Proof of Intrinsic Test Functions at a Maximiser of the Doubled Difference Linked through Tensor Powers

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 $\iota(n)$ 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. We also write $N$ for the real number $\iota(N)$; it satisfies $1\le N$ (Properties of the Canonical Map from the Natural Numbers to an Ordered Field), hence $0<N$, $N^{-1}$ exists and $0<N^{-1}\le N^{-1}N=1$ (claims 6 and 7 of Elementary Order Arithmetic in an Ordered Field, 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 $\beta$ of $\mathbb{R}^{d}$ or of $\mathbb{R}^{dN}$ also stands for the class of the constant map with value $\beta$ in the space $L^{2}(\rho;\mathbb{R}^{d})$ or $L^{2}(P;\mathbb{R}^{dN})$ at hand. Convergence in $\mathbb{R}^{q}$ is convergence in $(\mathbb{R}^{q},d_{E})$, where $d_{E}(a,a')=\lVert a-a'\rVert$ by claim 2 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$. Results and notions used at the configuration level are so read by N-Particle Systems on the Wasserstein Space: Particles, Configurations and the Configuration Level §configuration-level; in particular, for $P\in\mathcal{P}_{2}(\mathbb{R}^{dN})$, $M(P)\in\mathbb{R}^{dN}$ is the mean of The Mean of a Square-Integrable Probability Measure, Its Lift, Its Centring, and Functions of the Mean and Centred Integrals as Test Functions §mean read at the configuration level, and $m(\mu)\in\mathbb{R}^{d}$ is the mean of that clause for $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$. Dot products and Euclidean norms are handled with Bilinearity and Symmetry of the Dot Product on $\mathbb{R}^n$ and Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$, and finite sums with Properties of Finite Sums and Properties of Finite Sums of Vectors.

Step 0 (Block averages, the diagonal decomposition and two matrices). For $x\in\mathbb{R}^{dN}$ put

$$s(x)=\sum_{k=1}^{N}\mathfrak{p}_{k}(x)\in\mathbb{R}^{d},\qquad\bar{x}=N^{-1}s(x)\in\mathbb{R}^{d},\qquad x^{\perp}=x-\bar{x}^{\oplus}\in\mathbb{R}^{dN}.$$

Each $\mathfrak{p}_{k}$ is linear and $[y_{1}+ty'_{1},\dots,y_{N}+ty'_{N}]=[y_{1},\dots,y_{N}]+t[y'_{1},\dots,y'_{N}]$ (Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §linear), so $(a+ta')^{\oplus}=a^{\oplus}+ta'^{\oplus}$ for $a,a'\in\mathbb{R}^{d}$ and $t\in\mathbb{R}$, and the maps $x\mapsto s(x)$, $x\mapsto\bar{x}$ and $x\mapsto x^{\perp}$ are linear (claims 2 and 3 of Properties of Finite Sums of Vectors). Let $a\in\mathbb{R}^{d}$ and $x\in\mathbb{R}^{dN}$. Since $\mathfrak{p}_{k}(a^{\oplus})=a$ for every $k\in[N]$ (Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points §configuration), induction on $N$ with the recursion of claim 1 of Properties of Finite Sums of Vectors gives $s(a^{\oplus})=Na$, so $\overline{a^{\oplus}}=a$ and $(a^{\oplus})^{\perp}=0$. By Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §diagonal, $\mathfrak{p}_{k}(x+a^{\oplus})=\mathfrak{p}_{k}(x)+a$, $a^{\oplus}\cdot x=a\cdot s(x)$ and $\lVert a^{\oplus}\rVert^{2}=N\lVert a\rVert^{2}$; hence

$$a^{\oplus}\cdot x=N\,a\cdot\bar{x},\qquad\overline{x+a^{\oplus}}=\bar{x}+a,\qquad\lVert a^{\oplus}\rVert^{2}=N\lVert a\rVert^{2}.\qquad(0\mathrm{a})$$

By linearity $\overline{x^{\perp}}=\bar{x}-\overline{\bar{x}^{\oplus}}=0$, so $x^{\perp}\cdot a^{\oplus}=N\,a\cdot\overline{x^{\perp}}=0$ by (0a). As $x-a^{\oplus}=x^{\perp}+(\bar{x}-a)^{\oplus}$, expanding the square with this orthogonality and (0a) gives the diagonal decomposition

$$\lVert x-a^{\oplus}\rVert^{2}=\lVert x^{\perp}\rVert^{2}+N\lVert\bar{x}-a\rVert^{2},\qquad\text{in particular}\qquad\lVert x\rVert^{2}=\lVert x^{\perp}\rVert^{2}+N\lVert\bar{x}\rVert^{2}.\qquad(0\mathrm{b})$$

All terms being nonnegative, (0b) and $1\le N$ give $\lVert x^{\perp}\rVert^{2}\le\lVert x\rVert^{2}$ and $\lVert\bar{x}\rVert^{2}\le N\lVert\bar{x}\rVert^{2}\le\lVert x\rVert^{2}$, hence $\lVert x^{\perp}\rVert\le\lVert x\rVert$ and $\lVert\bar{x}\rVert\le\lVert x\rVert$ (claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field); likewise $\lVert a^{\oplus}\rVert^{2}=N\lVert a\rVert^{2}\le N^{2}\lVert a\rVert^{2}$ gives $\lVert a^{\oplus}\rVert\le N\lVert a\rVert$. Also, for $u\in\mathbb{R}^{dN}$, $u\cdot x^{\perp}=u\cdot x-N\,\bar{u}\cdot\bar{x}$ by (0a).

(0c) Means. For $P,P'\in\mathcal{P}_{2}(\mathbb{R}^{dN})$ put $\bar{m}(P)=\overline{M(P)}\in\mathbb{R}^{d}$. By linearity, the bound $\lVert\bar{x}\rVert\le\lVert x\rVert$ 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 at the configuration level, $\lVert\bar{m}(P)-\bar{m}(P')\rVert\le\lVert M(P)-M(P')\rVert\le W(P,P')$ and, for $\beta\in\mathbb{R}^{d}$, $M((\tau_{\beta^{\oplus}})_{\#}P)=M(P)+\beta^{\oplus}$, so $\bar{m}((\tau_{\beta^{\oplus}})_{\#}P)=\bar{m}(P)+\beta$ by (0a). Let $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$; then $\mu^{\otimes N}\in\mathcal{P}_{2}(\mathbb{R}^{dN})$ by Tensor Powers and One-Particle Marginals: Particle Laws, Product Integrals, Push-Forwards, Moments, Product Maps and Diagonal Shifts §moments. For $k\in[N]$ and $i\in[d]$ the coordinate of $x\in\mathbb{R}^{dN}$ with index $b(k,i)$ is the $i$th coordinate of $\mathfrak{p}_{k}(x)$ (Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points §blocks); the coordinate map $y\mapsto y_{i}$ is Borel and integrable against $\mu$ (The Mean of a Square-Integrable Probability Measure, Its Lift, Its Centring, and Functions of the Mean and Centred Integrals as Test Functions §mean), $\mathfrak{p}_{k}$ is Borel (Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §linear) and $(\mathfrak{p}_{k})_{\#}\mu^{\otimes N}=\mu$ (Tensor Powers and One-Particle Marginals: Particle Laws, Product Integrals, Push-Forwards, Moments, Product Maps and Diagonal Shifts §particle-laws), so the change-of-variables formula of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward gives that the coordinate of $M(\mu^{\otimes N})$ with index $b(k,i)$ is $\int\mathfrak{p}_{k}(x)_{i}\,\mu^{\otimes N}(dx)=\int y_{i}\,\mu(dy)=m(\mu)_{i}$, the coordinate of $m(\mu)^{\oplus}$ with that index (Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points §configuration). Every index in $[dN]$ being $b(k,i)$ for such $k,i$ (Block Indices: Enumerating an Initial Segment of Length qN by Blocks and Positions §bijection),

$$M(\mu^{\otimes N})=m(\mu)^{\oplus},\qquad\bar{m}(\mu^{\otimes N})=m(\mu).\qquad(0\mathrm{c})$$

