Proof of A Cross-Level Comparison Principle for First-Order Equations on a Fine Noise Wasserstein Space and a Family of Coarse Levels Linked by Mode Restrictions

Each result cited is universally quantified over the data in its own statement. For each $N\in\mathbb{N}$ the fine level and level $N$ are read as level 1 and level 2 of Two Noise Levels over Hilbert Spaces: a Fine and a Coarse Reading of the Noise Framework and Level-Indexed Notation §levels, as in the statement; every item cited for two levels (Properties of a Mode-Restriction Link: Isometric Embedding, Contraction of Noise Norms and Noise Wasserstein Distances, and Pull-Back of Cylindrical Functions and Tangent Fields, Noise Intrinsic Test Functions Pulled Back along a Mode Restriction, and the Squared Coarse Noise Wasserstein Distance as a Fine Test Function, Compatible Noise Penalty Pairs at a Fine and a Coarse Level, Cross-Level Data and Eta-Consistency of a Fine and a Coarse Level at (delta, R) and Bounds, Monotonicity and Borwein-Preiss Perturbed Maximisers of the Cross-Level Doubled Function in the Coarse Noise Wasserstein Distance for Compatible Noise-Closed Penalty Pairs) is applied with the link $(\kappa^{(N)},p^{(N)},j^{(N)})$ and the pairs $\mathcal{P}$ and $\mathcal{P}^{(N)}$, and an item of the single-level framework cited at level $N$ is read with the level-$N$ data. Once $N$ is fixed we write $p=p^{(N)}$ and $j=j^{(N)}$, $\mathrm{id}$ for the identity of $X^{(N)}$, and $\mathrm{id}-S=-(S-\mathrm{id})$ for a noise-optimal map $S$ at level $N$. At the fine level $W=W_{a}$ is a metric on $\mathcal{P}^{a}_{\rho}$ by The Noise Wasserstein Distance is a Metric on the Measures Noise-Connected to the Reference Measure: Existence of Noise-Optimal Couplings, Comparison with the Quadratic Wasserstein Distance and Lower Semicontinuity §metric, with $\rho\in\mathcal{P}^{a}_{\rho}$ by The Noise Wasserstein Distance is a Metric on the Measures Noise-Connected to the Reference Measure: Existence of Noise-Optimal Couplings, Comparison with the Quadratic Wasserstein Distance and Lower Semicontinuity §reference and $\mathcal{D}_{\Sigma}\subseteq\mathcal{D}\subseteq\mathcal{P}^{a}_{\rho}$ by Noise Penalty Pairs on the Noise Wasserstein Space: the Penalty, Its Score, and Their Domains §pair, and $\lVert\cdot\rVert_{\mu}$ is the norm of $L^{2}(\mu;X^{a})$; the same items give the same facts for $W^{(N)}_{a}$, $\rho^{(N)}$, $\mathcal{D}^{(N)}_{\Sigma}\subseteq\mathcal{D}^{(N)}\subseteq\mathcal{P}^{a}_{\rho^{(N)}}$ and $\lVert\cdot\rVert^{(N)}_{\nu}$ at level $N$. $|s|$ is the absolute value of $s\in\mathbb{R}$, $\alpha^{-1}$ the multiplicative inverse of a positive $\alpha$, and $J=\{\delta\in\mathbb{R}:0<\delta<1\}$. Elementary order arithmetic, finite sums, maxima and minima of finitely many reals and the Archimedean property of $\mathbb{R}$ are in force by The Real Numbers: Standing Notation and Background §background.

We prove claim 1 in Steps 1 to 14 and claim 2 in Step 15. The order of choice is: $b,b',\theta,c$ (given); $e_{0}$, $c_{0}$, $e_{1}$, $\sigma_{0}$, $A$, $R_{0}$, $\lambda$, $(\omega_{1},\omega_{2})$, $\epsilon$, $t_{1}$, $\beta_{0}$, $\eta_{1}$, $L_{0}$, $n$ and the strengths $\alpha_{i}$, the numbers $t^{(i)}_{2}$, $t_{2}$, $\eta_{2}$, $\delta_{*}$, $\delta_{**}$, $m$ and the weights $\delta_{j}$ (Step 1); for each cell $(i,j)$ of the resulting finite grid, $K_{ij}$, $B^{f}_{ij}$, $B^{c}_{ij}$, $R_{ij}$, $C^{f}_{ij}$, $C^{c}_{ij}$, $\bar{R}_{ij}$, $\gamma_{ij}$ and $N^{ij}_{0}$, and then $N_{1}$ (Step 2). None of these depends on $N$, $u$ or $v$. Then $N\ge N_{1}$, $u$, $v$ and a violating point $\mu_{0}$ are given (end of Step 2), a cell $(i,j)$ is selected (Step 4), the constants $K'$, $B_{1}$, $B_{2}$ of the doubling lemma are fixed and finally $\tau$, after which the perturbed maximiser is taken (Step 5).

Step 1 (Constants fixed by the data and by $b$, $b'$, $\theta$, $c$). Let $b,b',\theta,c$ be as in claim 1. By (H3) fix $e_{0}$ with $e_{0}\le\mathcal{E}^{(N)}(\nu)$ for every $N$ and $\nu\in\mathcal{D}^{(N)}$, and by (H8) fix $c_{0}\ge0$ such that $\mathcal{P}$ and $\mathcal{P}^{(N)}$ are compatible with constant $c_{0}$ for every $N$. Put $e_{1}=e_{0}-c_{0}$. For $N\in\mathbb{N}$ and $\mu\in\mathcal{D}$, Compatible Noise Penalty Pairs at a Fine and a Coarse Level §compatible gives $p^{(N)}_{\#}\mu\in\mathcal{D}^{(N)}$ and $\mathcal{E}^{(N)}(p^{(N)}_{\#}\mu)\le\mathcal{E}(\mu)+c_{0}$, so with (H3)

$$e_{0}\le\mathcal{E}^{(N)}(p^{(N)}_{\#}\mu)\le\mathcal{E}(\mu)+c_{0}\qquad\text{and hence}\qquad e_{1}\le\mathcal{E}(\mu)\qquad(\mu\in\mathcal{D},\ N\in\mathbb{N}).\qquad(1\mathrm{a})$$

By Noise Penalty Pairs on the Noise Wasserstein Space: the Penalty, Its Score, and Their Domains §nonempty fix $\sigma_{0}\in\mathcal{D}_{\Sigma}$; for $\delta\in J$ the zero field $0\in L^{2}(\sigma_{0};X^{a})$ gives $(\sigma_{0},0)\in\mathcal{V}^{a}(\mathcal{D}_{\Sigma})$, so the reals $F^{-}_{\delta}(\sigma_{0},0,0)$ and $F^{+}_{\delta}(\sigma_{0},0,0)$ are defined. Put

$$A=|b|+|b'|+|e_{0}|+|e_{1}|+c_{0}+1,\qquad R_{0}=2A,$$

both positive. By (H4) fix a positive $\lambda$ that is a properness constant for $F^{(N)}$ at $R_{0}$ for every $N$, and by (H6) a pair $(\omega_{1},\omega_{2})$ that is a structure pair for $F^{(N)}$ at $R_{0}$ for every $N$. Put $\epsilon=\lambda\theta/16$, positive. By The First-Order Structure Condition at Uniquely Noise-Mapped Pairs §pair $\omega_{1}$ is a modulus of continuity; by clause 2 of Modulus of Continuity fix a positive $t_{1}$ with $\omega_{1}(t)\le\epsilon$ whenever $0\le t\le t_{1}$, and put $\beta_{0}=1+2t_{1}^{-1}$ and $\eta_{1}=t_{1}/16$. Put $L_{0}=|b|+|b'|+|e_{0}|+|e_{1}|+1$, and fix $n\in\mathbb{N}$ with $2L_{0}\eta_{1}^{-1}\le n$. For $i\in\{0,\dots,n\}$ put $\alpha_{i}=2^{i}\beta_{0}$. Then $1<\alpha_{i}$, $\alpha_{i-1}=\alpha_{i}/2$ for $1\le i\le n$, and for $1\le i\le n$ one has $\alpha_{i}^{-1}\le(2\beta_{0})^{-1}<t_{1}/4$, as $4t_{1}^{-1}<2\beta_{0}$. For $1\le i\le n$, the function with value $\omega_{2}(t,\alpha_{i})$ at $t\ge0$ is a modulus of continuity by The First-Order Structure Condition at Uniquely Noise-Mapped Pairs §pair, as $1<\alpha_{i}$; by clause 2 of Modulus of Continuity fix a positive $t^{(i)}_{2}$ with $\omega_{2}(t,\alpha_{i})\le\epsilon$ whenever $0\le t\le t^{(i)}_{2}$. Let $t_{2}$ be the least of $t^{(1)}_{2},\dots,t^{(n)}_{2}$ and $\eta_{2}=t_{2}/8$. Put

