Proof of The Structure Estimate at a Maximiser of the Wasserstein-Doubled Difference on the Lift

Each result cited is universally quantified over the data in its own statement, and is applied to the data named here.

Step 1. The doubled difference on the lift. Write $A=\mathcal{D}^{\Lambda}$ and let $\hat{u},\hat{v}:A\to\mathbb{R}$ be the functions with values $\hat{u}(X)=u^{-}_{\delta}(\mathcal{L}(X))$ and $\hat{v}(X)=v^{+}_{\delta}(\mathcal{L}(X))$, defined because $\mathcal{L}(X)\in\mathcal{D}$ for $X\in A$ by Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation (iv); let $-\hat{v}:A\to\mathbb{R}$ have as value at $X$ the additive inverse of $\hat{v}(X)$. Let $L^{2}(\Omega;\mathbb{R}^{d})\times L^{2}(\Omega;\mathbb{R}^{d})$ be the product of the real Hilbert space $L^{2}(\Omega;\mathbb{R}^{d})$ with itself, a real Hilbert space by Properties of the Product of Two Real Inner Product Spaces (vi), with norm $|\cdot|$ and distance $d_{\times}$.

Read The Doubled Difference on the Lift of a Wasserstein-Coercive Penalty Pair: Bounds, Closed Superlevel Sets and the Least Penalty with the present $\delta$, $u$, $v$, $b$, $b'$ and $e_{0}$. Its claim 2 gives that $A$ is nonempty and that

$$\hat{u}(X)\le b-\delta e_{0},\qquad b'+\delta e_{0}\le\hat{v}(X)\qquad(X\in A),$$

the second because $-\hat{v}(X)\le-b'-\delta e_{0}$; its claim 3 gives that $\hat{u}$ and $-\hat{v}$ have closed superlevel sets in $L^{2}(\Omega;\mathbb{R}^{d})$.

Let $\Phi_{0}:A\times A\to\mathbb{R}$ have the value $\Phi_{0}(X,Y)=\hat{u}(X)-\hat{v}(Y)-\tfrac{\alpha}{2}\lVert X-Y\rVert_{L^{2}}^{2}$. This is the function of claim 4 of The Doubled Difference on the Lift of a Wasserstein-Coercive Penalty Pair: Bounds, Closed Superlevel Sets and the Least Penalty with the nonnegative multiplier there taken to be $0$ and with $q=(\hat{X},\hat{Y})$, the omitted term vanishing by claim 1 of Zero Products and Elementary Identities in a Field; so $\Phi_{0}$ is bounded above by $b-b'-2\delta e_{0}$ and has closed superlevel sets in the product space.

By Existence, Penalty Bounds and Optimal Realisation at a Maximiser of the Wasserstein-Doubled Difference on the Lift (iv), applied to the maximiser $(\hat{\mu},\hat{\nu})$ and to $\hat{X},\hat{Y}$,

$$\Phi_{0}(X,Y)\ \le\ u^{-}_{\delta}(\hat{\mu})-v^{+}_{\delta}(\hat{\nu})-\tfrac{\alpha}{2}\lVert\hat{X}-\hat{Y}\rVert_{L^{2}}^{2}\qquad(X,Y\in A).$$

Since $\lVert\hat{X}-\hat{Y}\rVert_{L^{2}}=W_{2}(\hat{\mu},\hat{\nu})$ the right-hand side is $\Psi_{\delta,\alpha}(\hat{\mu},\hat{\nu})=M(\delta,\alpha)$, and it equals $\Phi_{0}(\hat{X},\hat{Y})$ because $\mathcal{L}(\hat{X})=\hat{\mu}$ and $\mathcal{L}(\hat{Y})=\hat{\nu}$. Hence the supremum of the values of $\Phi_{0}$ is $M(\delta,\alpha)$, attained at $(\hat{X},\hat{Y})$. The same identity reads $\lVert\hat{X}-\hat{Y}\rVert_{L^{2}}^{2}=W_{2}(\mathcal{L}(\hat{X}),\mathcal{L}(\hat{Y}))^{2}$, so $(\hat{X},\hat{Y})$ is optimally coupled by Optimally Coupled Pairs of Square-Integrable Random Vectors (i).