(0d) Two matrices. Let $K\in\mathcal{M}_{dN\times d}(\mathbb{R})$ (Real Matrices, Symmetric Matrices and the Semidefinite Ordering: Standing Notation §matrices) have entry $1$ in row $b(k,i)$ and column $j$ if $i=j$ and entry $0$ otherwise ($k\in[N]$, $i,j\in[d]$), which is well defined by Block Indices: Enumerating an Initial Segment of Length qN by Blocks and Positions §bijection. For $a\in\mathbb{R}^{d}$, Matrix-Vector Product and claim 7 of Properties of Finite Sums give $(Ka)_{b(k,i)}=\sum_{j=1}^{d}K_{b(k,i)j}a_{j}=a_{i}$, so $Ka=a^{\oplus}$. For $x\in\mathbb{R}^{dN}$ and $a\in\mathbb{R}^{d}$, claims 1 and 5 of Elementary Properties of the Transpose of a Real Matrix and (0a) give $a\cdot(K^{\top}x)=(Ka)\cdot x=a^{\oplus}\cdot x=a\cdot s(x)$; with $a=K^{\top}x-s(x)$ this gives $\lVert K^{\top}x-s(x)\rVert^{2}=0$, so $K^{\top}x=s(x)=N\bar{x}$. For $\lambda\in\mathbb{R}$ and $B\in\mathcal{S}(d)$ put

$$Z_{\lambda,B}=\lambda\bigl(I_{dN}-N^{-1}(KK^{\top})\bigr)+N^{-2}\bigl((KB)K^{\top}\bigr)\in\mathcal{M}_{dN}(\mathbb{R}).$$

By claim 2 of Linearity, Compatibility with the Matrix Product, and a Norm Bound for the Matrix-Vector Product, $(KK^{\top})x=K(N\bar{x})=N\bar{x}^{\oplus}$ and $((KB)K^{\top})x=K(B(N\bar{x}))=N(B\bar{x})^{\oplus}$, the scalars being moved by claim 3 of Linearity of the Matrix-Vector Product and the Quadratic Form as a Double Sum and the linearity of $a\mapsto a^{\oplus}$; so claims 1 and 2 of Linearity of the Matrix-Vector Product and the Quadratic Form as a Double Sum give

$$Z_{\lambda,B}\,x=\lambda x^{\perp}+N^{-1}(B\bar{x})^{\oplus},\qquad u\cdot(Z_{\lambda,B}\,x)=\lambda\bigl(u\cdot x-N\,\bar{u}\cdot\bar{x}\bigr)+\bar{u}\cdot(B\bar{x})\qquad(u,x\in\mathbb{R}^{dN}),\qquad(0\mathrm{d})$$

the second by (0a). As $B^{\top}=B$, $\bar{u}\cdot(B\bar{x})=(B\bar{u})\cdot\bar{x}$ (claim 5 of Elementary Properties of the Transpose of a Real Matrix), so the right-hand side of the second identity is symmetric in $(u,x)$; with the entry formula of Vector, Entry and Comparison Bounds for the Norm of a Symmetric Real Matrix §entry-formula, $(Z_{\lambda,B})_{ij}=e_{i}\cdot(Z_{\lambda,B}e_{j})=e_{j}\cdot(Z_{\lambda,B}e_{i})=(Z_{\lambda,B})_{ji}$, so $Z_{\lambda,B}\in\mathcal{S}(dN)$ (Real Matrices, Symmetric Matrices and the Semidefinite Ordering: Standing Notation §symmetric). With $u=x$ and (0b),

$$x\cdot(Z_{\lambda,B}\,x)=\lambda\lVert x^{\perp}\rVert^{2}+\bar{x}\cdot(B\bar{x})\qquad(x\in\mathbb{R}^{dN}).\qquad(0\mathrm{e})$$

Put $\Pi^{\perp}=Z_{1,0_{d}}$; by (0d) and (0e), $\Pi^{\perp}x=x^{\perp}$ and $x\cdot(\Pi^{\perp}x)=\lVert x^{\perp}\rVert^{2}$.

Step 1 (The midpoint and two auxiliary functions). Put $\hat{Q}=\hat{\mu}^{\otimes N}$, which lies in $\mathcal{D}_{N}$ by hypothesis. By Existence of an Optimal Coupling of Two Probability Measures with Finite Second Moment fix an optimal coupling $\hat{\pi}\in\Pi(\hat{P},\hat{Q})$, and let $\hat{M}$, $c\in\mathbb{R}^{dN}$ and $A_{N}$ be the midpoint, the point and the function of The Displacement Midpoint and the Midpoint Split of the Squared Wasserstein Distance read at the configuration level for $\hat{P}$, $\hat{Q}$ and $\hat{\pi}$; thus $\hat{M}\in\mathcal{P}_{2}(\mathbb{R}^{dN})$ by The Displacement Midpoint and the Midpoint Split of the Squared Wasserstein Distance §midpoint and $0\le A_{N}(P)$ for every $P\in\mathcal{P}_{2}(\mathbb{R}^{dN})$ by The Displacement Midpoint and the Midpoint Split of the Squared Wasserstein Distance §nonnegative. For $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ put $\tilde{A}(\mu)=A_{N}(\mu^{\otimes N})$, so $0\le\tilde{A}(\mu)$. Put $\hat{\zeta}=\bar{m}(\hat{P})$, $\hat{\omega}=m(\hat{\mu})$ and $M_{0}=\Psi(\hat{P},\hat{\mu})$. By Basic Properties of a Wasserstein-Coercive Penalty Pair §bounded-below, applied to each pair (at the configuration level to the second), fix $e_{0},e_{N}\in\mathbb{R}$ with $e_{0}\le\mathcal{E}(\mu)$ for $\mu\in\mathcal{D}$ and $e_{N}\le\mathcal{E}_{N}(P)$ for $P\in\mathcal{D}_{N}$. For $P\in\mathcal{D}_{N}$ and $\mu\in\mathcal{D}$ put

$$\Theta(P)=U^{-}_{\delta}(P)-\alpha A_{N}(P)-\tfrac{\alpha}{2}\lVert M(P)^{\perp}\rVert^{2},\qquad\Xi(\mu)=N\,v^{+}_{\delta}(\mu)+\alpha\tilde{A}(\mu).$$

By The Delta-Envelopes of Bounded Functions and Their Monotonicity in the Weight, for a Wasserstein-Coercive Penalty Pair §bounded (at the configuration level for $U$), $U^{-}_{\delta}(P)\le b-\delta\mathcal{E}_{N}(P)$ and $b'+\delta\mathcal{E}(\mu)\le v^{+}_{\delta}(\mu)$; the subtracted and added terms $\alpha A_{N}(P)$, $\tfrac{\alpha}{2}\lVert M(P)^{\perp}\rVert^{2}$ and $\alpha\tilde{A}(\mu)$ are nonnegative, so by claims 2, 3 and 5 of Elementary Arithmetic in an Ordered Field

$$\Theta(P)\le b-\delta\,\mathcal{E}_{N}(P)\le b-\delta e_{N},\qquad Nb'+N\delta e_{0}\le Nb'+N\delta\,\mathcal{E}(\mu)\le\Xi(\mu).\qquad(1\mathrm{a})$$

Let $P\in\mathcal{D}_{N}$ and $\mu\in\mathcal{D}$. By The Displacement Midpoint and the Midpoint Split of the Squared Wasserstein Distance §split at the configuration level for the pair $(P,\mu^{\otimes N})$, with (0c) and (0b) for $x=M(P)$ and $a=m(\mu)$,

$$W(P,\mu^{\otimes N})^{2}\le2A_{N}(P)+2\tilde{A}(\mu)+\lVert M(P)-M(\mu^{\otimes N})\rVert^{2},\qquad\lVert M(P)-M(\mu^{\otimes N})\rVert^{2}=\lVert M(P)^{\perp}\rVert^{2}+N\lVert\bar{m}(P)-m(\mu)\rVert^{2}.$$

