2}-\lVert\bar{m}(P)-c\rVert^{2}\qquad\bigl(P\in\mathcal{P}_{2}(\mathbb{R}^{dN})\bigr).$$

Multiplying by $N^{-1}\ge0$ and subtracting $\lVert m(P^{[1]})-c\rVert^{2}=\lVert\bar{m}(P)-c\rVert^{2}$,

$$A(P^{[1]})\le B_{n}(P)\quad\bigl(P\in\mathcal{P}_{2}(\mathbb{R}^{dN})\bigr),\qquad A(\sigma_{n}^{[1]})=B_{n}(\sigma_{n}).\qquad(11\mathrm{a})$$

The test function. On $\mathcal{P}_{2}(\mathbb{R}^{dN})$ let $\psi_{n}(P)=W(P,\sigma_{n,T})^{2}$, $\kappa(P)=q_{c}^{\flat}(M(P))$ and $\kappa'_{n}(P)=(\chi'_{n})^{\flat}(M(P))$, with $q_{c}$ of Step 6 and $^{\flat}$ of Step 2; by (3a), $\kappa(P)=\lVert\bar{m}(P)-c\rVert^{2}$ and $\kappa'_{n}(P)=\chi'_{n}(\bar{m}(P))$. Let

$$\Phi_{n}=(-\alpha)\psi_{n}+(N\alpha)\kappa+\kappa'_{n},\qquad\text{so that}\qquad\Phi_{n}(P)=-N\alpha B_{n}(P)+\chi'_{n}(\bar{m}(P)).$$

Since $\mathcal{D}_{N}$ has the map property, The Squared Wasserstein Distance to a Fixed Measure and Functions of the Mean are Intrinsic Test Functions §distance at the configuration level (with $Q=\mathcal{D}_{N}$ and $\nu_{0}=\sigma_{n,T}$) shows that $\psi_{n}$ is an intrinsic test function on $\mathcal{D}_{N}$, with $\nabla\psi_{n}(\sigma_{n})=2(\mathrm{id}-S_{n})$ for any optimal map $S_{n}$ from $\sigma_{n}$ to $\sigma_{n,T}$, and $H_{\psi_{n}}(\sigma_{n})=2I_{dN}$. The pair $(\sigma_{n},\sigma_{n,T})$ is uniquely mapped, $\sigma_{n}$ lying in $\mathcal{D}_{N}$ (The Map Property of a Set of Probability Measures §map-property at the configuration level), so A Configuration-Level Majorant of the Distance from a One-Particle Marginal, Touching at a Given Configuration Law §map gives $\mathrm{id}-S_{n}=(\mathrm{id}-T_{n})^{\oplus}$ in $L^{2}(\sigma_{n};\mathbb{R}^{dN})$. By Step 2, $q_{c}^{\flat}$ and $(\chi'_{n})^{\flat}$ are of class $C^{2}$ on $\mathbb{R}^{dN}$, so by The Squared Wasserstein Distance to a Fixed Measure and Functions of the Mean are Intrinsic Test Functions §mean at the configuration level $\kappa$ and $\kappa'_{n}$ are intrinsic test functions on $\mathcal{D}_{N}$; as $\overline{M(\sigma_{n})}=\bar{m}(\sigma_{n})=\omega_{n}$ by (3a), (2a) gives

$$\nabla\kappa(\sigma_{n})=\bigl(N^{-1}2(\omega_{n}-c)\bigr)^{\oplus},\quad\nabla\kappa'_{n}(\sigma_{n})=\bigl(N^{-1}D\chi'_{n}(\omega_{n})\bigr)^{\oplus},\quad H_{\kappa}(\sigma_{n})=(2I_{d})^{\boxplus},\quad H_{\kappa'_{n}}(\sigma_{n})=\bigl(D^{2}\chi'_{n}(\omega_{n})\bigr)^{\boxplus}.$$