$$\delta_{*}=\theta\bigl(4|c|+2c_{0}+2\bigr)^{-1},\qquad\delta_{**}=\text{the least of }\tfrac{1}{2},\ \delta_{*}\text{ and }t_{2}\bigl(8|e_{0}|+4\bigr)^{-1},$$

fix $m\in\mathbb{N}$ with $2L_{0}\eta_{2}^{-1}\le m$, and for $j\in\{0,\dots,m\}$ put $\delta_{j}=2^{-j}\delta_{**}$. Then $\delta_{j}\in J$, $\delta_{j}\le\delta_{*}$, $\delta_{j+1}=\delta_{j}/2$ for $j<m$, $\delta_{j}(2|c|+c_{0})<\theta/2$ and $\delta_{j}(2|e_{0}|+1)\le t_{2}/4$. A cell is a pair $(i,j)$ with $1\le i\le n$ and $0\le j\le m-1$; there are $nm$ cells.

Step 2 (Constants attached to the cells, and $N_{1}$). Fix a cell $(i,j)$ and write $\alpha=\alpha_{i}$ and $\delta=\delta_{j}$ in this step. Put $K_{ij}=\delta^{-1}\bigl(|b|+|b'|+|e_{0}|+|e_{1}|+c_{0}\bigr)$, positive. As $\mathcal{P}$ is noise-closed (H1), Noise-Closed Noise Penalty Pairs §bounded with $c=K_{ij}$ gives a real number whose absolute value we call $B^{f}_{ij}$, so that $0\le B^{f}_{ij}$ and $W(\mu,\rho)\le B^{f}_{ij}$ for every $\mu\in\mathcal{D}$ with $\mathcal{E}(\mu)\le K_{ij}$; by (H3) with $c=K_{ij}$, let $B^{c}_{ij}$ be the absolute value of the number $B_{K_{ij}}$ provided there, so that $0\le B^{c}_{ij}$ and $W^{(N)}_{a}(\nu,\rho^{(N)})\le B^{c}_{ij}$ for every $N$ and every $\nu\in\mathcal{D}^{(N)}$ with $\mathcal{E}^{(N)}(\nu)\le K_{ij}$. Put

$$R_{ij}=(2\alpha+1)\bigl(B^{f}_{ij}+B^{c}_{ij}\bigr)+|e_{0}|+|e_{1}|+K_{ij}+A+2\epsilon+W(\sigma_{0},\rho)+|\mathcal{E}(\sigma_{0})|+\bigl|F^{-}_{\delta}(\sigma_{0},0,0)\bigr|+\bigl|F^{+}_{\delta}(\sigma_{0},0,0)\bigr|+1,$$

a positive real. Since $F$ satisfies the shift-coercivity condition (H1) and $\delta\in J$, fix by The Shift-Coercivity Condition for a First-Order Equation Operator on the Noise Wasserstein Space §coercivity a score bound $C^{f}_{ij}\ge0$ for $F$ at $(\delta,R_{ij})$; by (H5) fix $C^{c}_{ij}\ge0$ that is a score bound for $F^{(N)}$ at $(\delta,R_{ij})$ for every $N$. Put $\bar{R}_{ij}=R_{ij}+C^{f}_{ij}+C^{c}_{ij}$. Since $F$ has momentum-continuous shifts (H1), fix, applied with $\delta$ and $\bar{R}_{ij}$, the positive constant that First-Order Equation Operators on the Noise Wasserstein Space with Momentum-Continuous Shifts §momentum provides for $\eta=\epsilon$, which we call $\gamma^{f}_{ij}$; by (H7) with $\delta$, $\bar{R}_{ij}$ and $\eta=\epsilon$ fix the positive constant $\gamma^{c}_{ij}$ provided there, valid for every $N$; let $\gamma_{ij}$ be the lesser of $\gamma^{f}_{ij}$ and $\gamma^{c}_{ij}$. By (H9) with $\eta=\epsilon$, $\delta$ and $R=\bar{R}_{ij}$ fix $N^{ij}_{0}\in\mathbb{N}$ such that for every $N\ge N^{ij}_{0}$ the fine level with $F$ and $\mathcal{P}$ and level $N$ with $F^{(N)}$ and $\mathcal{P}^{(N)}$ are $\epsilon$-consistent at $(\delta,\bar{R}_{ij})$ (Cross-Level Data and Eta-Consistency of a Fine and a Coarse Level at (delta, R) §consistent).

Let $N_{1}$ be the greatest of the finitely many numbers $N^{ij}_{0}$, $(i,j)$ a cell. It depends only on the data of the statement and on $b,b',\theta,c$. Let $N\in\mathbb{N}$ with $N_{1}\le N$, let $u$ be a viscosity subsolution of $F$ relative to $\mathcal{P}$ with $u\le b$ on $\mathcal{D}$, and $v$ a viscosity supersolution of $F^{(N)}$ relative to $\mathcal{P}^{(N)}$ with $b'\le v$ on $\mathcal{D}^{(N)}$. We show $u(\mu)\le v(p_{\#}\mu)+\theta$ for every $\mu\in\mathcal{D}$ with $\mathcal{E}(\mu)\le c$; suppose instead that $\mu_{0}\in\mathcal{D}$ satisfies $\mathcal{E}(\mu_{0})\le c$ and $\theta<u(\mu_{0})-v(p_{\#}\mu_{0})$.

Step 3 (The doubled function and its corrected maximum). By Basic Properties of the Delta-Envelopes on the Noise Wasserstein Space, and the Envelopes of Bounded Functions for a Noise-Closed Penalty Pair §growth, $u$ has penalty-subordinate growth from above relative to $\mathcal{P}$, and, at level N, $v$ has penalty-subordinate growth from below relative to $\mathcal{P}^{(N)}$, both pairs being noise-closed by (H1) and (H2). By Basic Properties of the Delta-Envelopes on the Noise Wasserstein Space, and the Envelopes of Bounded Functions for a Noise-Closed Penalty Pair §duality at level N, $-v$ has penalty-subordinate growth from above and $(-v)^{-}_{\delta}=-v^{+}_{\delta}$ on $\mathcal{D}^{(N)}$. For $\delta\in J$ put $U_{\delta}=u^{-}_{\delta}$ on $\mathcal{D}$ and $V_{\delta}=-v^{+}_{\delta}$ on $\mathcal{D}^{(N)}$. By Basic Properties of the Delta-Envelopes on the Noise Wasserstein Space, and the Envelopes of Bounded Functions for a Noise-Closed Penalty Pair §semicontinuity, at the fine level for $u$ and at level N for $-v$, both are upper semicontinuous, and by Basic Properties of the Delta-Envelopes on the Noise Wasserstein Space, and the Envelopes of Bounded Functions for a Noise-Closed Penalty Pair §bounded, at the fine level for $u\le b$ and at level N for $b'\le v$,

$$U_{\delta}(\mu)\le b-\delta\,\mathcal{E}(\mu)\quad(\mu\in\mathcal{D}),\qquad V_{\delta}(\nu)\le-b'-\delta\,\mathcal{E}^{(N)}(\nu)\quad(\nu\in\mathcal{D}^{(N)}).$$