Writing $\tfrac{N\alpha}{2}$ for $N\cdot\tfrac{\alpha}{2}$, the definitions give

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

which is nonnegative and vanishes exactly when equality holds in The Displacement Midpoint and the Midpoint Split of the Squared Wasserstein Distance §split for $(P,\mu^{\otimes N})$, $\tfrac{\alpha}{2}$ 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 $(\hat{P},\hat{\mu})$,

$$\Theta(P)-\Xi(\mu)-\tfrac{N\alpha}{2}\lVert\bar{m}(P)-m(\mu)\rVert^{2}\le\Psi(P,\mu)\le M_{0}\qquad(P\in\mathcal{D}_{N},\ \mu\in\mathcal{D}),\qquad(1\mathrm{b})$$

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 $(P,\mu^{\otimes N})$. By The Displacement Midpoint and the Midpoint Split of the Squared Wasserstein Distance §endpoints this is so for $(\hat{P},\hat{Q})$:

$$\Theta(\hat{P})-\Xi(\hat{\mu})-\tfrac{N\alpha}{2}\lVert\hat{\zeta}-\hat{\omega}\rVert^{2}=M_{0}.\qquad(1\mathrm{c})$$

Step 2 (The fixed test functions). (2a) $A_{N}$. Let $q_{c}:\mathbb{R}^{dN}\to\mathbb{R}$, $q_{c}(x)=\lVert x-c\rVert^{2}=d_{E}(x,c)^{2}$. 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 $\mathbb{R}^{dN}$ with $Dq_{c}(x)=2(x-c)$ and $D^{2}q_{c}(x)=2I_{dN}$. Since $A_{N}(P)=W(P,\hat{M})^{2}-q_{c}(M(P))$ and $\mathcal{D}_{N}$ has the map property, The Squared Wasserstein Distance to a Fixed Measure and Functions of the Mean are Intrinsic Test Functions §distance (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, all at the configuration level, show that $A_{N}$ is an intrinsic test function on $\mathcal{D}_{N}$ at the configuration level, with

$$\nabla A_{N}(P)=2(\mathrm{id}-G_{P})-2\bigl(M(P)-c\bigr)\quad(P\in\mathcal{D}_{N}),\qquad H_{A_{N}}(P)=2I_{dN}-2I_{dN}=0_{dN}\quad\bigl(P\in\mathcal{P}_{2}(\mathbb{R}^{dN})\bigr),$$

where $G_{P}$ is any optimal map from $P$ to $\hat{M}$. By property (a) of Intrinsic Test Functions on the Wasserstein Space and Their Translation Hessians §test, $A_{N}$ is continuous on $\mathcal{P}_{2}(\mathbb{R}^{dN})$.

(2b) The off-diagonal part of the mean. Let $\phi^{\perp}(x)=\tfrac12\,x\cdot(\Pi^{\perp}x)=\tfrac12\lVert x^{\perp}\rVert^{2}$ (Step 0). By Quadratic and Affine Functions of Class $C^2$, Translation, and Quadratic Perturbation of Semiconvexity §quadratic, with $n=dN$, the matrix $\Pi^{\perp}\in\mathcal{S}(dN)$ and zero linear and constant terms, $\phi^{\perp}$ is of class $C^{2}$ on $\mathbb{R}^{dN}$ with $D\phi^{\perp}(x)=\Pi^{\perp}x=x^{\perp}$ and $D^{2}\phi^{\perp}(x)=\Pi^{\perp}$. So, by The Squared Wasserstein Distance to a Fixed Measure and Functions of the Mean are Intrinsic Test Functions §mean at the configuration level, $\Phi^{\perp}(P)=\phi^{\perp}(M(P))=\tfrac12\lVert M(P)^{\perp}\rVert^{2}$ defines an intrinsic test function on $\mathcal{D}_{N}$ with $\nabla\Phi^{\perp}(P)=M(P)^{\perp}$ ($P\in\mathcal{D}_{N}$) and $H_{\Phi^{\perp}}(P)=\Pi^{\perp}$; it is continuous by property (a).

(2c) Functions of the block average. Let $\chi:\mathbb{R}^{d}\to\mathbb{R}$ be of class $C^{2}$ on $\mathbb{R}^{d}$ and $g_{\chi}(x)=\chi(\bar{x})$ for $x\in\mathbb{R}^{dN}$. The $s$th coordinate of $\bar{x}$ is $e_{s}\cdot\bar{x}=(N^{-1}e_{s}^{\oplus})\cdot x$ by (0a) and Real Matrices, Symmetric Matrices and the Semidefinite Ordering: Standing Notation §basis, of class $C^{2}$ on $\mathbb{R}^{dN}$ by Quadratic and Affine Functions of Class $C^2$, Translation, and Quadratic Perturbation of Semiconvexity §quadratic (matrix $0_{dN}$, linear term $N^{-1}e_{s}^{\oplus}$); so $x\mapsto\bar{x}$ is of class $C^{2}$ by claim 1 of Coordinate Functions, the $C^k$ Hierarchy, and Partial Derivatives of a Smooth Map, and $g_{\chi}$ 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 $C^k$, $\mathbb{R}^{d}$ and $\mathbb{R}^{dN}$ being open (claim 1 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous). Fix $x\in\mathbb{R}^{dN}$ and put $p'=N^{-1}(D\chi(\bar{x}))^{\oplus}$ and $B'=Z_{0,D^{2}\chi(\bar{x})}\in\mathcal{S}(dN)$ (Step 0; $D^{2}\chi(\bar{x})\in\mathcal{S}(d)$ by Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation §derivatives). For $h\in\mathbb{R}^{dN}$, $p'\cdot h=D\chi(\bar{x})\cdot\bar{h}$ by (0a) and $h\cdot(B'h)=\bar{h}\cdot(D^{2}\chi(\bar{x})\bar{h})$ by (0e). Let $\varepsilon>0$. By Basic Properties of Twice Differentiability at a Point §c2, $\chi$ is twice differentiable at $\bar{x}$ with first-order coefficient $D\chi(\bar{x})$ and Hessian $D^{2}\chi(\bar{x})$, so there is $\delta'>0$ such that every $k\in\mathbb{R}^{d}$ with $\lVert k\rVert<\delta'$ satisfies $|\chi(\bar{x}+k)-\chi(\bar{x})-D\chi(\bar{x})\cdot k-\tfrac12k\cdot(D^{2}\chi(\bar{x})k)|\le\varepsilon\lVert k\rVert^{2}$. If $\lVert h\rVert<\delta'$ then $\lVert\bar{h}\rVert\le\lVert h\rVert<\delta'$ and $\overline{x+h}=\bar{x}+\bar{h}$ (Step 0), so

$$\Bigl|g_{\chi}(x+h)-g_{\chi}(x)-p'\cdot h-\tfrac12h\cdot(B'h)\Bigr|\le\varepsilon\lVert\bar{h}\rVert^{2}\le\varepsilon\lVert h\rVert^{2}.$$

Thus $g_{\chi}$ is twice differentiable at $x$ with first-order coefficient $p'$ and Hessian $B'$ (Twice Differentiability at a Point §twice-differentiable); it is also so with $Dg_{\chi}(x)$ and $D^{2}g_{\chi}(x)$ by Basic Properties of Twice Differentiability at a Point §c2, so A Symmetric Matrix is Determined by its Quadratic Form, and a Second-Order Expansion by its Coefficients §uniqueness gives

$$Dg_{\chi}(x)=N^{-1}\bigl(D\chi(\bar{x})\bigr)^{\oplus},\qquad D^{2}g_{\chi}(x)=Z_{0,D^{2}\chi(\bar{x})},\qquad h\cdot\bigl(D^{2}g_{\chi}(x)h\bigr)=\bar{h}\cdot\bigl(D^{2}\chi(\bar{x})\bar{h}\bigr).$$

By The Squared Wasserstein Distance to a Fixed Measure and Functions of the Mean are Intrinsic Test Functions §mean at the configuration level, $P\mapsto\chi(\bar{m}(P))=g_{\chi}(M(P))$ is an intrinsic test function on $\mathcal{D}_{N}$, with gradient $N^{-1}(D\chi(\bar{m}(P)))^{\oplus}$ at $P\in\mathcal{D}_{N}$ and translation Hessian $Z_{0,D^{2}\chi(\bar{m}(P))}$.