By Restrictions, Sums, Real Multiples and Differences of Intrinsic Test Functions on the Wasserstein Space §linear, applied twice, $\Phi_{n}$ is an intrinsic test function on $\mathcal{D}_{N}$ at the configuration level, and by (3d) and (6a) (with $G_{\sigma_{n}^{[1]}}=T_{n}$ and $m(\sigma_{n}^{[1]})=\omega_{n}$)

$$\nabla\Phi_{n}(\sigma_{n})=\bigl(-2\alpha(\mathrm{id}-T_{n})+2\alpha(\omega_{n}-c)+N^{-1}D\chi'_{n}(\omega_{n})\bigr)^{\oplus}=g_{n}^{\oplus},\qquad g_{n}=-\alpha\nabla A(\sigma_{n}^{[1]})+N^{-1}D\chi'_{n}(\omega_{n})\in L^{2}(\sigma_{n}^{[1]};\mathbb{R}^{d}),\qquad(11\mathrm{b})$$

and $H_{\Phi_{n}}(\sigma_{n})=(-\alpha)(2I_{dN})+(N\alpha)(2I_{d})^{\boxplus}+(D^{2}\chi'_{n}(\omega_{n}))^{\boxplus}$. Comparing with the definition of $\mathbb{Y}_{N}$ through claim 1 of Linearity of the Matrix-Vector Product and the Quadratic Form as a Double Sum, Bilinearity and Symmetry of the Dot Product on $\mathbb{R}^n$ and (1c), with $E_{n}=D^{2}\chi'_{n}(\omega_{n})-Y_{I}\in\mathcal{S}(d)$, every $x\in\mathbb{R}^{dN}$ satisfies

$$x\cdot\bigl((H_{\Phi_{n}}(\sigma_{n})-\mathbb{Y}_{N})x\bigr)=\bar{x}\cdot(D^{2}\chi'_{n}(\omega_{n})\bar{x})-\bar{x}\cdot(Y_{I}\bar{x})=\bar{x}\cdot(E_{n}\bar{x}),$$

whose absolute value is at most $\lVert E_{n}\rVert\lVert\bar{x}\rVert^{2}\le\lambda_{n}\lVert x\rVert^{2}$ with $\lambda_{n}=N^{-1}\lVert E_{n}\rVert\ge0$ (claims 1 and 2 of Properties of the Norm of a Symmetric Real Matrix, Step 1). By claim 6 of Properties of the Absolute Value in an Ordered Field and Vector, Entry and Comparison Bounds for the Norm of a Symmetric Real Matrix §identity, $-\lambda_{n}I_{dN}\preceq H_{\Phi_{n}}(\sigma_{n})-\mathbb{Y}_{N}\preceq\lambda_{n}I_{dN}$ (The Positive Semidefinite Ordering on Symmetric Matrices), the difference lying in $\mathcal{S}(dN)$ (Intrinsic Test Functions on the Wasserstein Space and Their Translation Hessians §hessian, Real Matrices, Symmetric Matrices and the Semidefinite Ordering: Standing Notation §symmetric); so claim 3 of Properties of the Norm of a Symmetric Real Matrix and $\lVert E_{n}\rVert<\tfrac1n$ give

$$\lVert H_{\Phi_{n}}(\sigma_{n})-\mathbb{Y}_{N}\rVert\le N^{-1}\lVert E_{n}\rVert<N^{-1}\tfrac1n\le\tfrac1n .\qquad(11\mathrm{c})$$

The local minimum. Let $P\in\mathcal{D}_{N}$ with $W(P,\sigma_{n})<r'_{n}$. Then $d_{E}(\bar{m}(P),\omega_{n})<r'_{n}$ by (3b). By (11a) and $N\alpha>0$, $N\alpha A(P^{[1]})\le N\alpha B_{n}(P)$; with $P\in\mathcal{D}_{N,\bar{m}(P)}$ and the choice of $\chi'_{n}$,