Step 2. A sequentially strict maximum nearby. For $n\in\mathbb{N}$ let $\epsilon_{n}$ be the quotient of $1$ by $n$, so that $0<\epsilon_{n}\le1$, and let $\epsilon_{n}^{2}=\epsilon_{n}\epsilon_{n}$ and $\epsilon_{n}^{3}=\epsilon_{n}\epsilon_{n}\epsilon_{n}$. The sequence whose $n$-th term is $\epsilon_{n}$ converges to $0$: given positive $\varepsilon\in\mathbb{R}$, The Archimedean Property of the Real Numbers provides $N\in\mathbb{N}$ with $\varepsilon^{-1}<N$, and for $n\in\mathbb{N}$ with $N\le n$ one has $\varepsilon^{-1}<n$, whence $\epsilon_{n}<\varepsilon$ on multiplying by the positive $\varepsilon\epsilon_{n}$ (claim 10 of Elementary Order Arithmetic in an Ordered Field); and $|\epsilon_{n}-0|=\epsilon_{n}$ by claim 1 of Properties of the Absolute Value in an Ordered Field.

Fix $n\in\mathbb{N}$ and apply A Perturbed Maximum Principle of Borwein-Preiss Type in a Real Hilbert Space in the real Hilbert space $L^{2}(\Omega;\mathbb{R}^{d})\times L^{2}(\Omega;\mathbb{R}^{d})$ to the nonempty set $A\times A$ and the function $\Phi_{0}$, with both of the positive parameters written $\mu$ and $\lambda$ there taken equal to $\epsilon_{n}$ and with $x_{0}=(\hat{X},\hat{Y})$; the hypothesis on $x_{0}$ holds because $\Phi_{0}$ attains its supremum there and $0<\epsilon_{n}\epsilon_{n}^{2}$. We obtain $\bar{q}_{n}$ in the product space and $(\bar{X}_{n},\bar{Y}_{n})\in A\times A$ such that, writing $\Psi_{n}:A\times A\to\mathbb{R}$ for the function with value $\Phi_{0}(X,Y)-\epsilon_{n}\,|(X,Y)-\bar{q}_{n}|^{2}$,

$$\bigl|\bar{q}_{n}-(\hat{X},\hat{Y})\bigr|\le4\epsilon_{n},\qquad\bigl|(\bar{X}_{n},\bar{Y}_{n})-(\hat{X},\hat{Y})\bigr|\le4\epsilon_{n},\qquad\bigl|(\bar{X}_{n},\bar{Y}_{n})-\bar{q}_{n}\bigr|\le8\epsilon_{n}$$

by its clause 1, that $\Psi_{n}$ attains a sequentially strict maximum on $A\times A$ at $(\bar{X}_{n},\bar{Y}_{n})$ by its clause 2, and that

$$M(\delta,\alpha)\ \le\ \Phi_{0}(\bar{X}_{n},\bar{Y}_{n})+2\epsilon_{n}^{3}$$

by its clause 3 together with Step 1.

Write $\bar{q}_{n}=(q^{1}_{n},q^{2}_{n})$. By Properties of the Product of Two Real Inner Product Spaces (i) differences in the product space are formed coordinatewise, and by Properties of the Product of Two Real Inner Product Spaces (ii)

$$\bigl|(X,Y)-\bar{q}_{n}\bigr|^{2}=\lVert X-q^{1}_{n}\rVert_{L^{2}}^{2}+\lVert Y-q^{2}_{n}\rVert_{L^{2}}^{2},$$

and each coordinate norm is at most the norm of the pair, so that

$$\lVert\bar{X}_{n}-\hat{X}\rVert_{L^{2}}\le4\epsilon_{n},\quad\lVert\bar{Y}_{n}-\hat{Y}\rVert_{L^{2}}\le4\epsilon_{n},\quad\lVert\bar{X}_{n}-q^{1}_{n}\rVert_{L^{2}}\le8\epsilon_{n},\quad\lVert\bar{Y}_{n}-q^{2}_{n}\rVert_{L^{2}}\le8\epsilon_{n}.$$

For $i\in[2]$ let $g^{i}_{n}:L^{2}(\Omega;\mathbb{R}^{d})\to\mathbb{R}$ have the value $\epsilon_{n}\lVert Z-q^{i}_{n}\rVert_{L^{2}}^{2}$ at $Z$. By claim 3 of Affine and Quadratic Functions on a Real Hilbert Space are of Class $C^2$, read with the multiplier $2\epsilon_{n}$ and the point $q^{i}_{n}$, the function $g^{i}_{n}$ is of class $C^{2}$ on $L^{2}(\Omega;\mathbb{R}^{d})$, hence continuous there by Basic Properties of Differentiability on an Open Subset of a Real Inner Product Space (ii). Let $u_{n},v_{n}:A\to\mathbb{R}$ have values $u_{n}(X)=\hat{u}(X)-g^{1}_{n}(X)$ and $v_{n}(Y)=\hat{v}(Y)+g^{2}_{n}(Y)$, so that $\Psi_{n}(X,Y)=u_{n}(X)-v_{n}(Y)-\tfrac{\alpha}{2}\lVert X-Y\rVert_{L^{2}}^{2}$. The values of $g^{1}_{n}$ and $g^{2}_{n}$ are nonnegative, so $u_{n}(X)\le b-\delta e_{0}$ and $-v_{n}(Y)\le-b'-\delta e_{0}$; and by Functions with Closed Superlevel Sets: Sequential Characterisation, Semicontinuity, Perturbation and Limits (iii), applied to $\hat{u}$ with $g^{1}_{n}$ and to $-\hat{v}$ with $g^{2}_{n}$, both $u_{n}$ and $-v_{n}$ have closed superlevel sets in $L^{2}(\Omega;\mathbb{R}^{d})$.