(2d) The pulled-back function $\tilde{A}$. By Pulling Back Intrinsic Test Functions along Tensor Powers §test, with $\mathcal{Q}=\mathcal{D}$, $\mathcal{Q}_{N}=\mathcal{D}_{N}$ (the hypothesis $\mu^{\otimes N}\in\mathcal{D}_{N}$ for $\mu\in\mathcal{D}$ being the one required there) and $\Phi=A_{N}$, $\tilde{A}$ is an intrinsic test function on $\mathcal{D}$ with $\nabla\tilde{A}(\mu)=N\,\Pi_{\mu^{\otimes N}}(\nabla A_{N}(\mu^{\otimes N}))$ for $\mu\in\mathcal{D}$; it is continuous by property (a). By Pulling Back Intrinsic Test Functions along Tensor Powers §hessian and (2a), $a\cdot(H_{\tilde{A}}(\mu)a)=a^{\oplus}\cdot(0_{dN}a^{\oplus})=0=a\cdot(0_{d}a)$ for all $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ and $a\in\mathbb{R}^{d}$ (Matrix-Vector Product, claim 7 of Properties of Finite Sums), so $H_{\tilde{A}}(\mu)=0_{d}$ by A Symmetric Matrix is Determined by its Quadratic Form, and a Second-Order Expansion by its Coefficients §polarization.

(2e) Projection of a constant field at a tensor power. Let $\sigma\in\mathcal{P}_{2}(\mathbb{R}^{d})$, $Q=\sigma^{\otimes N}$ and $\beta\in\mathbb{R}^{dN}$; then $Q^{[1]}=\sigma$ (Tensor Powers and One-Particle Marginals: Particle Laws, Product Integrals, Push-Forwards, Moments, Product Maps and Diagonal Shifts §marginal-of-tensor), and we show $\Pi_{Q}(\beta)=\bar{\beta}$. First, $\bar{\beta}\in T_{\sigma}$ 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 §constants. Let $\psi\in C_{c}^{\infty}(\mathbb{R}^{d})$. The product field $(\nabla\psi)^{\oplus}$ is the class of the product map of $\nabla\psi$ (Product Fields and the Projection onto One-Particle Tangent Fields §product-field), whose $k$th block at $x$ is $\nabla\psi(\mathfrak{p}_{k}(x))$ (Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §product-map), so $\beta\cdot(\nabla\psi)^{\oplus}(x)=\sum_{k=1}^{N}f_{k}(\mathfrak{p}_{k}(x))$ with $f_{k}(y)=\mathfrak{p}_{k}(\beta)\cdot\nabla\psi(y)$, by Particle Blocks: Linearity, Splitting of Inner Products, Product Maps and Diagonal Shifts §inner-product. Each $f_{k}$ is Borel and integrable against $\sigma$, being the pointwise dot product of two members of $L^{2}(\sigma;\mathbb{R}^{d})$ (Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation §fields); as $(\mathfrak{p}_{k})_{\#}Q=\sigma$ (Tensor Powers and One-Particle Marginals: Particle Laws, Product Integrals, Push-Forwards, Moments, Product Maps and Diagonal Shifts §particle-laws), the change-of-variables formula of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward makes $f_{k}\circ\mathfrak{p}_{k}$ integrable against $Q$ with $\int f_{k}\circ\mathfrak{p}_{k}\,dQ=\int f_{k}\,d\sigma$. By Linearity of the Lebesgue Integral over a Finite Sum of Integrable Functions (twice) and the bilinearity of the dot product,

$$\bigl\langle\beta,(\nabla\psi)^{\oplus}\bigr\rangle_{Q}=\sum_{k=1}^{N}\int f_{k}\,d\sigma=\int_{\mathbb{R}^{d}}s(\beta)\cdot\nabla\psi(y)\,\sigma(dy)=N\,\langle\bar{\beta},\nabla\psi\rangle_{\sigma}.$$

Hence $\langle\bar{\beta},\nabla\psi\rangle_{\sigma}=N^{-1}\langle\beta,(\nabla\psi)^{\oplus}\rangle_{Q}$ for every $\psi$, and the uniqueness in Product Fields and the Projection onto One-Particle Tangent Fields §projection gives $\Pi_{Q}(\beta)=\bar{\beta}$.

Step 3 (Compactness). We show:

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

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

For (K-u): by (1a), $\ell\le b-\delta\mathcal{E}_{N}(P_{n})$, so $\mathcal{E}_{N}(P_{n})\le c_{1}$ with $c_{1}=\delta^{-1}(b-\ell)$ (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 $\{P'\in\mathcal{D}_{N}:\mathcal{E}_{N}(P')\le c_{1}\}$ is sequentially compact (Wasserstein-Coercive Penalty Pairs §coercive at the configuration level), so there are a strictly increasing $(n_{j})_{j}$ and a point $P$ of that set, hence of $\mathcal{D}_{N}$, with $P_{n_{j}}\to P$ (Sequentially Compact Subset of a Metric Space). By (0c), $\lVert\bar{m}(P_{n_{j}})-\bar{m}(P)\rVert\le W(P_{n_{j}},P)$, so $\bar{m}(P_{n_{j}})\to\bar{m}(P)$; also $\bar{m}(P_{n_{j}})\to\zeta$ by A Subsequence of a Convergent Sequence Has the Same Limit, so $\bar{m}(P)=\zeta$ 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 $\mathcal{D}_{N}$ relative to $\mathcal{D}_{N}$ by Basic Properties of the Delta-Envelopes on the Wasserstein Space §semicontinuity, both at the configuration level. With the continuity of $A_{N}$ and $\Phi^{\perp}$ (Step 2), and as $\Theta=U^{-}_{\delta}-\alpha A_{N}-\alpha\Phi^{\perp}$ on $\mathcal{D}_{N}$, there is a positive $r$ such that every $P'\in\mathcal{D}_{N}$ with $W(P',P)<r$ satisfies $U^{-}_{\delta}(P')<U^{-}_{\delta}(P)+\tfrac{\varepsilon}{2}$, $|A_{N}(P')-A_{N}(P)|<\tfrac{\varepsilon}{4}\alpha^{-1}$ and $|\Phi^{\perp}(P')-\Phi^{\perp}(P)|<\tfrac{\varepsilon}{4}\alpha^{-1}$, where $\tfrac{\varepsilon}{4}=\tfrac12\cdot\tfrac{\varepsilon}{2}$ (the least of three radii, claim 9 of Elementary Order Arithmetic in an Ordered Field twice), hence $\Theta(P')<\Theta(P)+\varepsilon$ by claims 3 and 8 of Elementary Order Arithmetic in an Ordered Field and claim 3 of Properties of the Absolute Value in an Ordered Field. As $W(P_{n_{j}},P)<r$ for all large $j$, (K-u) follows. (K-v) is proved in the same way: (1a) gives $\mathcal{E}(\mu_{n})\le(N\delta)^{-1}(\ell'-Nb')$; the means converge 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; $v^{+}_{\delta}$ is lower semicontinuous on $\mathcal{D}$ relative to $\mathcal{D}$ by the second part of Basic Properties of the Delta-Envelopes on the Wasserstein Space §semicontinuity, $v$ 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 $\tilde{A}$ is continuous (2d), so a radius $r$ with $v^{+}_{\delta}(\mu)-\tfrac{\varepsilon}{2}N^{-1}<v^{+}_{\delta}(\mu')$ and $|\tilde{A}(\mu')-\tilde{A}(\mu)|<\tfrac{\varepsilon}{2}\alpha^{-1}$ for $\mu'\in\mathcal{D}$ with $W(\mu',\mu)<r$ gives $\Xi(\mu)-\varepsilon<\Xi(\mu')$.