$$U^{+}_{\delta}(P)-\Phi_{n}(P)=\Xi(P)-N\alpha A(P^{[1]})+N\alpha B_{n}(P)-\chi'_{n}(\bar{m}(P))\ge V(\bar{m}(P))-\chi'_{n}(\bar{m}(P))\ge V(\omega_{n})-\chi'_{n}(\omega_{n}),$$

and the right side equals $\Xi(\sigma_{n})-N\alpha A(\sigma_{n}^{[1]})+N\alpha B_{n}(\sigma_{n})-\chi'_{n}(\omega_{n})=U^{+}_{\delta}(\sigma_{n})-\Phi_{n}(\sigma_{n})$ by (11a). So $U^{+}_{\delta}-\Phi_{n}$ has a local minimum relative to $\mathcal{D}_{N}$ at $\sigma_{n}$ (Local Minimum of a Function Relative to a Subset of a Metric Space, in $(\mathcal{P}_{2}(\mathbb{R}^{dN}),W)$).

Step 12 (The limits $\rho^{*}$, $\sigma^{*}$; claim 1). We have $m(\rho_{n})=\zeta_{n}\to\hat{\zeta}$ and $\Theta(\rho_{n})=U(\zeta_{n})>U(\hat{\zeta})-1/n\ge U(\hat{\zeta})-1$. (K-u) gives a strictly increasing $(n_{j})_{j}$ and $\rho^{*}\in\mathcal{D}_{\hat{\zeta}}$ with $\rho_{n_{j}}\to\rho^{*}$; exactly as in (8a), $\Theta(\rho^{*})=U(\hat{\zeta})$, and then $|\Theta(\rho_{n_{j}})-\Theta(\rho^{*})|=|U(\zeta_{n_{j}})-U(\hat{\zeta})|<1/n_{j}$. Symmetrically, $\bar{m}(\sigma_{n})=\omega_{n}\to\hat{\omega}$ and $\Xi(\sigma_{n})=V(\omega_{n})<V(\hat{\omega})+1$, and (K-v) gives $(n'_{j})_{j}$ and $\sigma^{*}\in\mathcal{D}_{N,\hat{\omega}}$ with $\sigma_{n'_{j}}\to\sigma^{*}$, $\Xi(\sigma^{*})=V(\hat{\omega})$ and $|\Xi(\sigma_{n'_{j}})-\Xi(\sigma^{*})|<1/n'_{j}$. Then $\nu^{*}=(\sigma^{*})^{[1]}\in\mathcal{D}$ by hypothesis, with $m(\nu^{*})=\hat{\omega}$. By (5b) and (9b),

$$M_{0}=\Theta(\rho^{*})-\Xi(\sigma^{*})-\tfrac{N\alpha}{2}\lVert\hat{\zeta}-\hat{\omega}\rVert^{2}\le\Psi(\rho^{*},\sigma^{*})\le M_{0},$$

which is claim 1; and equality in the first inequality means, by Step 5, that equality holds in The Displacement Midpoint and the Midpoint Split of the Squared Wasserstein Distance §split for $(\rho^{*},\nu^{*})$, hence also for $(\nu^{*},\rho^{*})$, both sides of that inequality being symmetric in the pair by The Quadratic Wasserstein Distance is a Metric on the Wasserstein Space §symmetry and claim 5 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$.

