Proof of Intrinsic Test Functions at a Maximiser of the Wasserstein-Doubled Difference

Each result cited is universally quantified over the data in its own statement. We write $W$ for $W_{2}$, $1/n$ for the multiplicative inverse of the positive real $\iota(n)$ attached to $n\in\mathbb{N}$ (The Real Numbers: Standing Notation and Background (i)), and, as in The Squared Wasserstein Distance to a Fixed Measure and Functions of the Mean are Intrinsic Test Functions, a vector $a\in\mathbb{R}^{d}$ also for the class of the constant map with value $a$ in any $L^{2}(\rho;\mathbb{R}^{d})$. The real sequence $(1/n)_{n\in\mathbb{N}}$ converges to $0$ by The Archimedean Property of the Real Numbers. Convergence in $\mathbb{R}^{d}$ is convergence in the metric space $(\mathbb{R}^{d},d_{E})$, where $d_{E}(a,a')=\lVert a-a'\rVert$ by claim 2 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square.

Step 1 (The midpoint and two auxiliary functions). 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 $\hat{\zeta}=m(\hat{\mu})$, $\hat{\omega}=m(\hat{\nu})$ and $M=\Psi(\hat{\mu},\hat{\nu})$, 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 (i), and $0\le A(\rho)$ for every $\rho$ by The Displacement Midpoint and the Midpoint Split of the Squared Wasserstein Distance (ii). Fix $e_{0}\in\mathbb{R}$ with $e_{0}\le\mathcal{E}(\rho)$ for every $\rho\in\mathcal{D}$ (Basic Properties of a Wasserstein-Coercive Penalty Pair (ii)). For $\rho\in\mathcal{D}$ put

$$\Theta(\rho)=u^{-}_{\delta}(\rho)-\alpha A(\rho),\qquad\Xi(\rho)=v^{+}_{\delta}(\rho)+\alpha A(\rho).$$

As $u^{-}_{\delta}=u-\delta\mathcal{E}$ and $v^{+}_{\delta}=v+\delta\mathcal{E}$ on $\mathcal{D}$, $u\le b$, $b'\le v$, and $0\le\alpha A(\rho)$ and $\delta e_{0}\le\delta\mathcal{E}(\rho)$ by claim 5 of Elementary Arithmetic in an Ordered Field, for every $\rho\in\mathcal{D}$

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

For $\rho,\sigma\in\mathcal{D}$ one has $\Psi(\rho,\sigma)-\bigl(\Theta(\rho)-\Xi(\sigma)-\tfrac{\alpha}{2}\lVert m(\rho)-m(\sigma)\rVert^{2}\bigr)=\tfrac{\alpha}{2}\bigl(2A(\rho)+2A(\sigma)+\lVert m(\rho)-m(\sigma)\rVert^{2}-W(\rho,\sigma)^{2}\bigr)$, which is nonnegative by The Displacement Midpoint and the Midpoint Split of the Squared Wasserstein Distance (iii) and vanishes exactly when equality holds there, $\tfrac{\alpha}{2}$ 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 $(\hat{\mu},\hat{\nu})$,

$$\Theta(\rho)-\Xi(\sigma)-\tfrac{\alpha}{2}\lVert m(\rho)-m(\sigma)\rVert^{2}\le\Psi(\rho,\sigma)\le M\qquad(\rho,\sigma\in\mathcal{D}),\qquad(1\mathrm{b})$$

with equality in the first inequality exactly when equality holds in The Displacement Midpoint and the Midpoint Split of the Squared Wasserstein Distance (iii) for $(\rho,\sigma)$. By The Displacement Midpoint and the Midpoint Split of the Squared Wasserstein Distance (iv) this is so for $(\hat{\mu},\hat{\nu})$:

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

Step 2 ($A$ 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, applied 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 (i) (with $\nu_{0}=\hat{m}$), The Squared Wasserstein Distance to a Fixed Measure and Functions of the Mean are Intrinsic Test Functions (ii) (with $\phi=q_{c}$) and Restrictions, Sums, Real Multiples and Differences of Intrinsic Test Functions on the Wasserstein Space (iii) show that $A$ is an intrinsic test function on $\mathcal{D}$, with

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

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 (i), $A$ is continuous on $\mathcal{P}_{2}(\mathbb{R}^{d})$.

