Proof of Closure of Approximate Test Data for Viscosity Sub- and Supersolutions on the Wasserstein Space

Each result cited is universally quantified over the data in its own statement. We write $W$ for $W_{2}$; the $\delta$-envelopes are taken relative to the given penalty pair, and $\delta$ is the fixed real with $0<\delta<1$. For $n\in\mathbb{N}$ let $\epsilon_{n}$ be the multiplicative inverse of the positive real attached to $n$ (The Real Numbers: Standing Notation and Background §numbers); that real is at least $1$ (claim 2 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field), so $0<\epsilon_{n}\le1$ by Order Reversal under Reciprocals, and Summability of the Reciprocals of the Squares §reciprocal. The sequence $(\epsilon_{n})_{n}$ converges to $0$: given a positive $\varepsilon$, claim 3 of The Archimedean Property of the Real Numbers provides $N\in\mathbb{N}$ with $\epsilon_{N}<\varepsilon$, and for $n\ge N$ the real attached to $n$ is at least the one attached to $N$ (claims 3 and 6 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field), so $0<\epsilon_{n}\le\epsilon_{N}<\varepsilon$ by Order Reversal under Reciprocals, and Summability of the Reciprocals of the Squares §reciprocal, whence $|\epsilon_{n}-0|=\epsilon_{n}<\varepsilon$. Hence $(2\epsilon_{n})_{n}$ and $(4\epsilon_{n}^{2})_{n}$ converge to $0$ (claims 2 and 3 of Arithmetic of Limits of Real Sequences). By Basic Properties of a Wasserstein-Coercive Penalty Pair §bounded-below, the pair being Wasserstein-coercive, we fix $e_{0}\in\mathbb{R}$ with $e_{0}\le\mathcal{E}(\mu)$ for every $\mu\in\mathcal{D}$. By Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §nonempty we fix $\nu_{0}\in\mathcal{D}_{\Sigma}$; let $0_{\nu_{0}}\in L^{2}(\nu_{0};\mathbb{R}^{d})$ be the class of the zero map of $\mathbb{R}^{d}$, which is Borel, being constant, and has $\lVert0_{\nu_{0}}\rVert_{\nu_{0}}=\sqrt{0}=0$ (Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation §fields); so $(\nu_{0},0_{\nu_{0}})\in\mathcal{V}(\mathcal{D}_{\Sigma})$ by The Bundle of Vector Fields over a Set of Measures, Second-Order Equation Operators on the Wasserstein Space, and Their Delta-Shifts §bundle. The distance on $\mathcal{S}(d)$ is $d_{\mathcal{S}(d)}(P,P')=\lVert P-P'\rVert$, a metric (Real Matrices, Symmetric Matrices and the Semidefinite Ordering: Standing Notation §norm, The Set of Symmetric Real Matrices is a Metric Space), and $0_{d}\in\mathcal{S}(d)$ (Real Matrices, Symmetric Matrices and the Semidefinite Ordering: Standing Notation §symmetric); for $P\in\mathcal{S}(d)$ the matrices $P-0_{d}$ and $P$ have the same entries, so $\lVert P\rVert=d_{\mathcal{S}(d)}(P,0_{d})$, and the triangle inequality of the metric (Metric Space) gives $\lVert P\rVert\le\lVert P-P'\rVert+\lVert P'\rVert$ for $P,P'\in\mathcal{S}(d)$. For nonnegative real $t$, $\sqrt{t}$ is as in Probability Measures on Euclidean Space and Random Vectors: Standing Notation §spaces, a nonnegative real with square $t$. Sequences of objects indexed by $n$ below are chosen for all $n$ at once by Axiom of Countable Choice.