The pairs are noise-closed and compatible (H1, H2, H8), and $e_{1}$, $e_{0}$ are lower bounds of $\mathcal{E}$ on $\mathcal{D}$ and of $\mathcal{E}^{(N)}$ on $\mathcal{D}^{(N)}$ by (1a) and (H3). So Bounds, Monotonicity and Borwein-Preiss Perturbed Maximisers of the Cross-Level Doubled Function in the Coarse Noise Wasserstein Distance for Compatible Noise-Closed Penalty Pairs applies with $\delta$, $b_{1}=b$, $b_{2}=-b'$, $U=U_{\delta}$, $V=V_{\delta}$ and the lower bounds $e_{1}$ and $e_{0}$; write $\Psi_{\delta,\alpha}$ and $M(\delta,\alpha)$ for its $\Psi_{\alpha}$ and $M(\alpha)$, so that

$$\Psi_{\delta,\alpha}(\mu,\nu)=u^{-}_{\delta}(\mu)-v^{+}_{\delta}(\nu)-\tfrac{\alpha}{2}W^{(N)}_{a}(p_{\#}\mu,\nu)^{2}.$$

By Bounds, Monotonicity and Borwein-Preiss Perturbed Maximisers of the Cross-Level Doubled Function in the Coarse Noise Wasserstein Distance for Compatible Noise-Closed Penalty Pairs §bounds, for $\alpha>0$ and $(\mu,\nu)\in\mathcal{D}\times\mathcal{D}^{(N)}$,

$$\Psi_{\delta,\alpha}(\mu,\nu)\le b-b'-\delta\bigl(\mathcal{E}(\mu)+\mathcal{E}^{(N)}(\nu)\bigr)\le b-b'-\delta(e_{1}+e_{0}),\qquad(3\mathrm{a})$$

and $M(\delta,\alpha)$ is a real number. For $\alpha>0$ and $\delta\in J$ put

$$G(\alpha,\delta)=M(\delta,\alpha)+\delta(e_{1}+e_{0}),\qquad\text{so that}\qquad G(\alpha,\delta)\le b-b'.\qquad(3\mathrm{b})$$

Lower bound. Let $\delta\in J$ with $\delta\le\delta_{*}$ and $\alpha>0$. By Basic Properties of the Delta-Envelopes on the Noise Wasserstein Space, and the Envelopes of Bounded Functions for a Noise-Closed Penalty Pair §semicontinuity, $u(\mu_{0})-\delta\mathcal{E}(\mu_{0})\le u^{-}_{\delta}(\mu_{0})$ and, at level N, $v^{+}_{\delta}(p_{\#}\mu_{0})\le v(p_{\#}\mu_{0})+\delta\mathcal{E}^{(N)}(p_{\#}\mu_{0})$; with $\mathcal{E}(\mu_{0})\le c$ and $\mathcal{E}^{(N)}(p_{\#}\mu_{0})\le c+c_{0}$ by (1a), and Bounds, Monotonicity and Borwein-Preiss Perturbed Maximisers of the Cross-Level Doubled Function in the Coarse Noise Wasserstein Distance for Compatible Noise-Closed Penalty Pairs §diagonal at $\mu_{0}$,

$$M(\delta,\alpha)\ge U_{\delta}(\mu_{0})+V_{\delta}(p_{\#}\mu_{0})\ge u(\mu_{0})-v(p_{\#}\mu_{0})-\delta\bigl(2|c|+c_{0}\bigr)>\theta-\tfrac{\theta}{2}=\tfrac{\theta}{2},\qquad(3\mathrm{c})$$

using $\delta(2|c|+c_{0})\le\delta_{*}(2|c|+c_{0})<\theta/2$. As $\delta|e_{1}+e_{0}|\le|e_{0}|+|e_{1}|$, this and (3b) give $\theta/2-|e_{0}|-|e_{1}|<G(\alpha,\delta)\le b-b'$, so any two values of $G$ at such arguments differ by less than $b-b'-\theta/2+|e_{0}|+|e_{1}|<L_{0}$. (3d)

Decreasing the strength. By Bounds, Monotonicity and Borwein-Preiss Perturbed Maximisers of the Cross-Level Doubled Function in the Coarse Noise Wasserstein Distance for Compatible Noise-Closed Penalty Pairs §monotone, $M(\delta,\alpha)\le M(\delta,\alpha')$ for $0<\alpha'<\alpha$, so $G(\alpha,\delta)\le G(\alpha',\delta)$. (3e) Let $0<\alpha'<\alpha$, $\tau\ge0$ and $(\mu,\nu)\in\mathcal{D}\times\mathcal{D}^{(N)}$ with $M(\delta,\alpha)-\tau\le\Psi_{\delta,\alpha}(\mu,\nu)$. By Bounds, Monotonicity and Borwein-Preiss Perturbed Maximisers of the Cross-Level Doubled Function in the Coarse Noise Wasserstein Distance for Compatible Noise-Closed Penalty Pairs §strength,

$$\tfrac{\alpha-\alpha'}{2}\,W^{(N)}_{a}(p_{\#}\mu,\nu)^{2}\le M(\delta,\alpha')-M(\delta,\alpha)+\tau=G(\alpha',\delta)-G(\alpha,\delta)+\tau.\qquad(3\mathrm{f})$$

Decreasing the weight. Let $\alpha>0$, $\delta,\delta'\in J$ with $\delta'<\delta$, $\tau\ge0$ and $(\mu,\nu)\in\mathcal{D}\times\mathcal{D}^{(N)}$ with $M(\delta,\alpha)-\tau\le\Psi_{\delta,\alpha}(\mu,\nu)$. By Basic Properties of the Delta-Envelopes on the Noise Wasserstein Space, and the Envelopes of Bounded Functions for a Noise-Closed Penalty Pair §monotone, at the fine level for $u$ and at level N for $v$, $U_{\delta}(\mu)+(\delta-\delta')\mathcal{E}(\mu)\le U_{\delta'}(\mu)$ for $\mu\in\mathcal{D}$ and $v^{+}_{\delta'}(\nu)\le v^{+}_{\delta}(\nu)-(\delta-\delta')\mathcal{E}^{(N)}(\nu)$, that is $V_{\delta}(\nu)+(\delta-\delta')\mathcal{E}^{(N)}(\nu)\le V_{\delta'}(\nu)$, for $\nu\in\mathcal{D}^{(N)}$; the function with value $U_{\delta'}(\mu)+V_{\delta'}(\nu)-\tfrac{\alpha}{2}W^{(N)}_{a}(p_{\#}\mu,\nu)^{2}$ is $\Psi_{\delta',\alpha}$, bounded above with supremum $M(\delta',\alpha)$. So Bounds, Monotonicity and Borwein-Preiss Perturbed Maximisers of the Cross-Level Doubled Function in the Coarse Noise Wasserstein Distance for Compatible Noise-Closed Penalty Pairs §weight with $s=\delta-\delta'$, $U'=U_{\delta'}$ and $V'=V_{\delta'}$ gives $M(\delta,\alpha)-\tau+(\delta-\delta')(\mathcal{E}(\mu)+\mathcal{E}^{(N)}(\nu))\le M(\delta',\alpha)$; subtracting $(\delta-\delta')(e_{1}+e_{0})$ and using the definition of $G$,

$$(\delta-\delta')\bigl(\mathcal{E}(\mu)-e_{1}+\mathcal{E}^{(N)}(\nu)-e_{0}\bigr)\le G(\alpha,\delta')-G(\alpha,\delta)+\tau.\qquad(3\mathrm{g})$$