Step 3 (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 $(\rho_{n_{j}})_{j}$ converges 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) The same with $\ell'\in\mathbb{R}$, $\Xi(\sigma_{n})\le\ell'$ for every $n$ in place of the lower bound, and the conclusion $\Xi(\sigma)-\varepsilon<\Xi(\sigma_{n_{j}})$ for all sufficiently large $j$.

For (K-u): by (1a), $\ell\le b-\delta\mathcal{E}(\rho_{n})$, so $\mathcal{E}(\rho_{n})\le c_{1}$ 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 $\{\rho'\in\mathcal{D}:\mathcal{E}(\rho')\le c_{1}\}$ is sequentially compact (Wasserstein-Coercive Penalty Pairs (i)), 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 (i), $\lVert m(\rho_{n_{j}})-m(\rho)\rVert\le W(\rho_{n_{j}},\rho)$, 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 $u\le b$, $u$ is bounded above near each point (Upper and Lower Semicontinuous Envelopes of a Real-Valued Function (i)), so $u^{-}_{\delta}$ is upper semicontinuous on $\mathcal{D}$ relative to $\mathcal{D}$ by Basic Properties of the Delta-Envelopes on the Wasserstein Space (i); with the continuity of $A$ (Step 2) there is a positive $r$ such that every $\rho'\in\mathcal{D}$ with $W(\rho',\rho)<r$ satisfies $u^{-}_{\delta}(\rho')<u^{-}_{\delta}(\rho)+\tfrac{\varepsilon}{2}$ and $|A(\rho')-A(\rho)|<\tfrac{\varepsilon}{2}\alpha^{-1}$ (the least of two radii, claim 9 of Elementary Order Arithmetic in an Ordered Field), hence $\Theta(\rho')<\Theta(\rho)+\varepsilon$ by claim 3 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: (1a) gives $\mathcal{E}(\sigma_{n})\le\delta^{-1}(\ell'-b')$, and $v^{+}_{\delta}$ 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 (i), applied to $v$, which is bounded below near each point since $b'\le v$.

Step 4 (The fibre functions). For $\zeta\in\mathbb{R}^{d}$ let $\mathcal{D}_{\zeta}=\{\rho\in\mathcal{D}:m(\rho)=\zeta\}$. It is nonempty: $(\tau_{\zeta-\hat{\zeta}})_{\#}\hat{\mu}\in\mathcal{D}$ by Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains (iv), and its mean is $\hat{\zeta}+(\zeta-\hat{\zeta})=\zeta$ by The Mean of a Square-Integrable Probability Measure, Its Lift, Its Centring, and Functions of the Mean and Centred Integrals as Test Functions (i). By (1a) the set $\{\Theta(\rho):\rho\in\mathcal{D}_{\zeta}\}$ is bounded above and $\{\Xi(\sigma):\sigma\in\mathcal{D}_{\zeta}\}$ bounded below; let $U(\zeta)\in\mathbb{R}$ be the supremum of the first and $V(\zeta)\in\mathbb{R}$ 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 $\rho_{n}\in\mathcal{D}_{\zeta}$ 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 (i), $U(\zeta)\le\Theta(\rho)\le U(\zeta)$, 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 (ii), there is $\sigma\in\mathcal{D}_{\zeta}$ with $\Xi(\sigma)=V(\zeta)$.

(4b) $U$ 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 (v). 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 (4a) 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 (4a) and (K-v).

Step 5 (Ishii's lemma on the means; claim 2). Let $\zeta,\omega\in\mathbb{R}^{d}$ and, by (4a), $\rho\in\mathcal{D}_{\zeta}$ and $\sigma\in\mathcal{D}_{\omega}$ with $\Theta(\rho)=U(\zeta)$ and $\Xi(\sigma)=V(\omega)$. By (1b),