Step 0 (Norms along a coupling). Let $\nu,\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, $\pi\in\Pi(\nu,\mu)$, $q\in L^{2}(\nu;\mathbb{R}^{d})$ and $\eta\in L^{2}(\mu;\mathbb{R}^{d})$. The maps $q\circ\mathrm{pr}_{1}$ and $\eta\circ\mathrm{pr}_{2}$ (for representatives) are Borel, and by the change-of-variables formula and $(\mathrm{pr}_{1})_{\#}\pi=\nu$, $(\mathrm{pr}_{2})_{\#}\pi=\mu$ their classes in the real Hilbert space $L^{2}(\pi;\mathbb{R}^{d})$ (Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation §fields) have norms $\lVert q\rVert_{\nu}$ and $\lVert\eta\rVert_{\mu}$, while the norm of their difference is the square root of the discrepancy of $q$ and $\eta$ along $\pi$. The triangle inequality The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §triangle gives

With $q$ and $\eta$ the classes of $\mathrm{id}$ (Basic Properties of the Tangent Space: Closed Subspace, the Identity Map Belongs to It, Second-Moment Limits, and Representation of Bounded Functionals on Gradients §identity), the discrepancy is $I(\pi)$ (Couplings of Two Probability Measures on Euclidean Space and Their Quadratic Cost §cost) and $\lVert\mathrm{id}\rVert_{\nu}^{2}=M_{2}(\nu)$, so $\sqrt{M_{2}(\nu)}\le\sqrt{I(\pi)}+\sqrt{M_{2}(\mu)}$.

Step 1 (Composite couplings). Let $n\in\mathbb{N}$, let $\nu,\rho,\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$, $\gamma\in\Pi(\nu,\rho)$, $\pi\in\Pi(\rho,\mu)$, $q\in L^{2}(\nu;\mathbb{R}^{d})$, $\eta\in L^{2}(\rho;\mathbb{R}^{d})$ and $\theta\in L^{2}(\mu;\mathbb{R}^{d})$, and suppose that each of the four nonnegative reals $I(\gamma)$, $I(\pi)$, the discrepancy of $q$ and $\eta$ along $\gamma$, and the discrepancy of $\eta$ and $\theta$ along $\pi$, is less than $\epsilon_{n}^{2}$. By claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, applied to the square root of each of them and to $\epsilon_{n}$, each of the four square roots is less than $\epsilon_{n}$. Let $\beta$ be a gluing of $\gamma$ and $\pi$ (Gluing Two Couplings over a Common Middle Marginal, and the Composite Coupling §glued) and $\kappa=(\mathrm{q}_{1},\mathrm{q}_{3})_{\#}\beta$, which belongs to $\Pi(\nu,\mu)$ by Gluing Two Couplings over a Common Middle Marginal, and the Composite Coupling §composite. That clause gives $\sqrt{I(\kappa)}\le\sqrt{I(\gamma)}+\sqrt{I(\pi)}$, and A Triangle Inequality for Discrepancies Along a Composite Coupling §triangle gives the same bound for the square root of the discrepancy of $q$ and $\theta$ along $\kappa$ by the sum of the square roots of the two other discrepancies. With claims 2 and 3 of Elementary Order Arithmetic in an Ordered Field and $\epsilon_{n}+\epsilon_{n}=2\epsilon_{n}$,

and squaring (claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, with $(2\epsilon_{n})^{2}=4\epsilon_{n}^{2}$) gives $I(\kappa)<4\epsilon_{n}^{2}$ and that the discrepancy of $q$ and $\theta$ along $\kappa$ is less than $4\epsilon_{n}^{2}$.