Step 13 (The limiting gradients). As $\mathcal{D}$ has the map property and $\rho^{*},\nu^{*}\in\mathcal{D}$, the pairs $(\rho^{*},\hat{m})$, $(\rho^{*},\nu^{*})$, $(\nu^{*},\hat{m})$ and $(\nu^{*},\rho^{*})$ are uniquely mapped (The Map Property of a Set of Probability Measures §map-property). By The Displacement Midpoint and the Midpoint Split of the Squared Wasserstein Distance §equality for $(\rho^{*},\nu^{*})$ with the optimal maps $G_{\rho^{*}}$ and $S$, the constant $e$ there has value $\tfrac12(\hat{\zeta}+\hat{\omega})-c=0_{\mathbb{R}^{d}}$, so $2(\mathrm{id}-G_{\rho^{*}})=\mathrm{id}-S$ in $L^{2}(\rho^{*};\mathbb{R}^{d})$; likewise, for $(\nu^{*},\rho^{*})$ with $G_{\nu^{*}}$ and $S'$, $2(\mathrm{id}-G_{\nu^{*}})=\mathrm{id}-S'$ in $L^{2}(\nu^{*};\mathbb{R}^{d})$. As $2(\hat{\zeta}-c)=\hat{\zeta}-\hat{\omega}$ and $2(\hat{\omega}-c)=\hat{\omega}-\hat{\zeta}$, (6a) gives $\nabla A(\rho^{*})=(\mathrm{id}-S)-(\hat{\zeta}-\hat{\omega})$ and $\nabla A(\nu^{*})=(\mathrm{id}-S')-(\hat{\omega}-\hat{\zeta})$, hence, with $N^{-1}p=\alpha(\hat{\zeta}-\hat{\omega})$,

$$\alpha\nabla A(\rho^{*})+N^{-1}p=\alpha(\mathrm{id}-S)\ \text{in }L^{2}(\rho^{*};\mathbb{R}^{d}),\qquad-\alpha\nabla A(\nu^{*})+N^{-1}p=\alpha(S'-\mathrm{id})\ \text{in }L^{2}(\nu^{*};\mathbb{R}^{d}).\qquad(13\mathrm{a})$$

Step 14 (Claim 3). For each $j$ let $\pi_{j}\in\Pi(\rho_{n_{j}},\rho^{*})$ be optimal (Existence of an Optimal Coupling of Two Probability Measures with Finite Second Moment), so $I(\pi_{j})=W(\rho_{n_{j}},\rho^{*})^{2}\to0$ (Optimal Coupling of Two Probability Measures with Finite Second Moment §optimal, claim 2 of Arithmetic of Limits of Real Sequences). By property (c) of Intrinsic Test Functions on the Wasserstein Space and Their Translation Hessians §test for $A$ on $\mathcal{D}$, the discrepancy $D_{j}$ of $\nabla A(\rho_{n_{j}})$ and $\nabla A(\rho^{*})$ along $\pi_{j}$ converges to $0$. Let $E_{j}$ be the discrepancy of $\nabla\varphi_{n_{j}}(\rho_{n_{j}})$ and $\alpha(\mathrm{id}-S)$ along $\pi_{j}$, which is the integral in claim 3. It does not depend on representatives (The Discrepancy of Two Square-Integrable Vector Fields Along a Coupling of Their Base Measures §well-defined), so by (13a) and Step 10 its integrand may be taken to be $\lVert F_{1}(z)+F_{2}(z)\rVert^{2}$ with

$$F_{1}(z)=\alpha\bigl(\nabla A(\rho_{n_{j}})(x)-\nabla A(\rho^{*})(y)\bigr),\qquad F_{2}(z)=N^{-1}\bigl(D\chi_{n_{j}}(\zeta_{n_{j}})-p\bigr).$$

Both are Borel and square-integrable against $\pi_{j}$, with $\lVert F_{1}\rVert_{\pi_{j}}=\alpha\sqrt{D_{j}}$ (claim 5 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$) and $\lVert F_{2}\rVert_{\pi_{j}}=N^{-1}\lVert D\chi_{n_{j}}(\zeta_{n_{j}})-p\rVert<1/n_{j}$, so the triangle inequality The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §triangle in the real Hilbert space $L^{2}(\pi_{j};\mathbb{R}^{d})$ (Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation §fields) gives $\sqrt{E_{j}}<\alpha\sqrt{D_{j}}+1/n_{j}$. Moreover $u^{-}_{\delta}(\rho_{n_{j}})-u^{-}_{\delta}(\rho^{*})=N^{-1}\bigl(\Theta(\rho_{n_{j}})-\Theta(\rho^{*})\bigr)+\alpha\bigl(A(\rho_{n_{j}})-A(\rho^{*})\bigr)$, where the first term has absolute value below $N^{-1}/n_{j}\le1/n_{j}$ (Step 12, as $1\le N$) and the second converges to $0$ since $A$ is continuous (Step 6, Intrinsic Test Functions on the Wasserstein Space and Their Translation Hessians §continuity) and $\rho_{n_{j}}\to\rho^{*}$, by Continuity Between Metric Spaces is Equivalent to Sequential Continuity.