Step 3. Lions' lemma and the compression. Let $\gamma=(c_{e_{1}},\dots,c_{e_{d}})$ be the constant tuple, orthonormal by that clause, and let $N$ be the tail form it determines. By The Constant Tuple on $L^{2}(\Omega;\mathbb{R}^{d})$: Coordinates, Projection, Tail Form, and Translation-Closed Preimages (vi) the set $A$ is translation-closed along $\gamma$. Apply Lions' Lemma on a Hilbert Space over a Translation-Closed Set: Test Data and Form Bounds at a Sequentially Strict Maximum of a Quadratically Penalised Difference, read with $L^{2}(\Omega;\mathbb{R}^{d})$ in place of the space written $H$ there, with $d$ in place of the dimension written $m$, with $\gamma$ in place of the tuple written $e$, with the set $A$, with $u_{n}$ and $v_{n}$ in place of the functions written $u$ and $v$, and with the present $\alpha$; its hypotheses were verified in Step 2. Writing $p_{n}=\alpha(\bar{X}_{n}-\bar{Y}_{n})$ and $\mathrm{Sym}$ for the set of bounded symmetric bilinear forms on $L^{2}(\Omega;\mathbb{R}^{d})$, it yields $\mathbb{A}_{n},\mathbb{B}_{n}\in\mathrm{Sym}$ such that, by its clause 1, the quadruple $(\bar{X}_{n},u_{n}(\bar{X}_{n}),p_{n},\mathbb{A}_{n}+2\alpha N)$ is approximable by test data from above for $u_{n}$ on $A$ and the quadruple $(\bar{Y}_{n},v_{n}(\bar{Y}_{n}),p_{n},\mathbb{B}_{n}-2\alpha N)$ is approximable by test data from below for $v_{n}$ on $A$, while by its clauses 2, 3 and 4 the forms $\mathbb{A}_{n}$ and $\mathbb{B}_{n}$ satisfy $\mathbb{A}_{n}\preceq\mathbb{B}_{n}$, $\lVert\mathbb{A}_{n}\rVert\le6\alpha$, $\lVert\mathbb{B}_{n}\rVert\le6\alpha$ and

$$-3\alpha\bigl(\lVert Z\rVert_{L^{2}}^{2}+\lVert W\rVert_{L^{2}}^{2}\bigr)\ \le\ \mathbb{A}_{n}(Z,Z)-\mathbb{B}_{n}(W,W)\ \le\ 3\alpha\lVert Z-W\rVert_{L^{2}}^{2}$$

for all $Z,W\in L^{2}(\Omega;\mathbb{R}^{d})$.

Write $b^{\flat}$ for the compression along $\gamma$ of $b\in\mathrm{Sym}$. By the displayed bounds and Lifted Test Data from Second-Order Data on the Space of Square-Integrable Random Vectors: Compression along the Constant Tuple (vi) the pair $(\mathbb{A}_{n}^{\flat},\mathbb{B}_{n}^{\flat})$ is admitted at $\alpha$; in particular $\lVert\mathbb{A}_{n}^{\flat}\rVert\le6\alpha$ and $\lVert\mathbb{B}_{n}^{\flat}\rVert\le6\alpha$, these bounds being part of what that clause asserts.

Since $u_{n}=\hat{u}-g^{1}_{n}$ and $v_{n}=\hat{v}+g^{2}_{n}$, parts (a) and (b) of claim 3 of Lifted Test Data from Second-Order Data on the Space of Square-Integrable Random Vectors: Compression along the Constant Tuple, read with the multiplier $\epsilon_{n}$ and the points $q^{1}_{n}$ and $q^{2}_{n}$, convert the two quadruples above into: the quadruple

$$\bigl(\bar{X}_{n},\ \hat{u}(\bar{X}_{n}),\ p^{u}_{n},\ \mathbb{A}_{n}+2\alpha N+2\epsilon_{n}I\bigr),\qquad p^{u}_{n}=p_{n}+2\epsilon_{n}(\bar{X}_{n}-q^{1}_{n}),$$