Step 2 (Null sequences). (a) If $0\le a_{n}<4\epsilon_{n}^{2}$ for real $a_{n}$ and every $n$, then $|a_{n}-0|=a_{n}\le4\epsilon_{n}^{2}$, so $(a_{n})_{n}$ converges to $0$ by claim 3 of Order Properties of Limits of Real Sequences. (b) If $b_{n},b\in\mathbb{R}$ satisfy $|b_{n}-b|<2\epsilon_{n}$ for every $n$, then $(b_{n})_{n}$ converges to $b$ by the same claim. (c) If $Z_{n},Z\in\mathcal{S}(d)$ satisfy $\lVert Z_{n}-Z\rVert<2\epsilon_{n}$ for every $n$, then $(Z_{n})_{n}$ converges to $Z$ in $(\mathcal{S}(d),d_{\mathcal{S}(d)})$: given a positive $\varepsilon$ there is $N$ with $2\epsilon_{n}=|2\epsilon_{n}-0|<\varepsilon$ for $n\ge N$, so $d_{\mathcal{S}(d)}(Z_{n},Z)<\varepsilon$ for $n\ge N$ (claim 2 of Elementary Order Arithmetic in an Ordered Field). (d) For every positive $\varepsilon\in\mathbb{R}$ there is $N\in\mathbb{N}$ with $\epsilon_{n}<\varepsilon$ for every $n\ge N$, as shown above.

Step 3 (Clause 1: approximate viscosity data). Assume the hypotheses of clause 1 and fix $b\in\mathbb{R}$ with $u(\mu)\le b$ for every $\mu\in\mathcal{D}$; by The Delta-Envelopes of Bounded Functions and Their Monotonicity in the Weight, for a Wasserstein-Coercive Penalty Pair §growth, $u$ has penalty-subordinate growth from above. Put $s_{*}=u^{-}_{\delta}(\rho^{*})$. Let $n\in\mathbb{N}$. The hypothesis of clause 1 with $\varepsilon=\epsilon_{n}$ gives $\rho_{n}\in\mathcal{D}$, an intrinsic test function $\varphi_{n}$ on $\mathcal{D}$ such that the function with value $u^{-}_{\delta}(\rho')-\varphi_{n}(\rho')$ at $\rho'\in\mathcal{D}$ has a local maximum relative to $\mathcal{D}$ at $\rho_{n}$, and $\pi_{n}\in\Pi(\rho_{n},\rho^{*})$ with

As $u$ is a viscosity subsolution, Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space §subsolution with the positive $\delta$, $\varphi_{n}$, $\hat{\mu}=\rho_{n}$ and $\varepsilon=\epsilon_{n}$ gives $\nu_{n}\in\mathcal{D}_{\Sigma}$, $\gamma_{n}\in\Pi(\nu_{n},\rho_{n})$, $s_{n}\in\mathbb{R}$, $q_{n}\in L^{2}(\nu_{n};\mathbb{R}^{d})$ and $X_{n}\in\mathcal{S}(d)$ with

All measures here lie in $\mathcal{P}_{2}(\mathbb{R}^{d})$, as $\mathcal{D}_{\Sigma}\subseteq\mathcal{D}\subseteq\mathcal{P}_{2}(\mathbb{R}^{d})$ (Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §pair). Step 1 with $\nu=\nu_{n}$, $\rho=\rho_{n}$, $\mu=\rho^{*}$, $\gamma=\gamma_{n}$, $\pi=\pi_{n}$, $q=q_{n}$, $\eta=\nabla\varphi_{n}(\rho_{n})$ and $\theta=V_{*}$ gives $\kappa_{n}\in\Pi(\nu_{n},\rho^{*})$ with $\sqrt{I(\kappa_{n})}<2\epsilon_{n}$, $I(\kappa_{n})<4\epsilon_{n}^{2}$, and the discrepancy of $q_{n}$ and $V_{*}$ along $\kappa_{n}$ less than $4\epsilon_{n}^{2}$ with square root less than $2\epsilon_{n}$. By the triangle inequality of $|\cdot|$ (claim 5 of Properties of the Absolute Value in an Ordered Field), applied to $u^{-}_{\delta}(\nu_{n})-s_{*}=(u^{-}_{\delta}(\nu_{n})-u^{-}_{\delta}(\rho_{n}))+(u^{-}_{\delta}(\rho_{n})-s_{*})$ and to $s_{n}-s_{*}=(s_{n}-u^{-}_{\delta}(\rho_{n}))+(u^{-}_{\delta}(\rho_{n})-s_{*})$, by the triangle inequality of $d_{\mathcal{S}(d)}$ through $H_{\varphi_{n}}(\rho_{n})$, and by claims 2 and 3 of Elementary Order Arithmetic in an Ordered Field,