The left side is nonnegative by (1a) and (H3). For every positive $\tau$ such a pair $(\mu,\nu)$ exists by the approximation property of the supremum $M(\delta,\alpha)$, so $G(\alpha,\delta)\le G(\alpha,\delta')+\tau$ for every positive $\tau$, and Comparison of Real Numbers with Arbitrary Positive Slack §slack-above gives $G(\alpha,\delta)\le G(\alpha,\delta')$ whenever $\delta'<\delta$. (3h)

Step 4 (Choice of a cell by a finite pigeonhole). For $0\le i\le n$ and $0\le j\le m$ put $g(i,j)=G(\alpha_{i},\delta_{j})$; all $\delta_{j}$ lie in $J$ and are at most $\delta_{*}$. For a cell $(i,j)$ put $d_{1}(i,j)=g(i-1,j)-g(i,j)$ and $d_{2}(i,j)=g(i,j+1)-g(i,j)$; $d_{1}(i,j)\ge0$ by (3e), as $\alpha_{i-1}<\alpha_{i}$, and $d_{2}(i,j)\ge0$ by (3h), as $\delta_{j+1}<\delta_{j}$. We claim that some cell satisfies $d_{1}(i,j)<\eta_{1}$ and $d_{2}(i,j)<\eta_{2}$. Otherwise every cell has $d_{1}(i,j)\ge\eta_{1}$ or $d_{2}(i,j)\ge\eta_{2}$, so, both being nonnegative, $1\le d_{1}(i,j)\eta_{1}^{-1}+d_{2}(i,j)\eta_{2}^{-1}$. Summing over the $nm$ cells, the sums telescope: for fixed $j$, $\sum_{i=1}^{n}d_{1}(i,j)=g(0,j)-g(n,j)<L_{0}$, and for fixed $i$, $\sum_{j=0}^{m-1}d_{2}(i,j)=g(i,m)-g(i,0)<L_{0}$, both by (3d). Hence

$$nm<mL_{0}\eta_{1}^{-1}+nL_{0}\eta_{2}^{-1}\le\tfrac{mn}{2}+\tfrac{nm}{2}=nm,$$

by the choice of $n$ and $m$ in Step 1, which is absurd. Fix such a cell $(i,j)$; it depends on $N$, $u$, $v$ and $\mu_{0}$, but it is one of the finitely many cells of Step 1. From now on $\alpha=\alpha_{i}$, $\delta=\delta_{j}$, and $K=K_{ij}$, $B^{f}=B^{f}_{ij}$, $B^{c}=B^{c}_{ij}$, $R=R_{ij}$, $C^{f}=C^{f}_{ij}$, $C^{c}=C^{c}_{ij}$, $\bar{R}=\bar{R}_{ij}$, $\gamma=\gamma_{ij}$ are the constants of Step 2 for this cell. Since $\alpha/2=\alpha_{i-1}$ and $\delta/2=\delta_{j+1}$,

$$G(\tfrac{\alpha}{2},\delta)-G(\alpha,\delta)<\eta_{1},\qquad G(\alpha,\tfrac{\delta}{2})-G(\alpha,\delta)<\eta_{2}.\qquad(4\mathrm{a})$$

Moreover $1<\alpha$, $\alpha^{-1}<t_{1}/4$, $\omega_{2}(t,\alpha)\le\epsilon$ for $0\le t\le t_{2}$ (as $t_{2}\le t^{(i)}_{2}$), $\delta,\tfrac{\delta}{2}\in J$, $\delta\le\delta_{*}$ and $\delta(2|e_{0}|+1)\le t_{2}/4$ (Step 1); and since $N\ge N_{1}\ge N^{ij}_{0}$, the fine level and level $N$ are $\epsilon$-consistent at $(\delta,\bar{R})$ (Step 2). Write $\Psi=\Psi_{\delta,\alpha}$ and $M=M(\delta,\alpha)$; by (3c), $\theta/2<M(\delta,\alpha')$ for every $\alpha'>0$.

Step 5 (The perturbed maximiser). By Bounds, Monotonicity and Borwein-Preiss Perturbed Maximisers of the Cross-Level Doubled Function in the Coarse Noise Wasserstein Distance for Compatible Noise-Closed Penalty Pairs §radius with $m=\theta/2$, admissible by (3c), fix $K',B_{1},B_{2}\in\mathbb{R}$ with $0\le B_{1}$ and $0\le B_{2}$, not depending on the strength or the tolerance, with $W(\sigma,\rho)\le B_{1}$ for $\sigma\in\mathcal{D}$ with $\mathcal{E}(\sigma)\le K'$ and $W^{(N)}_{a}(\sigma,\rho^{(N)})\le B_{2}$ for $\sigma\in\mathcal{D}^{(N)}$ with $\mathcal{E}^{(N)}(\sigma)\le K'$. (These may depend on $N$, $u$, $v$; they enter only the choice of $\tau$.) Let $\gamma'$ be the lesser of $\gamma$ and $1$, and let $\tau$ be the least of $\tfrac{1}{2}$, $\tfrac{\theta}{16}$, $\eta_{1}$, $\eta_{2}$ and $\gamma'\bigl(8(B_{1}+B_{2})+2\bigr)^{-1}$. Then $0<\tau<1$, $\tau<\theta/2$, $\lambda\tau\le\epsilon$, $\tau\le\eta_{1}$, $\tau\le\eta_{2}$, and $4\tau B_{1}<\gamma'$ and $4\tau B_{2}<\gamma'$. By Bounds, Monotonicity and Borwein-Preiss Perturbed Maximisers of the Cross-Level Doubled Function in the Coarse Noise Wasserstein Distance for Compatible Noise-Closed Penalty Pairs §perturbed, with $\alpha$ and $\tau$, there are $(\hat{\mu},\hat{\nu})\in\mathcal{D}\times\mathcal{D}^{(N)}$, a sequence $(\mu_{k})_{k}$ in $\mathcal{D}$, a sequence $(\nu_{k})_{k}$ in $\mathcal{D}^{(N)}$ and positive reals $c_{k}$ whose series converges with $\sum_{k=1}^{\infty}c_{k}\le\tau$, such that: by Bounds, Monotonicity and Borwein-Preiss Perturbed Maximisers of the Cross-Level Doubled Function in the Coarse Noise Wasserstein Distance for Compatible Noise-Closed Penalty Pairs §localised, $\mathcal{E}(\hat{\mu})\le K'$, $\mathcal{E}(\mu_{k})\le K'$, $\mathcal{E}^{(N)}(\hat{\nu})\le K'$ and $\mathcal{E}^{(N)}(\nu_{k})\le K'$, so $W(\mu_{k},\rho)\le B_{1}$ and $W^{(N)}_{a}(\nu_{k},\rho^{(N)})\le B_{2}$, and by Basic Properties of a Noise-Closed Noise Penalty Pair: Lower Bound, Lower Semicontinuity, Complete Sublevel Sets and Bounded Distances §diameter (with $c=K'$ and $B_{1}$ at the fine level, and with $c=K'$ and $B_{2}$ at level N) $W(\hat{\mu},\mu_{k})\le2B_{1}$ and $W^{(N)}_{a}(\hat{\nu},\nu_{k})\le2B_{2}$; by Bounds, Monotonicity and Borwein-Preiss Perturbed Maximisers of the Cross-Level Doubled Function in the Coarse Noise Wasserstein Distance for Compatible Noise-Closed Penalty Pairs §near-maximiser, $M-\tau\le\Psi(\hat{\mu},\hat{\nu})$; and by Bounds, Monotonicity and Borwein-Preiss Perturbed Maximisers of the Cross-Level Doubled Function in the Coarse Noise Wasserstein Distance for Compatible Noise-Closed Penalty Pairs §strict-maximum, with $\Phi$ as there, $\Phi(\mu,\nu)<\Phi(\hat{\mu},\hat{\nu})$ for every $(\mu,\nu)\in\mathcal{D}\times\mathcal{D}^{(N)}$ other than $(\hat{\mu},\hat{\nu})$.