is approximable by test data from above for $\hat{u}$ on $A$, and the quadruple

$$\bigl(\bar{Y}_{n},\ \hat{v}(\bar{Y}_{n}),\ p^{v}_{n},\ \mathbb{B}_{n}-2\alpha N-2\epsilon_{n}I\bigr),\qquad p^{v}_{n}=p_{n}-2\epsilon_{n}(\bar{Y}_{n}-q^{2}_{n}),$$

is approximable by test data from below for $\hat{v}$ on $A$, where $I$ is the identity form of $\mathrm{Sym}$. By Compression of a Bounded Symmetric Bilinear Form along an Orthonormal Tuple (ii) and Compression of a Bounded Symmetric Bilinear Form along an Orthonormal Tuple (iii),

$$\bigl(\mathbb{A}_{n}+2\alpha N+2\epsilon_{n}I\bigr)^{\flat}=\mathbb{A}_{n}^{\flat}+2\epsilon_{n}I_{d},\qquad\bigl(\mathbb{B}_{n}-2\alpha N-2\epsilon_{n}I\bigr)^{\flat}=\mathbb{B}_{n}^{\flat}-2\epsilon_{n}I_{d}.$$

Apply claim 2 of Lifted Test Data from Second-Order Data on the Space of Square-Integrable Random Vectors: Compression along the Constant Tuple to each of these two quadruples, with $\epsilon_{n}$ in the role of the positive number written $\varepsilon$ there. There are $X_{n},Y_{n}\in A$ and lifted test functions $\Phi_{n}$ and $\Xi_{n}$ such that $\hat{u}-\Phi_{n}$ has a local maximum at $X_{n}$ relative to $A$, $\hat{v}-\Xi_{n}$ has a local minimum at $Y_{n}$ relative to $A$, and

$$\lVert X_{n}-\bar{X}_{n}\rVert_{L^{2}}<\epsilon_{n},\quad\bigl|\hat{u}(X_{n})-\hat{u}(\bar{X}_{n})\bigr|<\epsilon_{n},\quad\lVert D\Phi_{n}(X_{n})-p^{u}_{n}\rVert_{L^{2}}<\epsilon_{n},\quad\bigl\lVert H_{\Phi_{n}}(X_{n})-\mathbb{A}_{n}^{\flat}-2\epsilon_{n}I_{d}\bigr\rVert<\epsilon_{n},$$ $$\lVert Y_{n}-\bar{Y}_{n}\rVert_{L^{2}}<\epsilon_{n},\quad\bigl|\hat{v}(Y_{n})-\hat{v}(\bar{Y}_{n})\bigr|<\epsilon_{n},\quad\lVert D\Xi_{n}(Y_{n})-p^{v}_{n}\rVert_{L^{2}}<\epsilon_{n},\quad\bigl\lVert H_{\Xi_{n}}(Y_{n})-\mathbb{B}_{n}^{\flat}+2\epsilon_{n}I_{d}\bigr\rVert<\epsilon_{n}.$$

Step 4. The viscosity inequalities. The function on $\mathcal{D}^{\Lambda}$ with value $u^{-}_{\delta}(\mathcal{L}(X))-\Phi_{n}(X)$ at $X$ is $\hat{u}-\Phi_{n}$, which has a local maximum at $X_{n}\in\mathcal{D}^{\Lambda}$ relative to $\mathcal{D}^{\Lambda}$; so Viscosity Subsolution, Supersolution and Solution on the Lift of the Wasserstein Space (i), applied with $\Phi_{n}$, with $X_{n}$ and with $\epsilon_{n}$ in the role of the positive number written $\varepsilon$ there, provides $Z_{n}\in\mathcal{D}_{\Sigma}^{\Lambda}$, $V_{n}\in L^{2}(\Omega;\mathbb{R}^{d})$, $s_{n}\in\mathbb{R}$ and $\mathbb{X}_{n}\in\mathcal{S}(d)$ with

$$\lVert Z_{n}-X_{n}\rVert_{L^{2}}<\epsilon_{n},\quad\bigl|u^{-}_{\delta}(\mathcal{L}(Z_{n}))-\hat{u}(X_{n})\bigr|<\epsilon_{n},\quad\bigl|s_{n}-\hat{u}(X_{n})\bigr|<\epsilon_{n},$$ $$\lVert V_{n}-D\Phi_{n}(X_{n})\rVert_{L^{2}}<\epsilon_{n},\quad\bigl\lVert\mathbb{X}_{n}-H_{\Phi_{n}}(X_{n})\bigr\rVert<\epsilon_{n},\quad F^{-}_{\delta}(Z_{n},s_{n},V_{n},\mathbb{X}_{n})\le\epsilon_{n}.$$

Dually Viscosity Subsolution, Supersolution and Solution on the Lift of the Wasserstein Space (ii), applied with $\Xi_{n}$, with $Y_{n}$ and with $\epsilon_{n}$, provides $W_{n}\in\mathcal{D}_{\Sigma}^{\Lambda}$, $V'_{n}\in L^{2}(\Omega;\mathbb{R}^{d})$, $t_{n}\in\mathbb{R}$ and $\mathbb{Y}_{n}\in\mathcal{S}(d)$ with the five corresponding inequalities, in which $v^{+}_{\delta}$, $\hat{v}$, $Y_{n}$, $\Xi_{n}$ replace $u^{-}_{\delta}$, $\hat{u}$, $X_{n}$, $\Phi_{n}$, together with $-\epsilon_{n}\le F^{+}_{\delta}(W_{n},t_{n},V'_{n},\mathbb{Y}_{n})$.