By Step 2(a), $(\kappa_{n})_{n}$ is a sequence of couplings of vanishing cost from $(\nu_{n})_{n}$ to $\rho^{*}$ and $(q_{n})_{n}$ converges strongly to $V_{*}$ along it; by Step 2(b), $(s_{n})_{n}$ converges to $s_{*}$; by Step 2(c), $(X_{n})_{n}$ converges to $\mathbb{X}$ in $(\mathcal{S}(d),d_{\mathcal{S}(d)})$.

Step 4 (Clause 1: bounds and admissibility). Let $n\in\mathbb{N}$; recall $2\epsilon_{n}\le2$. Step 0 with $\kappa_{n}$ gives $\sqrt{M_{2}(\nu_{n})}\le\sqrt{I(\kappa_{n})}+\sqrt{M_{2}(\rho^{*})}<\sqrt{M_{2}(\rho^{*})}+2$ and, with $q=q_{n}$, $\eta=V_{*}$, $\lVert q_{n}\rVert_{\nu_{n}}<\lVert V_{*}\rVert_{\rho^{*}}+2$. By claim 5 of Properties of the Absolute Value in an Ordered Field, $|s_{n}|\le|s_{n}-s_{*}|+|s_{*}|<|s_{*}|+2$; by the preamble, $\lVert X_{n}\rVert\le\lVert X_{n}-\mathbb{X}\rVert+\lVert\mathbb{X}\rVert<\lVert\mathbb{X}\rVert+2$. For the penalty: $|u^{-}_{\delta}(\nu_{n})-s_{*}|<2\epsilon_{n}$ gives $s_{*}-2\epsilon_{n}<u^{-}_{\delta}(\nu_{n})$ (claim 9 of Properties of the Absolute Value in an Ordered Field and claim 1 of Elementary Order Arithmetic in an Ordered Field), and $-|s_{*}|-2\le s_{*}-2\epsilon_{n}$ (claim 3 of Properties of the Absolute Value in an Ordered Field), so $-|s_{*}|-2<u^{-}_{\delta}(\nu_{n})$; as $\nu_{n}\in\mathcal{D}$, The Delta-Envelopes of Bounded Functions and Their Monotonicity in the Weight, for a Wasserstein-Coercive Penalty Pair §bounded gives $u^{-}_{\delta}(\nu_{n})\le b-\delta\,\mathcal{E}(\nu_{n})$, hence $\delta\,\mathcal{E}(\nu_{n})<b+|s_{*}|+2\le|b|+|s_{*}|+2$, and multiplying by the positive $\delta^{-1}$ (claims 7 and 10 of Elementary Order Arithmetic in an Ordered Field) $\mathcal{E}(\nu_{n})<\delta^{-1}(|b|+|s_{*}|+2)$. Also $-|e_{0}|-1<-|e_{0}|\le e_{0}\le\mathcal{E}(\nu_{n})$. With

both $\delta^{-1}(|b|+|s_{*}|+2)$ and $|e_{0}|+1$ are at most $c_{\mathcal{E}}$, the other summands being nonnegative (claim 1 of Properties of the Absolute Value in an Ordered Field; claims 5 and 7 of Elementary Order Arithmetic in an Ordered Field for the product of $\delta^{-1}$ with the positive $|b|+|s_{*}|+2$); so $-c_{\mathcal{E}}<\mathcal{E}(\nu_{n})<c_{\mathcal{E}}$ and $|\mathcal{E}(\nu_{n})|<c_{\mathcal{E}}$ by claim 9 of Properties of the Absolute Value in an Ordered Field.