Step 6 (Bounds at the maximiser). Put $r_{1}=u^{-}_{\delta}(\hat{\mu})$ and $r_{2}=v^{+}_{\delta}(\hat{\nu})$. As $\Psi(\hat{\mu},\hat{\nu})=r_{1}-r_{2}-\tfrac{\alpha}{2}W^{(N)}_{a}(p_{\#}\hat{\mu},\hat{\nu})^{2}$, Step 5 and (3c) give

$$r_{1}-r_{2}\ge\Psi(\hat{\mu},\hat{\nu})\ge M-\tau>\tfrac{\theta}{2}-\tau>0.\qquad(6\mathrm{a})$$

By (3a) and $\Psi(\hat{\mu},\hat{\nu})>0$, with (1a) and (H3), $\delta\mathcal{E}(\hat{\mu})<b-b'-\delta\mathcal{E}^{(N)}(\hat{\nu})\le b-b'-\delta e_{0}\le|b|+|b'|+|e_{0}|$ and $\delta\mathcal{E}^{(N)}(\hat{\nu})<b-b'-\delta\mathcal{E}(\hat{\mu})\le|b|+|b'|+|e_{1}|$; by (1a) at $\hat{\mu}$, $\delta\mathcal{E}^{(N)}(p_{\#}\hat{\mu})\le\delta\mathcal{E}(\hat{\mu})+\delta c_{0}<|b|+|b'|+|e_{0}|+c_{0}$. From below, $\delta\mathcal{E}(\hat{\mu})\ge\delta e_{1}\ge-|e_{1}|$ and $\delta\mathcal{E}^{(N)}(\hat{\nu}),\delta\mathcal{E}^{(N)}(p_{\#}\hat{\mu})\ge\delta e_{0}\ge-|e_{0}|$. Hence

$$\delta|\mathcal{E}(\hat{\mu})|<A,\qquad\delta|\mathcal{E}^{(N)}(\hat{\nu})|<A,\qquad\delta|\mathcal{E}^{(N)}(p_{\#}\hat{\mu})|<A,\qquad(6\mathrm{b})$$

and, dividing the upper bounds by $\delta$, $\mathcal{E}(\hat{\mu})$, $\mathcal{E}^{(N)}(\hat{\nu})$ and $\mathcal{E}^{(N)}(p_{\#}\hat{\mu})$ are all less than $K$; with the lower bounds and $0<K$,

$$|\mathcal{E}(\hat{\mu})|\le|e_{1}|+K,\qquad|\mathcal{E}^{(N)}(\hat{\nu})|\le|e_{0}|+K,\qquad|\mathcal{E}^{(N)}(p_{\#}\hat{\mu})|\le|e_{0}|+K.\qquad(6\mathrm{c})$$

By the choice of $B^{f}$ and $B^{c}$ in Step 2 ($p_{\#}\hat{\mu}\in\mathcal{D}^{(N)}$ by (1a)), $W(\hat{\mu},\rho)\le B^{f}$, $W^{(N)}_{a}(\hat{\nu},\rho^{(N)})\le B^{c}$ and $W^{(N)}_{a}(p_{\#}\hat{\mu},\rho^{(N)})\le B^{c}$, so by The Noise Wasserstein Distance is a Metric on the Measures Noise-Connected to the Reference Measure: Existence of Noise-Optimal Couplings, Comparison with the Quadratic Wasserstein Distance and Lower Semicontinuity §triangle and The Noise Wasserstein Distance is a Metric on the Measures Noise-Connected to the Reference Measure: Existence of Noise-Optimal Couplings, Comparison with the Quadratic Wasserstein Distance and Lower Semicontinuity §symmetry at level N,

$$W^{(N)}_{a}(p_{\#}\hat{\mu},\hat{\nu})\le2B^{c}.\qquad(6\mathrm{d})$$

By Basic Properties of the Delta-Envelopes on the Noise Wasserstein Space, and the Envelopes of Bounded Functions for a Noise-Closed Penalty Pair §bounded, at the fine level and at level N, $r_{1}\le b-\delta\mathcal{E}(\hat{\mu})\le b-\delta e_{1}\le|b|+|e_{1}|$ and $r_{2}\ge b'+\delta\mathcal{E}^{(N)}(\hat{\nu})\ge-|b'|-|e_{0}|$; with (6a),

$$-A<r_{2}<r_{1}<A.\qquad(6\mathrm{e})$$

Step 7 (Noise-optimal maps and the coarse momentum). Since $\mathcal{D}^{(N)}$ has the noise map property (H2), $p_{\#}\hat{\mu},\hat{\nu}\in\mathcal{D}^{(N)}$ and $\nu_{k}\in\mathcal{P}^{a}_{\rho^{(N)}}$, the ordered pairs $(p_{\#}\hat{\mu},\hat{\nu})$, $(\hat{\nu},p_{\#}\hat{\mu})$ and $(\hat{\nu},\nu_{k})$ are uniquely noise-mapped at level N, and since $\mathcal{D}$ has the noise map property (H1), the ordered pairs $(\hat{\mu},\mu_{k})$ are uniquely noise-mapped at the fine level (The Noise Map Property of a Set of Probability Measures §map-property). By Noise-Optimal Maps and Uniquely Noise-Mapped Pairs §uniquely-mapped, at level N, fix a noise-optimal map $S$ from $p_{\#}\hat{\mu}$ to $\hat{\nu}$ and a noise-optimal map $S'$ from $\hat{\nu}$ to $p_{\#}\hat{\mu}$, and, by Axiom of Countable Choice, for each $k$ a noise-optimal map $S'_{k}$ from $\hat{\nu}$ to $\nu_{k}$ at level N and a noise-optimal map $S_{k}$ from $\hat{\mu}$ to $\mu_{k}$ at the fine level. Put

$$h=\alpha(\mathrm{id}-S)\in L^{2}(p_{\#}\hat{\mu};X^{(N)}_{a}).$$

By Squared Noise Wasserstein Distances, Their Convergent Series and Linear Combinations are Noise Intrinsic Test Functions §distance at level N, with $Q=\mathcal{D}^{(N)}$ and fixed measure $\hat{\nu}$, the function with value $W^{(N)}_{a}(\sigma,\hat{\nu})^{2}$ is a noise intrinsic test function on $\mathcal{D}^{(N)}$ whose gradient at $p_{\#}\hat{\mu}$ is $2(\mathrm{id}-S)$; by property (b) of Noise Intrinsic Test Functions on the Noise Wasserstein Space §differentiability this gradient lies in $T^{a,(N)}_{p_{\#}\hat{\mu}}$, which is a linear subspace by Linearity of the Noise Gradient, and the Noise Tangent Space is a Closed Linear Subspace §subspace, so $h=\tfrac{\alpha}{2}\cdot2(\mathrm{id}-S)$ and $-h=\alpha(S-\mathrm{id})$ lie in $T^{a,(N)}_{p_{\#}\hat{\mu}}$. By The Noise-Optimal Map: Transport Cost, Stability Along Nearly Optimal Couplings and Stability Under Perturbation of the Source §cost at level N, $\bigl(\lVert S-\mathrm{id}\rVert^{(N)}_{p_{\#}\hat{\mu}}\bigr)^{2}=W^{(N)}_{a}(p_{\#}\hat{\mu},\hat{\nu})^{2}$; by the scaling identity $d(\lambda x,\lambda y)=|\lambda|\,d(x,y)$ of The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §metric (with $y=0_{E}$, as $d(x,0_{E})=|x|$ by Real Inner Product Space §distance) in the real Hilbert space $L^{2}(p_{\#}\hat{\mu};X^{(N)}_{a})$ and claim 3 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, with (6d),

$$\lVert h\rVert^{(N)}_{p_{\#}\hat{\mu}}=\alpha W^{(N)}_{a}(p_{\#}\hat{\mu},\hat{\nu})\le2\alpha B^{c},\qquad\text{and likewise}\qquad\lVert\alpha(S'-\mathrm{id})\rVert^{(N)}_{\hat{\nu}}=\alpha W^{(N)}_{a}(\hat{\nu},p_{\#}\hat{\mu})\le2\alpha B^{c}.\qquad(7\mathrm{a})$$