Step 4 (The fibre functions). For $\zeta\in\mathbb{R}^{d}$ let $\mathcal{D}_{N}^{\zeta}=\{P\in\mathcal{D}_{N}:\bar{m}(P)=\zeta\}$ and $\mathcal{D}^{\zeta}=\{\mu\in\mathcal{D}:m(\mu)=\zeta\}$. Both are nonempty: $(\tau_{(\zeta-\hat{\zeta})^{\oplus}})_{\#}\hat{P}\in\mathcal{D}_{N}$ and $(\tau_{\zeta-\hat{\omega}})_{\#}\hat{\mu}\in\mathcal{D}$ by Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §translation (at the configuration level for the first), and by (0c) 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 their block average of the mean, respectively mean, is $\hat{\zeta}+(\zeta-\hat{\zeta})=\zeta$, respectively $\hat{\omega}+(\zeta-\hat{\omega})=\zeta$. By (1a) the set $\{\Theta(P):P\in\mathcal{D}_{N}^{\zeta}\}$ is bounded above and $\{\Xi(\mu):\mu\in\mathcal{D}^{\zeta}\}$ bounded below; let $\mathcal{U}(\zeta)\in\mathbb{R}$ be the supremum of the first and $\mathcal{V}(\zeta)\in\mathbb{R}$ the infimum of the second (Approximation Property of the Supremum and the Infimum in $\mathbb{R}$).

(4a) 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 $P_{n}\in\mathcal{D}_{N}^{\zeta}$ with $\mathcal{U}(\zeta)-1/n<\Theta(P_{n})$ for each $n$; then $\mathcal{U}(\zeta)-1\le\Theta(P_{n})$, as $1/n\le1$. (K-u), with the constant sequence $\zeta$ of block averages and $\ell=\mathcal{U}(\zeta)-1$, gives $(n_{j})_{j}$ and $P\in\mathcal{D}_{N}^{\zeta}$. For $\varepsilon>0$ take $j$ so large that $\Theta(P_{n_{j}})<\Theta(P)+\varepsilon$ 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 $\mathcal{U}(\zeta)<\Theta(P)+2\varepsilon$. By Comparison of Real Numbers with Arbitrary Positive Slack §slack-above, $\mathcal{U}(\zeta)\le\Theta(P)\le\mathcal{U}(\zeta)$, so $\Theta(P)=\mathcal{U}(\zeta)$. 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, there is $\mu\in\mathcal{D}^{\zeta}$ with $\Xi(\mu)=\mathcal{V}(\zeta)$.

(4b) $\mathcal{U}$ is upper and $\mathcal{V}$ 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 $\mathcal{U}$ 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 $\mathcal{U}(\zeta)+\varepsilon\le\mathcal{U}(\zeta_{n})$; so $\zeta_{n}\to\zeta$. By (4a) choose $P_{n}\in\mathcal{D}_{N}^{\zeta_{n}}$ with $\Theta(P_{n})=\mathcal{U}(\zeta_{n})$. (K-u) with $\ell=\mathcal{U}(\zeta)+\varepsilon$ gives $P\in\mathcal{D}_{N}^{\zeta}$ and, for large $j$, $\mathcal{U}(\zeta)+\varepsilon\le\Theta(P_{n_{j}})<\Theta(P)+\tfrac{\varepsilon}{2}\le\mathcal{U}(\zeta)+\tfrac{\varepsilon}{2}$, which is impossible as $\tfrac{\varepsilon}{2}<\varepsilon$ (claim 8 of Elementary Order Arithmetic in an Ordered Field). The lower semicontinuity of $\mathcal{V}$ follows in the same way from (4a) and (K-v).

Step 5 (Ishii's lemma in the particle dimension). Let $\zeta,\omega\in\mathbb{R}^{d}$ and, by (4a), $P\in\mathcal{D}_{N}^{\zeta}$ and $\mu\in\mathcal{D}^{\omega}$ with $\Theta(P)=\mathcal{U}(\zeta)$ and $\Xi(\mu)=\mathcal{V}(\omega)$. By (1b),

$$\mathcal{U}(\zeta)-\mathcal{V}(\omega)-\tfrac{N\alpha}{2}\lVert\zeta-\omega\rVert^{2}\le M_{0}.\qquad(5\mathrm{a})$$

Since $\hat{P}\in\mathcal{D}_{N}^{\hat{\zeta}}$ and $\hat{\mu}\in\mathcal{D}^{\hat{\omega}}$, $\Theta(\hat{P})\le\mathcal{U}(\hat{\zeta})$ and $\mathcal{V}(\hat{\omega})\le\Xi(\hat{\mu})$, so (1c) and (5a) give

$$\mathcal{U}(\hat{\zeta})-\mathcal{V}(\hat{\omega})-\tfrac{N\alpha}{2}\lVert\hat{\zeta}-\hat{\omega}\rVert^{2}=M_{0}.\qquad(5\mathrm{b})$$

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 $\mathcal{U}$ and $\mathcal{V}$ (Step 4), the positive number $N\alpha$ (claim 5 of Elementary Order Arithmetic in an Ordered Field) in place of its $\alpha$, $\hat{x}=\hat{\zeta}$, $\hat{y}=\hat{\omega}$ and radius $1$: its hypothesis holds by (5a) and (5b) at all points, $\tfrac{N\alpha}{2}$ being the product of $N\alpha$ with the inverse of $2$. Let $\mathbb{X}_{I},\mathbb{Y}_{I}\in\mathcal{S}(d)$ be the matrices it provides and $p=N\alpha(\hat{\zeta}-\hat{\omega})$. By its claims, $\mathbb{X}_{I}\preceq\mathbb{Y}_{I}$ (Ishii's Lemma: Test Data and Matrix Bounds at a Maximum of a Quadratically Penalised Difference §ordering), $\lVert\mathbb{X}_{I}\rVert\le6N\alpha$ and $\lVert\mathbb{Y}_{I}\rVert\le6N\alpha$ (Ishii's Lemma: Test Data and Matrix Bounds at a Maximum of a Quadratically Penalised Difference §norm-bound), and

$$-3N\alpha\bigl(\lVert\xi\rVert^{2}+\lVert\eta\rVert^{2}\bigr)\le\xi\cdot(\mathbb{X}_{I}\xi)-\eta\cdot(\mathbb{Y}_{I}\eta)\le3N\alpha\lVert\xi-\eta\rVert^{2}\qquad(\xi,\eta\in\mathbb{R}^{d})\qquad(5\mathrm{c})$$

(Ishii's Lemma: Test Data and Matrix Bounds at a Maximum of a Quadratically Penalised Difference §quadratic-bound); and by Ishii's Lemma: Test Data and Matrix Bounds at a Maximum of a Quadratically Penalised Difference §test-data, $(\hat{\zeta},\mathcal{U}(\hat{\zeta}),p,\mathbb{X}_{I})$ is approximable by test data from above for $\mathcal{U}$ and $(\hat{\omega},\mathcal{V}(\hat{\omega}),p,\mathbb{Y}_{I})$ from below for $\mathcal{V}$, with open set $\mathbb{R}^{d}$.

Step 6 (The matrices; claim 2). Put $\mathbb{X}=Z_{\alpha,\mathbb{X}_{I}}$ and $\mathbb{Y}_{N}=Z_{3\alpha,\mathbb{Y}_{I}}$, members of $\mathcal{S}(dN)$ by Step 0, and $\mathbb{Y}=N^{-1}\mathbb{Y}_{I}\in\mathcal{S}(d)$ (Real Matrices, Symmetric Matrices and the Semidefinite Ordering: Standing Notation §symmetric). By (0e), for $z,w\in\mathbb{R}^{dN}$,

$$z\cdot(\mathbb{X}z)=\alpha\lVert z^{\perp}\rVert^{2}+\bar{z}\cdot(\mathbb{X}_{I}\bar{z}),\qquad w\cdot(\mathbb{Y}_{N}w)=3\alpha\lVert w^{\perp}\rVert^{2}+\bar{w}\cdot(\mathbb{Y}_{I}\bar{w}).$$