The partner datum: $\eta_{0}=(\nu_{0},0,0_{\nu_{0}},\mathbb{X})$ is a test datum for $F$, as $(\nu_{0},0_{\nu_{0}})\in\mathcal{V}(\mathcal{D}_{\Sigma})$ and $\mathbb{X}\in\mathcal{S}(d)$; so $F^{+}_{\delta}(\eta_{0})$ is a real number. Now put

It is chosen after $b$, $e_{0}$, $\nu_{0}$ and $\eta_{0}$ and does not depend on $n$; all its summands except $3$ are nonnegative, so $R$ is positive, and $R$ minus each of $\sqrt{M_{2}(\rho^{*})}+2$, $c_{\mathcal{E}}$, $|s_{*}|+2$, $\lVert V_{*}\rVert_{\rho^{*}}+2$, $\lVert\mathbb{X}\rVert+2$, $\sqrt{M_{2}(\nu_{0})}$, $|\mathcal{E}(\nu_{0})|$, $0$ and $1+|F^{+}_{\delta}(\eta_{0})|$ is a sum of nonnegative reals and a positive real, hence positive (claims 1 and 3 of Elementary Order Arithmetic in an Ordered Field, claim 2 of Elementary Arithmetic in an Ordered Field). Hence, by the bounds above and claim 2 of Elementary Order Arithmetic in an Ordered Field, the test datum $\xi_{n}=(\nu_{n},s_{n},q_{n},X_{n})$ is $R$-bounded; and $\eta_{0}$ is $R$-bounded, since $\sqrt{M_{2}(\nu_{0})}<R$, $|\mathcal{E}(\nu_{0})|<R$, $|0|=0<R$, $\lVert0_{\nu_{0}}\rVert_{\nu_{0}}=0<R$ and $\lVert\mathbb{X}\rVert<R$. Finally $-F^{+}_{\delta}(\eta_{0})\le|F^{+}_{\delta}(\eta_{0})|$ (claims 2 and 3 of Properties of the Absolute Value in an Ordered Field) and $F^{-}_{\delta}(\xi_{n})\le\epsilon_{n}\le1$, so

So $\xi_{n}\in S^{-}_{\delta,R}$ for every $n$ (Test Data for an Intrinsic Second-Order Equation Operator on the Wasserstein Space and the Admissible Sets §admissible), with the $R$-bounded partner $\eta_{0}$.

Step 5 (Clause 1: score bound and closed score). As $0<\delta<1$ and $0<R$, the shift-coercivity condition (The Shift-Coercivity Condition for an Equation Operator on the Wasserstein Space §coercivity) gives a score bound $C\ge0$ for $F$ at $(\delta,R)$, so $\lVert\Sigma(\nu_{n})\rVert_{\nu_{n}}\le C$ for every $n$ by Step 4. Now $(\nu_{n})_{n}$ is a sequence in $\mathcal{D}_{\Sigma}$, $\rho^{*}\in\mathcal{D}$, and $(\kappa_{n})_{n}$ is a sequence of couplings of vanishing cost from $(\nu_{n})_{n}$ to $\rho^{*}$ (Step 3); so Penalty Pairs with Closed Score Along Couplings §closed, at the nonnegative level $C$, gives $\rho^{*}\in\mathcal{D}_{\Sigma}$ and that $(\Sigma(\nu_{n}))_{n}$ converges weakly to $\Sigma(\rho^{*})$ along $(\kappa_{n})_{n}$. As $V_{*}\in L^{2}(\rho^{*};\mathbb{R}^{d})$, $(\rho^{*},V_{*})\in\mathcal{V}(\mathcal{D}_{\Sigma})$ (The Bundle of Vector Fields over a Set of Measures, Second-Order Equation Operators on the Wasserstein Space, and Their Delta-Shifts §bundle), and $\xi_{*}=(\rho^{*},s_{*},V_{*},\mathbb{X})$ is a test datum for $F$ (Test Data for an Intrinsic Second-Order Equation Operator on the Wasserstein Space and the Admissible Sets §data).