The class $j\circ h\circ p\in L^{2}(\hat{\mu};X^{a})$ of Properties of a Mode-Restriction Link: Isometric Embedding, Contraction of Noise Norms and Noise Wasserstein Distances, and Pull-Back of Cylindrical Functions and Tangent Fields §fields satisfies, by Properties of a Mode-Restriction Link: Isometric Embedding, Contraction of Noise Norms and Noise Wasserstein Distances, and Pull-Back of Cylindrical Functions and Tangent Fields §fields-isometry,

$$\lVert j\circ h\circ p\rVert_{\hat{\mu}}=\lVert h\rVert^{(N)}_{p_{\#}\hat{\mu}}\le2\alpha B^{c}.\qquad(7\mathrm{b})$$

Step 8 (The fine test function and the fine exact inequality). Let $\varphi_{0},\chi:\mathcal{P}^{a}_{\rho}\to\mathbb{R}$ and $\psi_{0},\chi':\mathcal{P}^{a}_{\rho^{(N)}}\to\mathbb{R}$ be

$$\varphi_{0}(\mu)=W^{(N)}_{a}(p_{\#}\mu,\hat{\nu})^{2},\qquad\chi(\mu)=\sum_{k=1}^{\infty}c_{k}W(\mu,\mu_{k})^{2},\qquad\psi_{0}(\nu)=W^{(N)}_{a}(\nu,p_{\#}\hat{\mu})^{2},\qquad\chi'(\nu)=\sum_{k=1}^{\infty}c_{k}W^{(N)}_{a}(\nu,\nu_{k})^{2},$$

where $p_{\#}\mu\in\mathcal{P}^{a}_{\rho^{(N)}}$ by Properties of a Mode-Restriction Link: Isometric Embedding, Contraction of Noise Norms and Noise Wasserstein Distances, and Pull-Back of Cylindrical Functions and Tangent Fields §space, as $p_{\#}\rho\in\mathcal{P}^{a}_{\rho^{(N)}}$ by (H8), and the series converge by Squared Noise Wasserstein Distances, Their Convergent Series and Linear Combinations are Noise Intrinsic Test Functions §series-convergence (at the fine level with $Q=\mathcal{D}$, the bound $B_{1}$, the centres $\mu_{k}$ and $\beta_{k}=c_{k}$; at level N with $Q=\mathcal{D}^{(N)}$, the bound $B_{2}$, the centres $\nu_{k}$ and $\beta_{k}=c_{k}$; Step 5). By Noise Intrinsic Test Functions Pulled Back along a Mode Restriction, and the Squared Coarse Noise Wasserstein Distance as a Fine Test Function §distance-test, with $\mathcal{Q}_{1}=\mathcal{D}$ and $\mathcal{Q}_{2}=\mathcal{D}^{(N)}$ (which has the noise map property by (H2) and contains $p_{\#}\mu$ for $\mu\in\mathcal{D}$ by (1a)) and with $\hat{\nu}$ as its fixed measure, $\varphi_{0}$ is a noise intrinsic test function on $\mathcal{D}$ at the fine level, and by Noise Intrinsic Test Functions Pulled Back along a Mode Restriction, and the Squared Coarse Noise Wasserstein Distance as a Fine Test Function §distance-gradient, with the map $S$, $\nabla\varphi_{0}(\hat{\mu})=2\,j\circ(\mathrm{id}-S)\circ p$. By Squared Noise Wasserstein Distances, Their Convergent Series and Linear Combinations are Noise Intrinsic Test Functions §series-test at the fine level, $\chi$ is a noise intrinsic test function on $\mathcal{D}$, and by Squared Noise Wasserstein Distances, Their Convergent Series and Linear Combinations are Noise Intrinsic Test Functions §series-gradient, with the maps $S_{k}$, its gradient $q^{*}=\nabla\chi(\hat{\mu})$ satisfies

$$\lVert q^{*}\rVert_{\hat{\mu}}\le2\sum_{k=1}^{\infty}c_{k}W(\hat{\mu},\mu_{k})\le4B_{1}\sum_{k=1}^{\infty}c_{k}\le4\tau B_{1}<\gamma',$$

where the series $\sum_{k}c_{k}W(\hat{\mu},\mu_{k})$ converges by Squared Noise Wasserstein Distances, Their Convergent Series and Linear Combinations are Noise Intrinsic Test Functions §series-convergence, the middle step uses $W(\hat{\mu},\mu_{k})\le2B_{1}$ (Step 5) and the comparison and linearity of convergent series, and the last steps use $\sum_{k}c_{k}\le\tau$, $0\le B_{1}$ and Step 5. Put $\varphi=\tfrac{\alpha}{2}\varphi_{0}+\chi$. By Squared Noise Wasserstein Distances, Their Convergent Series and Linear Combinations are Noise Intrinsic Test Functions §linear at the fine level ($Q=\mathcal{D}$), $\varphi$ is a noise intrinsic test function on $\mathcal{D}$ and, by Properties of a Mode-Restriction Link: Isometric Embedding, Contraction of Noise Norms and Noise Wasserstein Distances, and Pull-Back of Cylindrical Functions and Tangent Fields §fields-linear,

$$q_{1}:=\nabla\varphi(\hat{\mu})=\tfrac{\alpha}{2}\cdot2\,j\circ(\mathrm{id}-S)\circ p+q^{*}=j\circ h\circ p+q^{*}.$$

For $(\mu,\nu)\in\mathcal{D}\times\mathcal{D}^{(N)}$ the series $\sum_{k}c_{k}\bigl(W(\mu,\mu_{k})^{2}+W^{(N)}_{a}(\nu,\nu_{k})^{2}\bigr)$ has sum $\chi(\mu)+\chi'(\nu)$ by Elementary Properties of Series of Real Numbers §linearity, so

$$\Phi(\mu,\nu)=u^{-}_{\delta}(\mu)-v^{+}_{\delta}(\nu)-\tfrac{\alpha}{2}W^{(N)}_{a}(p_{\#}\mu,\nu)^{2}-\chi(\mu)-\chi'(\nu).$$

Taking $\nu=\hat{\nu}$, Step 5 gives $\Phi(\mu,\hat{\nu})\le\Phi(\hat{\mu},\hat{\nu})$ for every $\mu\in\mathcal{D}$, that is, $u^{-}_{\delta}(\mu)-\varphi(\mu)\le u^{-}_{\delta}(\hat{\mu})-\varphi(\hat{\mu})$ after cancelling $v^{+}_{\delta}(\hat{\nu})+\chi'(\hat{\nu})$; so the function with value $u^{-}_{\delta}(\mu)-\varphi(\mu)$ has a local maximum at $\hat{\mu}$ relative to $\mathcal{D}$ (with radius $1$). As $\mathcal{P}$ is noise-closed with closed score along noise couplings, $F$ satisfies the shift-coercivity and shift-semicontinuity conditions (H1), $\delta\in J$, $u$ has penalty-subordinate growth from above (Step 3) and is a viscosity subsolution, Exact Inequalities at a Touching Point of a Penalised Viscosity Subsolution or Supersolution on the Noise Wasserstein Space §subsolution at the fine level with $\varphi$ gives $\hat{\mu}\in\mathcal{D}_{\Sigma}$ and

$$F^{-}_{\delta}(\hat{\mu},r_{1},q_{1})\le0.\qquad(8\mathrm{a})$$

By The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §triangle, (7b) and $\gamma'\le1$, $\lVert q_{1}\rVert_{\hat{\mu}}<2\alpha B^{c}+1<R$, and $\lVert q_{1}-j\circ h\circ p\rVert_{\hat{\mu}}=\lVert q^{*}\rVert_{\hat{\mu}}<\gamma$. (8b)