Ordering. As $\alpha\le3\alpha$ and $\bar{z}\cdot(\mathbb{X}_{I}\bar{z})\le\bar{z}\cdot(\mathbb{Y}_{I}\bar{z})$ (The Positive Semidefinite Ordering on Symmetric Matrices), $z\cdot(\mathbb{X}z)\le z\cdot(\mathbb{Y}_{N}z)$ for every $z$, that is $\mathbb{X}\preceq\mathbb{Y}_{N}$. Norms. By claim 2 of Properties of the Norm of a Symmetric Real Matrix, $|\bar{z}\cdot(\mathbb{X}_{I}\bar{z})|\le6N\alpha\lVert\bar{z}\rVert^{2}$ and likewise for $\mathbb{Y}_{I}$, so by (0b) and claim 3 of Properties of the Absolute Value in an Ordered Field

$$|z\cdot(\mathbb{X}z)|\le\alpha\lVert z^{\perp}\rVert^{2}+6\alpha N\lVert\bar{z}\rVert^{2}\le6\alpha\lVert z\rVert^{2},\qquad|z\cdot(\mathbb{Y}_{N}z)|\le3\alpha\lVert z^{\perp}\rVert^{2}+6\alpha N\lVert\bar{z}\rVert^{2}\le6\alpha\lVert z\rVert^{2}.$$

As $z\cdot((\pm6\alpha)I_{dN}z)=\pm6\alpha\lVert z\rVert^{2}$ (Vector, Entry and Comparison Bounds for the Norm of a Symmetric Real Matrix §identity), this is $-6\alpha I_{dN}\preceq\mathbb{X}\preceq6\alpha I_{dN}$ and the same for $\mathbb{Y}_{N}$, so $\lVert\mathbb{X}\rVert\le6\alpha$ and $\lVert\mathbb{Y}_{N}\rVert\le6\alpha$ by claim 3 of Properties of the Norm of a Symmetric Real Matrix, $6\alpha$ being nonnegative. Quadratic bounds. Let $z,w\in\mathbb{R}^{dN}$ and write $z\cdot(\mathbb{X}z)-w\cdot(\mathbb{Y}_{N}w)=T_{\perp}+T_{I}$ with $T_{\perp}=\alpha\lVert z^{\perp}\rVert^{2}-3\alpha\lVert w^{\perp}\rVert^{2}$ and $T_{I}=\bar{z}\cdot(\mathbb{X}_{I}\bar{z})-\bar{w}\cdot(\mathbb{Y}_{I}\bar{w})$. By (5c), $-3N\alpha(\lVert\bar{z}\rVert^{2}+\lVert\bar{w}\rVert^{2})\le T_{I}\le3N\alpha\lVert\bar{z}-\bar{w}\rVert^{2}$. For $a=z^{\perp}$ and $b=w^{\perp}$, expanding the squares gives

$$3\lVert a-b\rVert^{2}-\lVert a\rVert^{2}+3\lVert b\rVert^{2}=2\lVert a\rVert^{2}-6\,a\cdot b+6\lVert b\rVert^{2}=\tfrac12\bigl(\lVert2a-3b\rVert^{2}+3\lVert b\rVert^{2}\bigr)\ge0,$$

so $T_{\perp}\le3\alpha\lVert z^{\perp}-w^{\perp}\rVert^{2}$; and $T_{\perp}\ge-3\alpha\lVert w^{\perp}\rVert^{2}\ge-3\alpha(\lVert z^{\perp}\rVert^{2}+\lVert w^{\perp}\rVert^{2})$. Since $z^{\perp}-w^{\perp}=(z-w)^{\perp}$ and $\bar{z}-\bar{w}=\overline{z-w}$ (Step 0), (0b) gives

$$-3\alpha\bigl(\lVert z\rVert^{2}+\lVert w\rVert^{2}\bigr)\le z\cdot(\mathbb{X}z)-w\cdot(\mathbb{Y}_{N}w)\le3\alpha\bigl(\lVert(z-w)^{\perp}\rVert^{2}+N\lVert\overline{z-w}\rVert^{2}\bigr)=3\alpha\lVert z-w\rVert^{2}.$$

Hence $(\mathbb{X},\mathbb{Y}_{N})$ is admitted at $\alpha$ at the configuration level (The Second-Order Structure Condition at Optimally Coupled Pairs on the Lift of the Wasserstein Space §admitted in $\mathcal{S}(dN)$). Finally, for $a\in\mathbb{R}^{d}$, $(a^{\oplus})^{\perp}=0$ and $\overline{a^{\oplus}}=a$ (Step 0), so $a^{\oplus}\cdot(\mathbb{Y}_{N}a^{\oplus})=a\cdot(\mathbb{Y}_{I}a)=N\,a\cdot(\mathbb{Y}a)$, the last because $a\cdot(\mathbb{Y}a)=N^{-1}\,a\cdot(\mathbb{Y}_{I}a)$ by claim 1 of Linearity of the Matrix-Vector Product and the Quadratic Form as a Double Sum. This is claim 2.

Step 7 (Test functions at fibre maximisers and minimisers). 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 $\mathcal{U}(\zeta)-\chi_{n}(\zeta)\le\mathcal{U}(\zeta_{n})-\chi_{n}(\zeta_{n})$ whenever $d_{E}(\zeta,\zeta_{n})<r_{n}$, and

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

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 (4a) choose $\rho_{n}\in\mathcal{D}_{N}^{\zeta_{n}}$ with $\Theta(\rho_{n})=\mathcal{U}(\zeta_{n})$, and let $\varphi_{n}=\alpha A_{N}+\alpha\Phi^{\perp}+G_{n}$, where $G_{n}(P)=\chi_{n}(\bar{m}(P))$. By (2a), (2b), (2c) and Restrictions, Sums, Real Multiples and Differences of Intrinsic Test Functions on the Wasserstein Space §linear (twice, at the configuration level), $\varphi_{n}$ is an intrinsic test function on $\mathcal{D}_{N}$ at the configuration level, with

$$\nabla\varphi_{n}(\rho_{n})=\alpha\nabla A_{N}(\rho_{n})+\alpha M(\rho_{n})^{\perp}+N^{-1}\bigl(D\chi_{n}(\zeta_{n})\bigr)^{\oplus},\qquad H_{\varphi_{n}}(\rho_{n})=\alpha0_{dN}+\alpha\Pi^{\perp}+Z_{0,D^{2}\chi_{n}(\zeta_{n})}.$$

For $h\in\mathbb{R}^{dN}$, claim 1 of Linearity of the Matrix-Vector Product and the Quadratic Form as a Double Sum, (0e) and Step 6 give $h\cdot(H_{\varphi_{n}}(\rho_{n})h)=\alpha\lVert h^{\perp}\rVert^{2}+\bar{h}\cdot(D^{2}\chi_{n}(\zeta_{n})\bar{h})$ and hence $h\cdot((H_{\varphi_{n}}(\rho_{n})-\mathbb{X})h)=\bar{h}\cdot((D^{2}\chi_{n}(\zeta_{n})-\mathbb{X}_{I})\bar{h})$; with $\lambda_{n}=\lVert D^{2}\chi_{n}(\zeta_{n})-\mathbb{X}_{I}\rVert$, claim 2 of Properties of the Norm of a Symmetric Real Matrix and $\lVert\bar{h}\rVert\le\lVert h\rVert$ bound its absolute value by $\lambda_{n}\lVert h\rVert^{2}$, so, as in Step 6, claim 3 of that lemma gives

$$\lVert H_{\varphi_{n}}(\rho_{n})-\mathbb{X}\rVert\le\lambda_{n}<\tfrac1n.$$

For $P'\in\mathcal{D}_{N}$ with $W(P',\rho_{n})<r_{n}$ we have $d_{E}(\bar{m}(P'),\zeta_{n})\le W(P',\rho_{n})<r_{n}$ by (0c), so, by the definition of $\mathcal{U}$ and $P'\in\mathcal{D}_{N}^{\bar{m}(P')}$,

$$U^{-}_{\delta}(P')-\varphi_{n}(P')=\Theta(P')-\chi_{n}(\bar{m}(P'))\le\mathcal{U}(\bar{m}(P'))-\chi_{n}(\bar{m}(P'))\le\mathcal{U}(\zeta_{n})-\chi_{n}(\zeta_{n})=U^{-}_{\delta}(\rho_{n})-\varphi_{n}(\rho_{n}).$$