Step 6 (Clause 1: shift-semicontinuity). Put $R''=R+C$, positive as $0<R$ and $0\le C$. Each $\xi_{n}$ is $R$-bounded, hence $R''$-bounded ($R\le R''$ and claim 2 of Elementary Order Arithmetic in an Ordered Field), and $\lVert\Sigma(\nu_{n})\rVert_{\nu_{n}}\le C\le R''$. With Steps 3 and 5 this says that $(\xi_{n})_{n}$ converges to $\xi_{*}$ along $(\kappa_{n})_{n}$ with score bounded by $R''$: $(\kappa_{n})_{n}$ has vanishing cost, $(q_{n})_{n}$ converges strongly to $V_{*}$ and $(\Sigma(\nu_{n}))_{n}$ weakly to $\Sigma(\rho^{*})$ along it, $(s_{n})_{n}$ converges to $s_{*}$ and $(X_{n})_{n}$ to $\mathbb{X}$. By the shift-semicontinuity condition (The Shift-Semicontinuity Condition for an Equation Operator on the Wasserstein Space §semicontinuity), $F$ is shift-semicontinuous at $(\delta,R'')$. Let $\varepsilon$ be positive; Step 2(d) gives $N$ with $\epsilon_{n}<\varepsilon$ for $n\ge N$, so $F^{-}_{\delta}(\xi_{n})\le\epsilon_{n}<0+\varepsilon$ for $n\ge N$. The first implication of The Shift-Semicontinuity Condition for an Equation Operator on the Wasserstein Space §level with $c=0$ therefore gives

which, with $\rho^{*}\in\mathcal{D}_{\Sigma}$ from Step 5, is clause 1.

Step 7 (Clause 2: approximate viscosity data). Assume the hypotheses of clause 2 (the objects named in Steps 3 to 6 play no role here, and $\gamma_{n}$ now denotes the coupling supplied by the hypothesis of clause 2) and fix $b'\in\mathbb{R}$ with $b'\le v(\mu)$ for every $\mu\in\mathcal{D}$; by The Delta-Envelopes of Bounded Functions and Their Monotonicity in the Weight, for a Wasserstein-Coercive Penalty Pair §growth, $v$ has penalty-subordinate growth from below. Put $t_{*}=v^{+}_{\delta}(\sigma^{*})$. Let $n\in\mathbb{N}$. The hypothesis of clause 2 with $\varepsilon=\epsilon_{n}$ gives $\sigma_{n}\in\mathcal{D}$, an intrinsic test function $\psi_{n}$ on $\mathcal{D}$ such that the function with value $v^{+}_{\delta}(\sigma')-\psi_{n}(\sigma')$ at $\sigma'\in\mathcal{D}$ has a local minimum relative to $\mathcal{D}$ at $\sigma_{n}$, and $\gamma_{n}\in\Pi(\sigma_{n},\sigma^{*})$ with

As $v$ is a viscosity supersolution, Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space §supersolution with the positive $\delta$, $\varphi=\psi_{n}$, $\hat{\mu}=\sigma_{n}$ and $\varepsilon=\epsilon_{n}$ gives $\nu'_{n}\in\mathcal{D}_{\Sigma}$, $\theta_{n}\in\Pi(\nu'_{n},\sigma_{n})$, $t_{n}\in\mathbb{R}$, $q'_{n}\in L^{2}(\nu'_{n};\mathbb{R}^{d})$ and $Y_{n}\in\mathcal{S}(d)$ with