Step 9 (Fine score bound through a fixed partner). The triple $\xi=(\hat{\mu},r_{1},q_{1})$ is a test datum for $F$, as $\hat{\mu}\in\mathcal{D}_{\Sigma}$, and it is $R$-bounded: $W(\hat{\mu},\rho)\le B^{f}<R$, $|\mathcal{E}(\hat{\mu})|\le|e_{1}|+K<R$ by (6c), $|r_{1}|<A<R$ by (6e), and $\lVert q_{1}\rVert_{\hat{\mu}}<R$ by (8b). The triple $\zeta_{0}=(\sigma_{0},0,0)$ of Step 1 is a test datum, $R$-bounded since $W(\sigma_{0},\rho)<R$, $|\mathcal{E}(\sigma_{0})|<R$, $|0|<R$ and $\lVert0\rVert_{\sigma_{0}}=0<R$. By (8a), $F^{-}_{\delta}(\xi)-F^{+}_{\delta}(\zeta_{0})\le-F^{+}_{\delta}(\sigma_{0},0,0)\le|F^{+}_{\delta}(\sigma_{0},0,0)|<R$. So $\xi\in S^{-}_{\delta,R}$ by Test Data for a First-Order Equation Operator on the Noise Wasserstein Space and the Admissible Sets §admissible, with witness $\zeta_{0}$, and, $C^{f}$ being a score bound for $F$ at $(\delta,R)$ (The Shift-Coercivity Condition for a First-Order Equation Operator on the Noise Wasserstein Space §bound),

$$\lVert\Sigma(\hat{\mu})\rVert_{\hat{\mu}}\le C^{f}.\qquad(9\mathrm{a})$$

Step 10 (Removing the perturbation, and passing to level $N$). By the choice of $\gamma^{f}\ge\gamma$ in Step 2 (First-Order Equation Operators on the Noise Wasserstein Space with Momentum-Continuous Shifts §momentum), applied at $\hat{\mu}$ with $r_{1}$, $q=q_{1}$ and $q'=j\circ h\circ p$, which satisfy the bounds there ($\lVert\Sigma(\hat{\mu})\rVert_{\hat{\mu}}\le C^{f}\le\bar{R}$ by (9a), $|r_{1}|<A\le\bar{R}$, both fields of norm below $R\le\bar{R}$ by (8b) and (7b), their difference of norm below $\gamma$ by (8b)), (8a) gives

$$F^{-}_{\delta}(\hat{\mu},r_{1},j\circ h\circ p)<\epsilon.\qquad(10\mathrm{a})$$

The triple $(\hat{\mu},r_{1},h)$ is a cross-level datum, as $\hat{\mu}\in\mathcal{D}_{\Sigma}$ and $h\in T^{a,(N)}_{p_{\#}\hat{\mu}}$ (Step 7), and it is $\bar{R}$-bounded: $W(\hat{\mu},\rho)\le B^{f}<\bar{R}$, $|\mathcal{E}(\hat{\mu})|<R\le\bar{R}$, $\lVert\Sigma(\hat{\mu})\rVert_{\hat{\mu}}\le C^{f}<\bar{R}$ (as $0<R$), $|r_{1}|<\bar{R}$ and $\lVert h\rVert^{(N)}_{p_{\#}\hat{\mu}}\le2\alpha B^{c}<\bar{R}$ by (7a). By the $\epsilon$-sub-consistency at $(\delta,\bar{R})$ (Step 4, Cross-Level Data and Eta-Consistency of a Fine and a Coarse Level at (delta, R) §sub-consistent) and (10a),

$$F^{(N),-}_{\delta}(p_{\#}\hat{\mu},r_{1},h)\le F^{-}_{\delta}(\hat{\mu},r_{1},j\circ h\circ p)+\epsilon<2\epsilon.\qquad(10\mathrm{b})$$

Step 11 (The coarse test function, the coarse exact inequality and the coarse score bound). By Squared Noise Wasserstein Distances, Their Convergent Series and Linear Combinations are Noise Intrinsic Test Functions §distance at level N, with $Q=\mathcal{D}^{(N)}$ and $p_{\#}\hat{\mu}$ as its fixed measure, $\psi_{0}$ is a noise intrinsic test function on $\mathcal{D}^{(N)}$ with $\nabla\psi_{0}(\hat{\nu})=2(\mathrm{id}-S')$; by Squared Noise Wasserstein Distances, Their Convergent Series and Linear Combinations are Noise Intrinsic Test Functions §series-test and Squared Noise Wasserstein Distances, Their Convergent Series and Linear Combinations are Noise Intrinsic Test Functions §series-gradient at level N, with the maps $S'_{k}$, $\chi'$ is a noise intrinsic test function on $\mathcal{D}^{(N)}$ and $q'^{*}=\nabla\chi'(\hat{\nu})$ satisfies $\lVert q'^{*}\rVert^{(N)}_{\hat{\nu}}\le4\tau B_{2}<\gamma'$, exactly as in Step 8 with $W^{(N)}_{a}(\hat{\nu},\nu_{k})\le2B_{2}$. Put $\psi=\bigl(-\tfrac{\alpha}{2}\bigr)\psi_{0}+(-1)\chi'$; by Squared Noise Wasserstein Distances, Their Convergent Series and Linear Combinations are Noise Intrinsic Test Functions §linear at level N it is a noise intrinsic test function on $\mathcal{D}^{(N)}$ with

$$q_{2}:=\nabla\psi(\hat{\nu})=\alpha(S'-\mathrm{id})-q'^{*}.$$

Taking $\mu=\hat{\mu}$ in Step 5 and using $W^{(N)}_{a}(p_{\#}\hat{\mu},\nu)=W^{(N)}_{a}(\nu,p_{\#}\hat{\mu})$ (The Noise Wasserstein Distance is a Metric on the Measures Noise-Connected to the Reference Measure: Existence of Noise-Optimal Couplings, Comparison with the Quadratic Wasserstein Distance and Lower Semicontinuity §symmetry at level N), $\Phi(\hat{\mu},\nu)\le\Phi(\hat{\mu},\hat{\nu})$ for $\nu\in\mathcal{D}^{(N)}$ reads $v^{+}_{\delta}(\hat{\nu})-\psi(\hat{\nu})\le v^{+}_{\delta}(\nu)-\psi(\nu)$; so the function with value $v^{+}_{\delta}(\nu)-\psi(\nu)$ has a local minimum at $\hat{\nu}$ relative to $\mathcal{D}^{(N)}$. The operator $F^{(N)}$ satisfies the shift-coercivity condition relative to $\mathcal{P}^{(N)}$, since by (H5) there is a score bound for it at every $(\delta'',R'')$ with $\delta''\in J$ and $0<R''$ (The Shift-Coercivity Condition for a First-Order Equation Operator on the Noise Wasserstein Space §coercivity); with (H2), $\delta\in J$, $v$ having penalty-subordinate growth from below and being a viscosity supersolution, Exact Inequalities at a Touching Point of a Penalised Viscosity Subsolution or Supersolution on the Noise Wasserstein Space §supersolution at level N with $\psi$ gives $\hat{\nu}\in\mathcal{D}^{(N)}_{\Sigma}$ and