Step 5. The limits. Put $V_{*}=\alpha(\hat{X}-\hat{Y})$. Using the triangle inequality of The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity (i) and the estimates of Steps 2, 3 and 4,

$$\lVert Z_{n}-\hat{X}\rVert_{L^{2}}\le6\epsilon_{n},\qquad\lVert W_{n}-\hat{Y}\rVert_{L^{2}}\le6\epsilon_{n},$$ $$\lVert p_{n}-V_{*}\rVert_{L^{2}}\le8\alpha\epsilon_{n},\qquad\lVert p^{u}_{n}-p_{n}\rVert_{L^{2}}\le16\epsilon_{n}^{2},\qquad\lVert p^{v}_{n}-p_{n}\rVert_{L^{2}}\le16\epsilon_{n}^{2},$$

the second line by The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity (iii) and the bounds on $\lVert\bar{X}_{n}-\hat{X}\rVert_{L^{2}}$, $\lVert\bar{Y}_{n}-\hat{Y}\rVert_{L^{2}}$, $\lVert\bar{X}_{n}-q^{1}_{n}\rVert_{L^{2}}$ and $\lVert\bar{Y}_{n}-q^{2}_{n}\rVert_{L^{2}}$; hence

$$\lVert V_{n}-V_{*}\rVert_{L^{2}}\le2\epsilon_{n}+16\epsilon_{n}^{2}+8\alpha\epsilon_{n},\qquad\lVert V'_{n}-V_{*}\rVert_{L^{2}}\le2\epsilon_{n}+16\epsilon_{n}^{2}+8\alpha\epsilon_{n}.$$

Also $\lVert I_{d}\rVert\le1$: by Norm of a Symmetric Real Matrix the norm is the supremum of the numbers $|\xi\cdot(I_{d}\xi)|$ over $\lVert\xi\rVert\le1$, and $\xi\cdot(I_{d}\xi)=\xi\cdot\xi=\lVert\xi\rVert^{2}\le1$ by claim 2 of Linearity of the Matrix-Vector Product and the Quadratic Form as a Double Sum and claim 1 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square. Therefore

$$\lVert\mathbb{X}_{n}-\mathbb{A}_{n}^{\flat}\rVert\le2\epsilon_{n}+2\epsilon_{n}\lVert I_{d}\rVert\le4\epsilon_{n},\qquad\lVert\mathbb{Y}_{n}-\mathbb{B}_{n}^{\flat}\rVert\le4\epsilon_{n}.$$

Since the sequence whose $n$-th term is $\epsilon_{n}$ converges to $0$, each right-hand side above converges to $0$ by Arithmetic of Limits of Real Sequences, and therefore, directly from the definition of convergence, the sequences whose $n$-th terms are $Z_{n}$, $W_{n}$, $V_{n}$ and $V'_{n}$ converge in $L^{2}(\Omega;\mathbb{R}^{d})$ to $\hat{X}$, $\hat{Y}$, $V_{*}$ and $V_{*}$.

Next, the value bounds. Since $\tfrac{\alpha}{2}\lVert\bar{X}_{n}-\bar{Y}_{n}\rVert_{L^{2}}^{2}$ is nonnegative, Step 2 gives $M(\delta,\alpha)-2\epsilon_{n}^{3}\le\hat{u}(\bar{X}_{n})-\hat{v}(\bar{Y}_{n})$. Combining this with the bounds of Step 1 and with $0\le M(\delta,\alpha)$,

$$b'+\delta e_{0}-2\epsilon_{n}^{3}\ \le\ \hat{u}(\bar{X}_{n})\ \le\ b-\delta e_{0},\qquad b'+\delta e_{0}\ \le\ \hat{v}(\bar{Y}_{n})\ \le\ b-\delta e_{0}+2\epsilon_{n}^{3}.$$