Let $\varepsilon>0$. Each of the real sequences $(I(\pi_{j}))_{j}$, $(\sqrt{D_{j}})_{j}$ (square roots by Existence and Uniqueness of the Nonnegative Square Root, convergence by claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field), $(1/n_{j})_{j}$ and $(A(\rho_{n_{j}})-A(\rho^{*}))_{j}$ converges to $0$, so we may fix $j$ with $I(\pi_{j})<\varepsilon^{2}$, $1/n_{j}<\tfrac{\varepsilon}{2}$, $\alpha\sqrt{D_{j}}<\tfrac{\varepsilon}{2}$ and $\alpha|A(\rho_{n_{j}})-A(\rho^{*})|<\tfrac{\varepsilon}{2}$. Put $\rho=\rho_{n_{j}}$, $\varphi=\varphi_{n_{j}}$ and $\pi=\pi_{j}$. By Step 10, $u^{-}_{\delta}-\varphi$ has a local maximum relative to $\mathcal{D}$ at $\rho$; $|u^{-}_{\delta}(\rho)-u^{-}_{\delta}(\rho^{*})|<\varepsilon$ by claim 5 of Properties of the Absolute Value in an Ordered Field; $\sqrt{E_{j}}<\varepsilon$, so $E_{j}<\varepsilon^{2}$ (claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field); and $\lVert H_{\varphi}(\rho)-\mathbb{X}\rVert<1/n_{j}<\varepsilon$ by (10a). This is claim 3.

Step 15 (Claim 4). For each $j$ let $\gamma_{j}\in\Pi(\sigma_{n'_{j}},\sigma^{*})$ be optimal (Existence of an Optimal Coupling of Two Probability Measures with Finite Second Moment at the configuration level), so $I(\gamma_{j})=W(\sigma_{n'_{j}},\sigma^{*})^{2}\to0$, and let $\bar{\gamma}_{j}\in\Pi(\sigma_{n'_{j}}^{[1]},\nu^{*})$ be the averaged coupling of Step 4, so $I(\bar{\gamma}_{j})=N^{-1}I(\gamma_{j})\to0$ (claim 3 of Arithmetic of Limits of Real Sequences). The measures $\sigma_{n'_{j}}^{[1]}$ and $\nu^{*}$ lie in $\mathcal{D}$ by hypothesis, so property (c) of Intrinsic Test Functions on the Wasserstein Space and Their Translation Hessians §test for the fixed intrinsic test function $A$ on $\mathcal{D}$, along the couplings $\bar{\gamma}_{j}$, shows that the discrepancy $\bar{D}_{j}$ of $\nabla A(\sigma_{n'_{j}}^{[1]})$ and $\nabla A(\nu^{*})$ along $\bar{\gamma}_{j}$ converges to $0$. Let $g=\alpha(S'-\mathrm{id})\in L^{2}(\nu^{*};\mathbb{R}^{d})$; by (3d), $\alpha(S'-\mathrm{id})^{\oplus}=g^{\oplus}$, and by (11b), $\nabla\Phi_{n'_{j}}(\sigma_{n'_{j}})=g_{n'_{j}}^{\oplus}$. So the integral $E_{j}$ in claim 4, for $\sigma=\sigma_{n'_{j}}$, $\Phi=\Phi_{n'_{j}}$ and $\gamma=\gamma_{j}$, is the discrepancy of $g_{n'_{j}}^{\oplus}$ and $g^{\oplus}$ along $\gamma_{j}$, and by (4a) $E_{j}=N\bar{E}_{j}$, where $\bar{E}_{j}$ is the discrepancy of $g_{n'_{j}}$ and $g$ along $\bar{\gamma}_{j}$. By (13a), (11b) and the independence of representatives (The Discrepancy of Two Square-Integrable Vector Fields Along a Coupling of Their Base Measures §well-defined), the integrand of $\bar{E}_{j}$ may be taken to be $\lVert F'_{1}(z')+F'_{2}(z')\rVert^{2}$ with