Step 1 with $\nu=\nu'_{n}$, $\rho=\sigma_{n}$, $\mu=\sigma^{*}$, $\gamma=\theta_{n}$, $\pi=\gamma_{n}$, $q=q'_{n}$, $\eta=\nabla\psi_{n}(\sigma_{n})$ and $\theta=W_{*}$ gives $\kappa'_{n}\in\Pi(\nu'_{n},\sigma^{*})$ with $\sqrt{I(\kappa'_{n})}<2\epsilon_{n}$, $I(\kappa'_{n})<4\epsilon_{n}^{2}$, and the discrepancy of $q'_{n}$ and $W_{*}$ along $\kappa'_{n}$ less than $4\epsilon_{n}^{2}$ with square root less than $2\epsilon_{n}$. Exactly as in Step 3 (claim 5 of Properties of the Absolute Value in an Ordered Field through $v^{+}_{\delta}(\sigma_{n})$, the triangle inequality of $d_{\mathcal{S}(d)}$ through $H_{\psi_{n}}(\sigma_{n})$, claims 2 and 3 of Elementary Order Arithmetic in an Ordered Field),

By Step 2(a)–(c), $(\kappa'_{n})_{n}$ is a sequence of couplings of vanishing cost from $(\nu'_{n})_{n}$ to $\sigma^{*}$, $(q'_{n})_{n}$ converges strongly to $W_{*}$ along it, $(t_{n})_{n}$ converges to $t_{*}$, and $(Y_{n})_{n}$ converges to $\mathbb{Y}$ in $(\mathcal{S}(d),d_{\mathcal{S}(d)})$.

Step 8 (Clause 2: bounds and admissibility). Let $n\in\mathbb{N}$. As in Step 4, Step 0 with $\kappa'_{n}$ gives $\sqrt{M_{2}(\nu'_{n})}<\sqrt{M_{2}(\sigma^{*})}+2$ and $\lVert q'_{n}\rVert_{\nu'_{n}}<\lVert W_{*}\rVert_{\sigma^{*}}+2$, and $|t_{n}|<|t_{*}|+2$, $\lVert Y_{n}\rVert<\lVert\mathbb{Y}\rVert+2$. The penalty bound changes direction: $|v^{+}_{\delta}(\nu'_{n})-t_{*}|<2\epsilon_{n}$ gives $v^{+}_{\delta}(\nu'_{n})<t_{*}+2\epsilon_{n}\le|t_{*}|+2$ (claim 9 of Properties of the Absolute Value in an Ordered Field, claim 1 of Elementary Order Arithmetic in an Ordered Field, claim 3 of Properties of the Absolute Value in an Ordered Field); as $\nu'_{n}\in\mathcal{D}$, The Delta-Envelopes of Bounded Functions and Their Monotonicity in the Weight, for a Wasserstein-Coercive Penalty Pair §bounded gives $b'+\delta\,\mathcal{E}(\nu'_{n})\le v^{+}_{\delta}(\nu'_{n})$, hence $\delta\,\mathcal{E}(\nu'_{n})<|t_{*}|+2-b'\le|b'|+|t_{*}|+2$ (as $-b'\le|b'|$ by claims 2 and 3 of Properties of the Absolute Value in an Ordered Field), so $\mathcal{E}(\nu'_{n})<\delta^{-1}(|b'|+|t_{*}|+2)$ (claims 7 and 10 of Elementary Order Arithmetic in an Ordered Field); with $-|e_{0}|-1<e_{0}\le\mathcal{E}(\nu'_{n})$ and $c'_{\mathcal{E}}=\delta^{-1}(|b'|+|t_{*}|+2)+|e_{0}|+1$ we get $|\mathcal{E}(\nu'_{n})|<c'_{\mathcal{E}}$ exactly as in Step 4.