From $|s_{n}-\hat{u}(X_{n})|<\epsilon_{n}$ and $|\hat{u}(X_{n})-\hat{u}(\bar{X}_{n})|<\epsilon_{n}$ we get $|s_{n}-\hat{u}(\bar{X}_{n})|<2\epsilon_{n}$, and likewise $|t_{n}-\hat{v}(\bar{Y}_{n})|<2\epsilon_{n}$; hence

$$b'+\delta e_{0}-2\epsilon_{n}^{3}-2\epsilon_{n}\le s_{n}\le b-\delta e_{0}+2\epsilon_{n},\qquad b'+\delta e_{0}-2\epsilon_{n}\le t_{n}\le b-\delta e_{0}+2\epsilon_{n}^{3}+2\epsilon_{n},$$ $$M(\delta,\alpha)-2\epsilon_{n}^{3}-4\epsilon_{n}\ \le\ s_{n}-t_{n}.$$

As $0<\epsilon_{n}\le1$ and $0<\delta<1$, the sequences whose $n$-th terms are $s_{n}$ and $t_{n}$ are bounded, a bound for both being $|b|+|b'|+|e_{0}|+4$.

By Lifted Test Data from Second-Order Data on the Space of Square-Integrable Random Vectors: Compression along the Constant Tuple (viii) there are a strictly increasing $\varphi_{1}:\mathbb{N}\to\mathbb{N}$ and $\mathbb{X}_{*},\mathbb{Y}_{*}\in\mathcal{S}(d)$ such that the sequences whose $k$-th terms are $\mathbb{A}^{\flat}_{\varphi_{1}(k)}$ and $\mathbb{B}^{\flat}_{\varphi_{1}(k)}$ converge to $\mathbb{X}_{*}$ and to $\mathbb{Y}_{*}$. Applying Bolzano-Weierstrass Theorem for Real Sequences twice, first to the sequence whose $k$-th term is $s_{\varphi_{1}(k)}$ and then to the corresponding sequence of the $t$'s, and composing the index maps by claim 2 of A Subsequence of a Subsequence is a Subsequence, we obtain a strictly increasing $\varphi:\mathbb{N}\to\mathbb{N}$, each of whose values is a value of $\varphi_{1}$, such that the sequences whose $k$-th terms are $s_{\varphi(k)}$ and $t_{\varphi(k)}$ converge, to $s_{*}$ and $t_{*}$ say. By A Subsequence of a Convergent Sequence Has the Same Limit every convergence established above persists along $\varphi$; in particular $\mathbb{A}^{\flat}_{\varphi(k)}\to\mathbb{X}_{*}$ and $\mathbb{B}^{\flat}_{\varphi(k)}\to\mathbb{Y}_{*}$, and since $\lVert\mathbb{X}_{n}-\mathbb{A}^{\flat}_{n}\rVert\le4\epsilon_{n}$ and $\lVert\mathbb{Y}_{n}-\mathbb{B}^{\flat}_{n}\rVert\le4\epsilon_{n}$ also $\mathbb{X}_{\varphi(k)}\to\mathbb{X}_{*}$ and $\mathbb{Y}_{\varphi(k)}\to\mathbb{Y}_{*}$. Each pair $(\mathbb{A}^{\flat}_{\varphi(k)},\mathbb{B}^{\flat}_{\varphi(k)})$ is admitted at $\alpha$, so $(\mathbb{X}_{*},\mathbb{Y}_{*})$ is admitted at $\alpha$ by Lifted Test Data from Second-Order Data on the Space of Square-Integrable Random Vectors: Compression along the Constant Tuple (ix).

Passing to the limit along $\varphi$ in the three displayed inequalities for $s_{n}$, $t_{n}$ and $s_{n}-t_{n}$, using Arithmetic of Limits of Real Sequences and Order Properties of Limits of Real Sequences,

$$M(\delta,\alpha)\le s_{*}-t_{*},\qquad b'+\delta e_{0}\le s_{*}\le b-\delta e_{0},\qquad b'+\delta e_{0}\le t_{*}\le b-\delta e_{0}.$$

Since $0<\delta<1$, the last two give $|s_{*}|\le|b|+|b'|+|e_{0}|\le B$ and $|t_{*}|\le B$.

Step 6. Shift-coercivity, closed score and shift-semicontinuity. For $k\in\mathbb{N}$ put $\xi_{k}=(Z_{\varphi(k)},s_{\varphi(k)},V_{\varphi(k)},\mathbb{X}_{\varphi(k)})$ and $\eta_{k}=(W_{\varphi(k)},t_{\varphi(k)},V'_{\varphi(k)},\mathbb{Y}_{\varphi(k)})$; these are test data for $F$.