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

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

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

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), the functions $U$ and $V$ (Step 4), $\hat{x}=\hat{\zeta}$, $\hat{y}=\hat{\omega}$ and radius $1$: its hypothesis holds by (5a) and (5b) at all points. Let $\mathbb{X},\mathbb{Y}\in\mathcal{S}(d)$ be the matrices it provides and $p=\alpha(\hat{\zeta}-\hat{\omega})$. Its claims Ishii's Lemma: Test Data and Matrix Bounds at a Maximum of a Quadratically Penalised Difference (iv), Ishii's Lemma: Test Data and Matrix Bounds at a Maximum of a Quadratically Penalised Difference (v) and Ishii's Lemma: Test Data and Matrix Bounds at a Maximum of a Quadratically Penalised Difference (iii) are the four conditions of The Second-Order Structure Condition at Optimally Coupled Pairs on the Lift of the Wasserstein Space (i); this is claim 2. By Ishii's Lemma: Test Data and Matrix Bounds at a Maximum of a Quadratically Penalised Difference (i), $(\hat{\zeta},U(\hat{\zeta}),p,\mathbb{X})$ is approximable by test data from above for $U$ and $(\hat{\omega},V(\hat{\omega}),p,\mathbb{Y})$ from below for $V$, with open set $\mathbb{R}^{d}$.

Step 6 (Test functions at fibre maximisers and minimisers). Let $n\in\mathbb{N}$. By Quadruple Approximable by Test-Function Data (i) 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

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

the last because $d_{\mathcal{S}(d)}(P,P')=\lVert P-P'\rVert$ (Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation (iii)). By (4a) choose $\rho_{n}\in\mathcal{D}_{\zeta_{n}}$ with $\Theta(\rho_{n})=U(\zeta_{n})$, and let $\varphi_{n}=\alpha A+\chi_{n}\circ m$. By Step 2, The Squared Wasserstein Distance to a Fixed Measure and Functions of the Mean are Intrinsic Test Functions (ii) and Restrictions, Sums, Real Multiples and Differences of Intrinsic Test Functions on the Wasserstein Space (ii), $\varphi_{n}$ is an intrinsic test function on $\mathcal{D}$ with

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

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 (i), so, by the definition of $U$ and $\rho'\in\mathcal{D}_{m(\rho')}$,

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

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

Symmetrically, Quadruple Approximable by Test-Function Data (ii) with $\varepsilon=1/n$ gives $\omega_{n}$, $\chi'_{n}$ 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 the four bounds with $\omega_{n},\hat{\omega},V,\chi'_{n},\mathbb{Y}$ in place of $\zeta_{n},\hat{\zeta},U,\chi_{n},\mathbb{X}$. Choose $\sigma_{n}\in\mathcal{D}_{\omega_{n}}$ with $\Xi(\sigma_{n})=V(\omega_{n})$ and let $\psi_{n}=(-\alpha)A+\chi'_{n}\circ m$, an intrinsic test function on $\mathcal{D}$ with $\nabla\psi_{n}(\sigma_{n})=-\alpha\nabla A(\sigma_{n})+D\chi'_{n}(\omega_{n})$ and $H_{\psi_{n}}(\sigma_{n})=D^{2}\chi'_{n}(\omega_{n})$. For $\sigma'\in\mathcal{D}$ with $W(\sigma',\sigma_{n})<r'_{n}$, $v^{+}_{\delta}(\sigma')-\psi_{n}(\sigma')=\Xi(\sigma')-\chi'_{n}(m(\sigma'))\ge V(m(\sigma'))-\chi'_{n}(m(\sigma'))\ge V(\omega_{n})-\chi'_{n}(\omega_{n})=v^{+}_{\delta}(\sigma_{n})-\psi_{n}(\sigma_{n})$, so $v^{+}_{\delta}-\psi_{n}$ 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 7 (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 (4a), $\Theta(\rho^{*})=U(\hat{\zeta})$, and then $|\Theta(\rho_{n_{j}})-\Theta(\rho^{*})|=|U(\zeta_{n_{j}})-U(\hat{\zeta})|<1/n_{j}$. Symmetrically, (K-v) gives $(n'_{j})_{j}$ and $\sigma^{*}\in\mathcal{D}_{\hat{\omega}}$ with $\sigma_{n'_{j}}\to\sigma^{*}$, $\Xi(\sigma^{*})=V(\hat{\omega})$ and $|\Xi(\sigma_{n'_{j}})-\Xi(\sigma^{*})|<1/n'_{j}$. By (1b) and (5b),

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

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 (iii) for $(\rho^{*},\sigma^{*})$, hence also for $(\sigma^{*},\rho^{*})$, both sides of that inequality being symmetric in the pair by The Quadratic Wasserstein Distance is a Metric on the Wasserstein Space (i) and claim 5 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square.