The partner datum is now on the subsolution side: $\xi_{0}=(\nu_{0},0,0_{\nu_{0}},\mathbb{Y})$ is a test datum for $F$, so $F^{-}_{\delta}(\xi_{0})$ is a real number. Put

chosen after $b'$, $e_{0}$, $\nu_{0}$ and $\xi_{0}$ and independent of $n$; exactly as for $R$ in Step 4, $R'$ is positive and exceeds each of $\sqrt{M_{2}(\sigma^{*})}+2$, $c'_{\mathcal{E}}$, $|t_{*}|+2$, $\lVert W_{*}\rVert_{\sigma^{*}}+2$, $\lVert\mathbb{Y}\rVert+2$, $\sqrt{M_{2}(\nu_{0})}$, $|\mathcal{E}(\nu_{0})|$, $0$ and $|F^{-}_{\delta}(\xi_{0})|+1$. So $\eta_{n}=(\nu'_{n},t_{n},q'_{n},Y_{n})$ and $\xi_{0}$ are $R'$-bounded test data. From $-\epsilon_{n}\le F^{+}_{\delta}(\eta_{n})$ we get $-F^{+}_{\delta}(\eta_{n})\le\epsilon_{n}\le1$ (claim 4 of Elementary Order Arithmetic in an Ordered Field), and $F^{-}_{\delta}(\xi_{0})\le|F^{-}_{\delta}(\xi_{0})|$ (claim 3 of Properties of the Absolute Value in an Ordered Field), so

So $\eta_{n}\in S^{+}_{\delta,R'}$ for every $n$ (Test Data for an Intrinsic Second-Order Equation Operator on the Wasserstein Space and the Admissible Sets §admissible), with the $R'$-bounded partner $\xi_{0}$.

Step 9 (Clause 2: closed score and shift-semicontinuity). The shift-coercivity condition gives a score bound $C'\ge0$ for $F$ at $(\delta,R')$ (The Shift-Coercivity Condition for an Equation Operator on the Wasserstein Space §coercivity, The Shift-Coercivity Condition for an Equation Operator on the Wasserstein Space §bound), and this bound applies to elements of $S^{+}_{\delta,R'}$, so $\lVert\Sigma(\nu'_{n})\rVert_{\nu'_{n}}\le C'$ for every $n$. By Penalty Pairs with Closed Score Along Couplings §closed at the level $C'$, applied to $(\nu'_{n})_{n}$ in $\mathcal{D}_{\Sigma}$, $\sigma^{*}\in\mathcal{D}$ and $(\kappa'_{n})_{n}$, we get $\sigma^{*}\in\mathcal{D}_{\Sigma}$ and that $(\Sigma(\nu'_{n}))_{n}$ converges weakly to $\Sigma(\sigma^{*})$ along $(\kappa'_{n})_{n}$. So $(\sigma^{*},W_{*})\in\mathcal{V}(\mathcal{D}_{\Sigma})$ and $\eta_{*}=(\sigma^{*},t_{*},W_{*},\mathbb{Y})$ is a test datum. With $R'''=R'+C'$, positive, each $\eta_{n}$ is $R'''$-bounded and $\lVert\Sigma(\nu'_{n})\rVert_{\nu'_{n}}\le R'''$, so by Step 7 and the above $(\eta_{n})_{n}$ converges to $\eta_{*}$ along $(\kappa'_{n})_{n}$ with score bounded by $R'''$ (The Shift-Semicontinuity Condition for an Equation Operator on the Wasserstein Space §converging), and $F$ is shift-semicontinuous at $(\delta,R''')$ (The Shift-Semicontinuity Condition for an Equation Operator on the Wasserstein Space §semicontinuity). Let $\varepsilon$ be positive; Step 2(d) gives $N$ with $\epsilon_{n}<\varepsilon$ for $n\ge N$, so $0-\varepsilon<-\epsilon_{n}\le F^{+}_{\delta}(\eta_{n})$ for $n\ge N$ (claims 4 and 2 of Elementary Order Arithmetic in an Ordered Field). The second implication of The Shift-Semicontinuity Condition for an Equation Operator on the Wasserstein Space §level with $c=0$ gives

which, with $\sigma^{*}\in\mathcal{D}_{\Sigma}$, is clause 2.