Each of the ten quantities occurring in the boundedness condition is bounded above independently of $n$. Indeed $\lVert Z_{n}\rVert_{L^{2}}\le\lVert\hat{X}\rVert_{L^{2}}+6$ and $\lVert W_{n}\rVert_{L^{2}}\le\lVert\hat{Y}\rVert_{L^{2}}+6$; $|s_{n}|\le B+4$ and $|t_{n}|\le B+4$; $\lVert V_{n}\rVert_{L^{2}}\le\alpha\lVert\hat{X}-\hat{Y}\rVert_{L^{2}}+8\alpha+18$ and the same for $\lVert V'_{n}\rVert_{L^{2}}$; $\lVert\mathbb{X}_{n}\rVert\le6\alpha+4$ and $\lVert\mathbb{Y}_{n}\rVert\le6\alpha+4$. For the penalties, Step 5 gives $|\hat{u}(\bar{X}_{n})|\le B+2$ and $|\hat{v}(\bar{Y}_{n})|\le B+2$, whence $|u^{-}_{\delta}(\mathcal{L}(Z_{n}))|\le B+4$ and $|v^{+}_{\delta}(\mathcal{L}(W_{n}))|\le B+4$; since $u^{-}_{\delta}=u-\delta\mathcal{E}$ and $v^{+}_{\delta}=v+\delta\mathcal{E}$ on $\mathcal{D}$, with $u\le b$ and $b'\le v$, this gives $\delta\mathcal{E}(\mathcal{L}(Z_{n}))\le b+B+4$ and $\delta\mathcal{E}(\mathcal{L}(W_{n}))\le B+4-b'$, while $\delta e_{0}\le\delta\mathcal{E}(\mathcal{L}(Z_{n}))$ and $\delta e_{0}\le\delta\mathcal{E}(\mathcal{L}(W_{n}))$; multiplying by the positive $\delta^{-1}$,

$$\bigl|\mathcal{E}(\mathcal{L}(Z_{n}))\bigr|\le\delta^{-1}\bigl(|b|+|b'|+B+|e_{0}|+4\bigr),\qquad\bigl|\mathcal{E}(\mathcal{L}(W_{n}))\bigr|\le\delta^{-1}\bigl(|b|+|b'|+B+|e_{0}|+4\bigr).$$

Put

$$R'=\lVert\hat{X}\rVert_{L^{2}}+\lVert\hat{Y}\rVert_{L^{2}}+2\alpha\lVert\hat{X}-\hat{Y}\rVert_{L^{2}}+28\alpha+2\delta^{-1}\bigl(|b|+|b'|+B+|e_{0}|+4\bigr)+2B+60,$$

a sum of nonnegative terms exceeding each of the ten bounds just listed, and exceeding $2$. Every $\xi_{k}$ and every $\eta_{k}$ is therefore $R'$-bounded. Moreover $F^{-}_{\delta}(\xi_{k})-F^{+}_{\delta}(\eta_{k})\le\epsilon_{\varphi(k)}+\epsilon_{\varphi(k)}\le2<R'$ by Step 4, so $\xi_{k}\in S^{-}_{\delta,R'}$ and $\eta_{k}\in S^{+}_{\delta,R'}$ by Test Data for a Second-Order Equation Operator on the Lift of the Wasserstein Space and the Admissible Sets (iii), each of the two serving as the witness required for the other. The shift-coercivity condition provides a nonnegative score bound $C$ for $F$ at $(\delta,R')$, so that

$$\bigl\lVert\Sigma(\mathcal{L}(Z_{\varphi(k)}))\bigr\rVert_{\mathcal{L}(Z_{\varphi(k)})}\le C,\qquad\bigl\lVert\Sigma(\mathcal{L}(W_{\varphi(k)}))\bigr\rVert_{\mathcal{L}(W_{\varphi(k)})}\le C\qquad(k\in\mathbb{N}).$$

Apply Penalty Pairs with Closed Score (i) with the nonnegative $C$ to the sequence whose $k$-th term is $Z_{\varphi(k)}$, which converges to $\hat{X}$: it gives $\mathcal{L}(\hat{X})\in\mathcal{D}_{\Sigma}$, that is $\hat{X}\in\mathcal{D}_{\Sigma}^{\Lambda}$, and that the sequence whose $k$-th term is $\Sigma(\mathcal{L}(Z_{\varphi(k)}))\circ Z_{\varphi(k)}$ converges weakly to $\Sigma(\mathcal{L}(\hat{X}))\circ\hat{X}$. Likewise $\hat{Y}\in\mathcal{D}_{\Sigma}^{\Lambda}$, with the corresponding weak convergence. In particular $\xi=(\hat{X},s_{*},V_{*},\mathbb{X}_{*})$ and $\eta=(\hat{Y},t_{*},V_{*},\mathbb{Y}_{*})$ are test data for $F$.