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

Symmetrically, Quadruple Approximable by Test-Function Data §below with $\varepsilon=1/n$ gives $\omega_{n}$, $\chi'_{n}$ of class $C^{2}$ on $\mathbb{R}^{d}$ and $r'_{n}>0$ with $\mathcal{V}(\omega)-\chi'_{n}(\omega)\ge\mathcal{V}(\omega_{n})-\chi'_{n}(\omega_{n})$ whenever $d_{E}(\omega,\omega_{n})<r'_{n}$, and the four bounds with $\omega_{n},\hat{\omega},\mathcal{V},\chi'_{n},\mathbb{Y}_{I}$ in place of $\zeta_{n},\hat{\zeta},\mathcal{U},\chi_{n},\mathbb{X}_{I}$. Choose $\sigma_{n}\in\mathcal{D}^{\omega_{n}}$ with $\Xi(\sigma_{n})=\mathcal{V}(\omega_{n})$ and let $\psi_{n}=(-\alpha N^{-1})\tilde{A}+N^{-1}(\chi'_{n}\circ m)$. By (2d), 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, $\psi_{n}$ is an intrinsic test function on $\mathcal{D}$ with

$$\nabla\psi_{n}(\sigma_{n})=-\alpha N^{-1}\nabla\tilde{A}(\sigma_{n})+N^{-1}D\chi'_{n}(\omega_{n}),\qquad H_{\psi_{n}}(\sigma_{n})=(-\alpha N^{-1})0_{d}+N^{-1}D^{2}\chi'_{n}(\omega_{n})=N^{-1}D^{2}\chi'_{n}(\omega_{n}),$$

so, the matrix identity $N^{-1}D^{2}\chi'_{n}(\omega_{n})-N^{-1}\mathbb{Y}_{I}=N^{-1}(D^{2}\chi'_{n}(\omega_{n})-\mathbb{Y}_{I})$ holding entrywise (Real Matrices, Symmetric Matrices and the Semidefinite Ordering: Standing Notation §matrices), claim 5 of Properties of the Norm of a Symmetric Real Matrix and $N^{-1}\le1$ give $\lVert H_{\psi_{n}}(\sigma_{n})-\mathbb{Y}\rVert=N^{-1}\lVert D^{2}\chi'_{n}(\omega_{n})-\mathbb{Y}_{I}\rVert<\tfrac1n$. For $\sigma'\in\mathcal{D}$ with $W(\sigma',\sigma_{n})<r'_{n}$, $d_{E}(m(\sigma'),\omega_{n})\le W(\sigma',\sigma_{n})<r'_{n}$ (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 $N\,v^{+}_{\delta}(\sigma')=\Xi(\sigma')-\alpha\tilde{A}(\sigma')$, so, multiplying by the positive $N^{-1}$ (claim 5 of Elementary Arithmetic in an Ordered Field),

$$v^{+}_{\delta}(\sigma')-\psi_{n}(\sigma')=N^{-1}\bigl(\Xi(\sigma')-\chi'_{n}(m(\sigma'))\bigr)\ge N^{-1}\bigl(\mathcal{V}(m(\sigma'))-\chi'_{n}(m(\sigma'))\bigr)\ge N^{-1}\bigl(\mathcal{V}(\omega_{n})-\chi'_{n}(\omega_{n})\bigr)=v^{+}_{\delta}(\sigma_{n})-\psi_{n}(\sigma_{n}),$$

and $v^{+}_{\delta}-\psi_{n}$ has a local minimum relative to $\mathcal{D}$ at $\sigma_{n}$ (Local Minimum of a Function Relative to a Subset of a Metric Space).

Step 8 (The limits $\rho^{*}$, $\sigma^{*}$; claim 1). We have $\bar{m}(\rho_{n})=\zeta_{n}\to\hat{\zeta}$ and $\Theta(\rho_{n})=\mathcal{U}(\zeta_{n})>\mathcal{U}(\hat{\zeta})-1/n\ge\mathcal{U}(\hat{\zeta})-1$. (K-u) gives a strictly increasing $(n_{j})_{j}$ and $\rho^{*}\in\mathcal{D}_{N}^{\hat{\zeta}}$ with $\rho_{n_{j}}\to\rho^{*}$; exactly as in (4a), $\Theta(\rho^{*})=\mathcal{U}(\hat{\zeta})$, and then $|\Theta(\rho_{n_{j}})-\Theta(\rho^{*})|=|\mathcal{U}(\zeta_{n_{j}})-\mathcal{U}(\hat{\zeta})|<1/n_{j}$. Symmetrically, (K-v) gives $(n'_{j})_{j}$ and $\sigma^{*}\in\mathcal{D}^{\hat{\omega}}$ with $\sigma_{n'_{j}}\to\sigma^{*}$, $\Xi(\sigma^{*})=\mathcal{V}(\hat{\omega})$ and $|\Xi(\sigma_{n'_{j}})-\Xi(\sigma^{*})|<1/n'_{j}$. Put $Q^{*}=(\sigma^{*})^{\otimes N}\in\mathcal{D}_{N}$. By (1b) and (5b),

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

which is claim 1; and equality in the first inequality means, by Step 1, that equality holds in The Displacement Midpoint and the Midpoint Split of the Squared Wasserstein Distance §split for $(\rho^{*},Q^{*})$, hence also for $(Q^{*},\rho^{*})$, 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 $\mathbb{R}^n$. Moreover $\bar{m}(\rho^{*})=\hat{\zeta}$ and $m(\sigma^{*})=\hat{\omega}$, so $M(Q^{*})=\hat{\omega}^{\oplus}$ by (0c) and $M(\rho^{*})=M(\rho^{*})^{\perp}+\hat{\zeta}^{\oplus}$.

Step 9 (The limiting gradients). As $\mathcal{D}_{N}$ has the map property and $\rho^{*},Q^{*}\in\mathcal{D}_{N}$, the pairs $(\rho^{*},\hat{M})$, $(\rho^{*},Q^{*})$, $(Q^{*},\hat{M})$ and $(Q^{*},\rho^{*})$ are uniquely mapped (The Map Property of a Set of Probability Measures §map-property at the configuration level). By The Displacement Midpoint and the Midpoint Split of the Squared Wasserstein Distance §equality at the configuration level, for $(\rho^{*},Q^{*})$ with the optimal maps $G_{\rho^{*}}$ and $S$, $\mathrm{id}-G_{\rho^{*}}=\tfrac12(\mathrm{id}-S)+e$ in $L^{2}(\rho^{*};\mathbb{R}^{dN})$ with $e=\tfrac12(M(\rho^{*})+M(Q^{*}))-c$; so (2a) gives $\nabla A_{N}(\rho^{*})=(\mathrm{id}-S)+2e-2(M(\rho^{*})-c)=(\mathrm{id}-S)-(M(\rho^{*})-M(Q^{*}))$. Likewise, for $(Q^{*},\rho^{*})$ with $G_{Q^{*}}$ and $S'$, $\nabla A_{N}(Q^{*})=(\mathrm{id}-S')+(M(\rho^{*})-M(Q^{*}))$ in $L^{2}(Q^{*};\mathbb{R}^{dN})$. With Step 8 and $N^{-1}p^{\oplus}=\alpha(\hat{\zeta}-\hat{\omega})^{\oplus}=\alpha\hat{\zeta}^{\oplus}-\alpha\hat{\omega}^{\oplus}$,

$$\alpha\nabla A_{N}(\rho^{*})+\alpha M(\rho^{*})^{\perp}+N^{-1}p^{\oplus}=\alpha(\mathrm{id}-S)-\alpha\bigl(M(\rho^{*})-\hat{\omega}^{\oplus}\bigr)+\alpha\bigl(M(\rho^{*})-\hat{\zeta}^{\oplus}\bigr)+\alpha\bigl(\hat{\zeta}^{\oplus}-\hat{\omega}^{\oplus}\bigr)=\alpha(\mathrm{id}-S)\qquad(9\mathrm{a})$$

in $L^{2}(\rho^{*};\mathbb{R}^{dN})$. On the other side, $\Pi_{Q^{*}}$ is linear (The Projection onto One-Particle Tangent Fields: Pairing with Product Fields, Contraction, and Product Fields of Tangent Fields §contraction) and $\Pi_{Q^{*}}(M(\rho^{*})-\hat{\omega}^{\oplus})=\overline{M(\rho^{*})-\hat{\omega}^{\oplus}}=\hat{\zeta}-\hat{\omega}$ by (2e) and (0a); so, by (2d), $\nabla\tilde{A}(\sigma^{*})=N\,\Pi_{Q^{*}}(\mathrm{id}-S')+N(\hat{\zeta}-\hat{\omega})$, and, as $N^{-1}p=\alpha(\hat{\zeta}-\hat{\omega})$ and $S'-\mathrm{id}=-(\mathrm{id}-S')$,

$$-\alpha N^{-1}\nabla\tilde{A}(\sigma^{*})+N^{-1}p=-\alpha\,\Pi_{Q^{*}}(\mathrm{id}-S')=\alpha\,\Pi_{Q^{*}}(S'-\mathrm{id})\qquad\text{in }L^{2}(\sigma^{*};\mathbb{R}^{d}).\qquad(9\mathrm{b})$$