$$F'_{1}(z')=-\alpha\bigl(\nabla A(\sigma_{n'_{j}}^{[1]})(x')-\nabla A(\nu^{*})(y')\bigr),\qquad F'_{2}(z')=N^{-1}\bigl(D\chi'_{n'_{j}}(\omega_{n'_{j}})-p\bigr),$$

so, as in Step 14, $\sqrt{\bar{E}_{j}}<\alpha\sqrt{\bar{D}_{j}}+1/n'_{j}$. Moreover $U^{+}_{\delta}(\sigma_{n'_{j}})-U^{+}_{\delta}(\sigma^{*})=\bigl(\Xi(\sigma_{n'_{j}})-\Xi(\sigma^{*})\bigr)-N\alpha\bigl(A(\sigma_{n'_{j}}^{[1]})-A(\nu^{*})\bigr)$, where the first difference has absolute value below $1/n'_{j}$ (Step 12) and the second converges to $0$ since $P\mapsto A(P^{[1]})$ is continuous (Step 6) and $\sigma_{n'_{j}}\to\sigma^{*}$, by Continuity Between Metric Spaces is Equivalent to Sequential Continuity.

Let $\varepsilon>0$, and let $\eta$ be the positive square root of the positive real $N^{-1}\varepsilon^{2}$ (Existence and Uniqueness of the Nonnegative Square Root, claim 5 of Elementary Order Arithmetic in an Ordered Field). Each of the real sequences $(I(\gamma_{j}))_{j}$, $(\sqrt{\bar{D}_{j}})_{j}$, $(1/n'_{j})_{j}$ and $(A(\sigma_{n'_{j}}^{[1]})-A(\nu^{*}))_{j}$ converges to $0$, so we may fix $j$ with $I(\gamma_{j})<\varepsilon^{2}$, $1/n'_{j}<\tfrac{\eta}{2}$, $1/n'_{j}<\tfrac{\varepsilon}{2}$, $\alpha\sqrt{\bar{D}_{j}}<\tfrac{\eta}{2}$ and $N\alpha|A(\sigma_{n'_{j}}^{[1]})-A(\nu^{*})|<\tfrac{\varepsilon}{2}$. Put $\sigma=\sigma_{n'_{j}}$, $\Phi=\Phi_{n'_{j}}$ and $\gamma=\gamma_{j}$. By Step 11, $\Phi$ is an intrinsic test function on $\mathcal{D}_{N}$ at the configuration level and $U^{+}_{\delta}-\Phi$ has a local minimum relative to $\mathcal{D}_{N}$ at $\sigma$; $\gamma\in\Pi(\sigma,\sigma^{*})$ and $I(\gamma)<\varepsilon^{2}$; $|U^{+}_{\delta}(\sigma)-U^{+}_{\delta}(\sigma^{*})|<\varepsilon$ by claim 5 of Properties of the Absolute Value in an Ordered Field; $\sqrt{\bar{E}_{j}}<\eta$, so $\bar{E}_{j}<\eta^{2}=N^{-1}\varepsilon^{2}$ (claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field) and $E_{j}=N\bar{E}_{j}<\varepsilon^{2}$ (claim 10 of Elementary Order Arithmetic in an Ordered Field); and $\lVert H_{\Phi}(\sigma)-\mathbb{Y}_{N}\rVert<1/n'_{j}<\varepsilon$ by (11c). This is claim 4, with $\rho^{*}$, $\sigma^{*}$, $\mathbb{X}$ and $\mathbb{Y}_{N}$ as constructed, $\mathbb{Y}$ being the matrix of claim 2.