Put $R''=R'+C$, a positive real. Every $\xi_{k}$ and $\eta_{k}$ is $R''$-bounded and the displayed scores are at most $R''$; with the convergences of Step 5 this says that the sequence whose $k$-th term is $\xi_{k}$ converges to $\xi$ with score bounded by $R''$, and likewise that the sequence of the $\eta_{k}$ converges to $\eta$ with score bounded by $R''$. The operator $F$ is shift-semicontinuous at $(\delta,R'')$. In the first implication of The Shift-Semicontinuity Condition for an Equation Operator on the Lift of the Wasserstein Space (ii) take $c=0$: given positive $\varepsilon\in\mathbb{R}$, the convergence of the $\epsilon_{n}$ to $0$ and the strict increase of $\varphi$ furnish $N\in\mathbb{N}$ with $\epsilon_{\varphi(k)}\le\varepsilon$ for $k\ge N$, and then $F^{-}_{\delta}(\xi_{k})\le\epsilon_{\varphi(k)}\le0+\varepsilon$; hence $F^{-}_{\delta}(\xi)\le0$. The second implication, again with $c=0$, gives $0\le F^{+}_{\delta}(\eta)$. Therefore

$$F^{-}_{\delta}(\xi)-F^{+}_{\delta}(\eta)\ \le\ 0 .$$

Step 7. Properness and the structure condition. We have $\hat{X},\hat{Y}\in\mathcal{D}_{\Sigma}^{\Lambda}$, the pair $(\hat{X},\hat{Y})$ is optimally coupled by Step 1, $\delta(|\mathcal{E}(\hat{\mu})|+|\mathcal{E}(\hat{\nu})|)\le2B\le R$, and $-R\le t_{*}\le R$ because $|t_{*}|\le B\le R$; the pair $(\mathbb{X}_{*},\mathbb{Y}_{*})$ is admitted at $\alpha$, and $V_{*}=\alpha(\hat{X}-\hat{Y})$. Clause 2 of The Second-Order Structure Condition at Optimally Coupled Pairs on the Lift of the Wasserstein Space, applied to the structure pair $(\omega_{1},\omega_{2})$ at $R$ with the value slot $t_{*}$, therefore gives

$$-\omega_{1}\bigl(\alpha\lVert\hat{X}-\hat{Y}\rVert_{L^{2}}^{2}+\alpha^{-1}\bigr)-\omega_{2}\bigl(\delta(|\mathcal{E}(\hat{\mu})|+|\mathcal{E}(\hat{\nu})|+1),\alpha\bigr)\ \le\ F^{-}_{\delta}(\hat{X},t_{*},V_{*},\mathbb{X}_{*})-F^{+}_{\delta}(\eta).$$

By Second-Order Equation Operators on the Lift of the Wasserstein Space and Their Delta-Shifts (iii), $F^{-}_{\delta}(\hat{X},s,V_{*},\mathbb{X}_{*})=F(\hat{X},s+\delta\mathcal{E}(\hat{\mu}),V_{*}+\delta\,\Sigma(\hat{\mu})\circ\hat{X},\mathbb{X}_{*})$ for every $s\in\mathbb{R}$. From $0\le M(\delta,\alpha)\le s_{*}-t_{*}$ we get $t_{*}\le s_{*}$, and $|s_{*}+\delta\mathcal{E}(\hat{\mu})|\le B+B=2B\le R$, likewise with $t_{*}$ in place of $s_{*}$. Hence Locally Strictly Proper Second-Order Equation Operator on the Lift of the Wasserstein Space (i), applied to the properness constant $\lambda$ at $R$ with the two value slots $t_{*}+\delta\mathcal{E}(\hat{\mu})\le s_{*}+\delta\mathcal{E}(\hat{\mu})$,

$$\lambda\,(s_{*}-t_{*})\ \le\ F^{-}_{\delta}(\xi)-F^{-}_{\delta}(\hat{X},t_{*},V_{*},\mathbb{X}_{*}).$$

Adding the two displays and using $F^{-}_{\delta}(\xi)-F^{+}_{\delta}(\eta)\le0$,

$$\lambda\,(s_{*}-t_{*})\ \le\ \omega_{1}\bigl(\alpha\lVert\hat{X}-\hat{Y}\rVert_{L^{2}}^{2}+\alpha^{-1}\bigr)+\omega_{2}\bigl(\delta(|\mathcal{E}(\hat{\mu})|+|\mathcal{E}(\hat{\nu})|+1),\alpha\bigr).$$

Finally $M(\delta,\alpha)\le s_{*}-t_{*}$ and $0<\lambda$ give $\lambda M(\delta,\alpha)\le\lambda(s_{*}-t_{*})$ by claim 5 of Elementary Arithmetic in an Ordered Field, and $\lVert\hat{X}-\hat{Y}\rVert_{L^{2}}^{2}=W_{2}(\hat{\mu},\hat{\nu})^{2}$. This is the asserted estimate.