Step 8 (The limiting gradients). As $\mathcal{D}$ has the map property and $\rho^{*},\sigma^{*}\in\mathcal{D}$, the pairs $(\rho^{*},\hat{m})$, $(\rho^{*},\sigma^{*})$, $(\sigma^{*},\hat{m})$ and $(\sigma^{*},\rho^{*})$ are uniquely mapped (The Map Property of a Set of Probability Measures (i)). By The Displacement Midpoint and the Midpoint Split of the Squared Wasserstein Distance (v) for $(\rho^{*},\sigma^{*})$ 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 $(\sigma^{*},\rho^{*})$ with $G_{\sigma^{*}}$ and $S'$, $2(\mathrm{id}-G_{\sigma^{*}})=\mathrm{id}-S'$ in $L^{2}(\sigma^{*};\mathbb{R}^{d})$. As $2(\hat{\zeta}-c)=\hat{\zeta}-\hat{\omega}$ and $2(\hat{\omega}-c)=\hat{\omega}-\hat{\zeta}$, Step 2 gives $\nabla A(\rho^{*})=(\mathrm{id}-S)-(\hat{\zeta}-\hat{\omega})$ and $\nabla A(\sigma^{*})=(\mathrm{id}-S')-(\hat{\omega}-\hat{\zeta})$, hence, with $p=\alpha(\hat{\zeta}-\hat{\omega})$,

$$\alpha\nabla A(\rho^{*})+p=\alpha(\mathrm{id}-S)\ \text{in }L^{2}(\rho^{*};\mathbb{R}^{d}),\qquad-\alpha\nabla A(\sigma^{*})+p=\alpha(S'-\mathrm{id})\ \text{in }L^{2}(\sigma^{*};\mathbb{R}^{d}).\qquad(8\mathrm{a})$$

Step 9 (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 (i)). By property (c) of Intrinsic Test Functions on the Wasserstein Space and Their Translation Hessians (i) 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 (i)), so by (8a) and Step 6 its integrand may be taken to be $\lVert F_{1}(z)+F_{2}(z)\rVert^{2}$ with

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

Both are Borel and square-integrable against $\pi_{j}$, with $\lVert F_{1}\rVert_{\pi_{j}}=\alpha\sqrt{D_{j}}$ (claim 5 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square) and $\lVert F_{2}\rVert_{\pi_{j}}=\lVert D\chi_{n_{j}}(\zeta_{n_{j}})-p\rVert<1/n_{j}$, so the triangle inequality The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity (i) 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 (ii)) gives $\sqrt{E_{j}}<\alpha\sqrt{D_{j}}+1/n_{j}$. Moreover $u^{-}_{\delta}(\rho_{n_{j}})-u^{-}_{\delta}(\rho^{*})=\bigl(\Theta(\rho_{n_{j}})-\Theta(\rho^{*})\bigr)+\alpha\bigl(A(\rho_{n_{j}})-A(\rho^{*})\bigr)$, where the first difference has absolute value below $1/n_{j}$ (Step 7) and the second converges to $0$ by the continuity of $A$.

Let $\varepsilon>0$. 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), $(1/n_{j})_{j}$ 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 6, $u^{-}_{\delta}-\varphi$ 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 $\lVert H_{\varphi}(\rho)-\mathbb{X}\rVert=\lVert D^{2}\chi_{n_{j}}(\zeta_{n_{j}})-\mathbb{X}\rVert<1/n_{j}<\varepsilon$. This is claim 3.

Step 10 (Claim 4). The same argument, with $\sigma_{n'_{j}}$, $\sigma^{*}$, optimal couplings $\gamma_{j}\in\Pi(\sigma_{n'_{j}},\sigma^{*})$, $\psi_{n'_{j}}$, $\Xi$, $v^{+}_{\delta}=\Xi-\alpha A$, the second identity of (8a), the fields $F_{1}(z)=-\alpha\bigl(\nabla A(\sigma_{n'_{j}})(x)-\nabla A(\sigma^{*})(y)\bigr)$ and $F_{2}(z)=D\chi'_{n'_{j}}(\omega_{n'_{j}})-p$, the local minimum of Step 6 and $H_{\psi_{n'_{j}}}(\sigma_{n'_{j}})=D^{2}\chi'_{n'_{j}}(\omega_{n'_{j}})$, gives claim 4.