Step 10 (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). By property (c) of Intrinsic Test Functions on the Wasserstein Space and Their Translation Hessians §test for $A_{N}$ on $\mathcal{D}_{N}$ at the configuration level, the discrepancy $D_{j}$ of $\nabla A_{N}(\rho_{n_{j}})$ and $\nabla A_{N}(\rho^{*})$ 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 (9a) and Step 7 its integrand may be taken to be $\lVert F_{1}(z)+F_{2}(z)\rVert^{2}$ with

$$F_{1}(z)=\alpha\bigl(\nabla A_{N}(\rho_{n_{j}})(x)-\nabla A_{N}(\rho^{*})(y)\bigr),\qquad F_{2}(z)=\beta_{j}=\alpha\bigl(M(\rho_{n_{j}})-M(\rho^{*})\bigr)^{\perp}+N^{-1}\bigl(D\chi_{n_{j}}(\zeta_{n_{j}})-p\bigr)^{\oplus},$$

where the linearity of $x\mapsto x^{\perp}$ and $a\mapsto a^{\oplus}$ was used. By Step 0 and (0c), $\lVert\beta_{j}\rVert\le\alpha W(\rho_{n_{j}},\rho^{*})+\lVert D\chi_{n_{j}}(\zeta_{n_{j}})-p\rVert<\alpha W(\rho_{n_{j}},\rho^{*})+1/n_{j}$ (claim 6 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$). Both fields 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 $\lVert F_{2}\rVert_{\pi_{j}}=\lVert\beta_{j}\rVert$, so the triangle inequality The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §triangle in the real Hilbert space $L^{2}(\pi_{j};\mathbb{R}^{dN})$ (Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation §fields) gives $\sqrt{E_{j}}\le\alpha\sqrt{D_{j}}+\lVert\beta_{j}\rVert$. Moreover

$$U^{-}_{\delta}(\rho_{n_{j}})-U^{-}_{\delta}(\rho^{*})=\bigl(\Theta(\rho_{n_{j}})-\Theta(\rho^{*})\bigr)+\alpha\bigl(A_{N}(\rho_{n_{j}})-A_{N}(\rho^{*})\bigr)+\alpha\bigl(\Phi^{\perp}(\rho_{n_{j}})-\Phi^{\perp}(\rho^{*})\bigr),$$

where the first difference has absolute value below $1/n_{j}$ (Step 8) and the other two converge to $0$ since $A_{N}$ and $\Phi^{\perp}$ are continuous (Step 2, 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 $\varepsilon>0$; put $\tfrac{\varepsilon}{4}=\tfrac12\cdot\tfrac{\varepsilon}{2}$. Each of the real sequences $(I(\pi_{j}))_{j}$, $(\sqrt{D_{j}})_{j}$ (claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field), $(1/n_{j})_{j}$, $(W(\rho_{n_{j}},\rho^{*}))_{j}$, $(A_{N}(\rho_{n_{j}})-A_{N}(\rho^{*}))_{j}$ and $(\Phi^{\perp}(\rho_{n_{j}})-\Phi^{\perp}(\rho^{*}))_{j}$ converges to $0$, so we may fix $j$ with $I(\pi_{j})<\varepsilon^{2}$, $1/n_{j}<\tfrac{\varepsilon}{4}$, $\alpha W(\rho_{n_{j}},\rho^{*})<\tfrac{\varepsilon}{4}$, $\alpha\sqrt{D_{j}}<\tfrac{\varepsilon}{2}$, $\alpha|A_{N}(\rho_{n_{j}})-A_{N}(\rho^{*})|<\tfrac{\varepsilon}{4}$ and $\alpha|\Phi^{\perp}(\rho_{n_{j}})-\Phi^{\perp}(\rho^{*})|<\tfrac{\varepsilon}{4}$. Put $\rho=\rho_{n_{j}}$, $\varphi=\varphi_{n_{j}}$ and $\pi=\pi_{j}$. By Step 7, $U^{-}_{\delta}-\varphi$ has a local maximum relative to $\mathcal{D}_{N}$ at $\rho$; $|U^{-}_{\delta}(\rho)-U^{-}_{\delta}(\rho^{*})|<\tfrac{\varepsilon}{4}+\tfrac{\varepsilon}{4}+\tfrac{\varepsilon}{4}<\varepsilon$ by claim 5 of Properties of the Absolute Value in an Ordered Field and claims 3 and 8 of Elementary Order Arithmetic in an Ordered Field; $\sqrt{E_{j}}<\tfrac{\varepsilon}{2}+\tfrac{\varepsilon}{4}+\tfrac{\varepsilon}{4}=\varepsilon$, 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 Step 7. This is claim 3.

Step 11 (Claim 4). The same argument gives claim 4, with $\sigma_{n'_{j}}$, $\sigma^{*}$, optimal couplings $\gamma_{j}\in\Pi(\sigma_{n'_{j}},\sigma^{*})$, the discrepancy $D'_{j}$ of $\nabla\tilde{A}(\sigma_{n'_{j}})$ and $\nabla\tilde{A}(\sigma^{*})$ along $\gamma_{j}$, which converges to $0$ by property (c) of Intrinsic Test Functions on the Wasserstein Space and Their Translation Hessians §test for $\tilde{A}$ on $\mathcal{D}$ (2d), the test functions $\psi_{n'_{j}}$ and the local minima of Step 7, the identity (9b), and the fields

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

for which $\lVert F_{1}\rVert_{\gamma_{j}}=\alpha N^{-1}\sqrt{D'_{j}}\le\alpha\sqrt{D'_{j}}$ and $\lVert F_{2}\rVert_{\gamma_{j}}\le\lVert D\chi'_{n'_{j}}(\omega_{n'_{j}})-p\rVert<1/n'_{j}$, as $0<N^{-1}\le1$; with the identity

$$v^{+}_{\delta}(\sigma_{n'_{j}})-v^{+}_{\delta}(\sigma^{*})=N^{-1}\Bigl(\bigl(\Xi(\sigma_{n'_{j}})-\Xi(\sigma^{*})\bigr)-\alpha\bigl(\tilde{A}(\sigma_{n'_{j}})-\tilde{A}(\sigma^{*})\bigr)\Bigr),$$

whose first difference has absolute value below $1/n'_{j}$ (Step 8) and whose second converges to $0$ since $\tilde{A}$ is continuous ((2d), Intrinsic Test Functions on the Wasserstein Space and Their Translation Hessians §continuity) and $\sigma_{n'_{j}}\to\sigma^{*}$, by Continuity Between Metric Spaces is Equivalent to Sequential Continuity; and with $\lVert H_{\psi_{n'_{j}}}(\sigma_{n'_{j}})-\mathbb{Y}\rVert<1/n'_{j}$ from Step 7. The integral in claim 4 is the discrepancy of $\nabla\psi_{n'_{j}}(\sigma_{n'_{j}})$ and $\alpha\,\Pi_{Q^{*}}(S'-\mathrm{id})$ along $\gamma_{j}$, whose integrand may be taken to be $\lVert F_{1}(z)+F_{2}(z)\rVert^{2}$ by (9b), Step 7 and The Discrepancy of Two Square-Integrable Vector Fields Along a Coupling of Their Base Measures §well-defined.