$$0\le F^{(N),+}_{\delta}(\hat{\nu},r_{2},q_{2}).\qquad(11\mathrm{a})$$

By The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §triangle and (7a), $\lVert q_{2}\rVert^{(N)}_{\hat{\nu}}<2\alpha B^{c}+1<R$ and $\lVert q_{2}-\alpha(S'-\mathrm{id})\rVert^{(N)}_{\hat{\nu}}=\lVert q'^{*}\rVert^{(N)}_{\hat{\nu}}<\gamma$. The triple $\zeta=(\hat{\nu},r_{2},q_{2})$ is a test datum for $F^{(N)}$ at level N, $R$-bounded: $W^{(N)}_{a}(\hat{\nu},\rho^{(N)})\le B^{c}<R$, $|\mathcal{E}^{(N)}(\hat{\nu})|\le|e_{0}|+K<R$, $|r_{2}|<A<R$ and $\lVert q_{2}\rVert^{(N)}_{\hat{\nu}}<R$. The triple $\xi'=(p_{\#}\hat{\mu},r_{1},h)$ is a test datum for $F^{(N)}$ at level N, as $p_{\#}\hat{\mu}\in\mathcal{D}^{(N)}_{\Sigma}$ by Compatible Noise Penalty Pairs at a Fine and a Coarse Level §compatible, and it is $R$-bounded: $W^{(N)}_{a}(p_{\#}\hat{\mu},\rho^{(N)})\le B^{c}<R$, $|\mathcal{E}^{(N)}(p_{\#}\hat{\mu})|\le|e_{0}|+K<R$ by (6c), $|r_{1}|<R$ and $\lVert h\rVert^{(N)}_{p_{\#}\hat{\mu}}<R$ by (7a). By (10b) and (11a), $F^{(N),-}_{\delta}(\xi')-F^{(N),+}_{\delta}(\zeta)<2\epsilon<R$. So $\zeta\in S^{+}_{\delta,R}$ at level N (Test Data for a First-Order Equation Operator on the Noise Wasserstein Space and the Admissible Sets §admissible), with witness $\xi'$, and, $C^{c}$ being a score bound for $F^{(N)}$ at $(\delta,R)$ by (H5), $\lVert\Sigma^{(N)}(\hat{\nu})\rVert^{(N)}_{\hat{\nu}}\le C^{c}\le\bar{R}$. By the choice of $\gamma^{c}\ge\gamma$ in Step 2 ((H7)), applied at $\hat{\nu}$ with $r_{2}$, $q=q_{2}$ and $q'=\alpha(S'-\mathrm{id})$, all within the bounds there, (11a) gives

$$-\epsilon<F^{(N),+}_{\delta}\bigl(\hat{\nu},r_{2},\alpha(S'-\mathrm{id})\bigr).\qquad(11\mathrm{b})$$

Step 12 (Properness and the structure condition at level $N$). By The Bundle of Noise Fields over a Set of Measures, First-Order Equation Operators on the Noise Wasserstein Space, and Their Delta-Shifts §shifted at level N, $F^{(N),-}_{\delta}(p_{\#}\hat{\mu},r,h)$ is $F^{(N)}$ evaluated at $p_{\#}\hat{\mu}$, $r+\delta\mathcal{E}^{(N)}(p_{\#}\hat{\mu})$ and $h+\delta\Sigma^{(N)}(p_{\#}\hat{\mu})$ for every $r\in\mathbb{R}$, the first and last arguments not depending on $r$, and $(p_{\#}\hat{\mu},h+\delta\Sigma^{(N)}(p_{\#}\hat{\mu}))\in\mathcal{V}^{a,(N)}(\mathcal{D}^{(N)}_{\Sigma})$. By (6a), (6b) and (6e), $r_{2}+\delta\mathcal{E}^{(N)}(p_{\#}\hat{\mu})\le r_{1}+\delta\mathcal{E}^{(N)}(p_{\#}\hat{\mu})$ and both have absolute value below $A+A=R_{0}$, so the properness constant $\lambda$ for $F^{(N)}$ at $R_{0}$ (Step 1, Locally Strictly Proper First-Order Equation Operators on the Noise Wasserstein Space §constant with $Q=\mathcal{D}^{(N)}_{\Sigma}$) gives

$$\lambda(r_{1}-r_{2})\le F^{(N),-}_{\delta}(p_{\#}\hat{\mu},r_{1},h)-F^{(N),-}_{\delta}(p_{\#}\hat{\mu},r_{2},h).\qquad(12\mathrm{a})$$

We have $1<\alpha$, $\delta\in J$, $p_{\#}\hat{\mu},\hat{\nu}\in\mathcal{D}^{(N)}_{\Sigma}$ with both ordered pairs $(p_{\#}\hat{\mu},\hat{\nu})$ and $(\hat{\nu},p_{\#}\hat{\mu})$ uniquely noise-mapped, $S$ a noise-optimal map from $p_{\#}\hat{\mu}$ to $\hat{\nu}$ and $S'$ one from $\hat{\nu}$ to $p_{\#}\hat{\mu}$ (Step 7), $\delta(|\mathcal{E}^{(N)}(p_{\#}\hat{\mu})|+|\mathcal{E}^{(N)}(\hat{\nu})|)<2A=R_{0}$ by (6b), and $-R_{0}\le r_{2}\le R_{0}$ by (6e). So The First-Order Structure Condition at Uniquely Noise-Mapped Pairs §pair at level N, for the structure pair $(\omega_{1},\omega_{2})$ of Step 1 at $R_{0}$, applied with its measures $\mu$ and $\nu$ being $p_{\#}\hat{\mu}$ and $\hat{\nu}$, its maps $S$ and $S'$ being $S$ and $S'$, and the value slot $r_{2}$, gives, writing $\Omega_{1}=\omega_{1}\bigl(\alpha W^{(N)}_{a}(p_{\#}\hat{\mu},\hat{\nu})^{2}+\alpha^{-1}\bigr)$ and $\Omega_{2}=\omega_{2}\bigl(\delta(|\mathcal{E}^{(N)}(p_{\#}\hat{\mu})|+|\mathcal{E}^{(N)}(\hat{\nu})|+1),\alpha\bigr)$,

$$-\Omega_{1}-\Omega_{2}\le F^{(N),-}_{\delta}(p_{\#}\hat{\mu},r_{2},h)-F^{(N),+}_{\delta}\bigl(\hat{\nu},r_{2},\alpha(S'-\mathrm{id})\bigr).$$

Adding this to (12a) and using (10b) and (11b),

$$\lambda(r_{1}-r_{2})\le F^{(N),-}_{\delta}(p_{\#}\hat{\mu},r_{1},h)-F^{(N),+}_{\delta}\bigl(\hat{\nu},r_{2},\alpha(S'-\mathrm{id})\bigr)+\Omega_{1}+\Omega_{2}<3\epsilon+\Omega_{1}+\Omega_{2}.\qquad(12\mathrm{b})$$

Step 13 (The moduli). By (3f) with $\alpha'=\tfrac{\alpha}{2}$, whose coefficient is $\tfrac{\alpha}{4}$, applied to $(\hat{\mu},\hat{\nu})$ (admissible by Step 5), then (4a) and $\tau\le\eta_{1}$,

$$\tfrac{\alpha}{4}W^{(N)}_{a}(p_{\#}\hat{\mu},\hat{\nu})^{2}\le G(\tfrac{\alpha}{2},\delta)-G(\alpha,\delta)+\tau<2\eta_{1},$$

so $\alpha W^{(N)}_{a}(p_{\#}\hat{\mu},\hat{\nu})^{2}<8\eta_{1}=t_{1}/2$, and with $\alpha^{-1}<t_{1}/4$ (Step 4) the argument of $\omega_{1}$ lies in $[0,t_{1}]$; hence $\Omega_{1}\le\epsilon$. By (3g) with $\delta'=\tfrac{\delta}{2}\in J$, applied to $(\hat{\mu},\hat{\nu})$, then (4a) and $\tau\le\eta_{2}$,

$$\tfrac{\delta}{2}\bigl(\mathcal{E}(\hat{\mu})-e_{1}+\mathcal{E}^{(N)}(\hat{\nu})-e_{0}\bigr)\le G(\alpha,\tfrac{\delta}{2})-G(\alpha,\delta)+\tau<2\eta_{2}=\tfrac{t_{2}}{4}.$$

By (1a) at $\hat{\mu}$, $\mathcal{E}^{(N)}(p_{\#}\hat{\mu})-e_{0}\ge0$, so the triangle inequality for $|\cdot|$ applied to $\mathcal{E}^{(N)}(p_{\#}\hat{\mu})=(\mathcal{E}^{(N)}(p_{\#}\hat{\mu})-e_{0})+e_{0}$ and (1a) give $|\mathcal{E}^{(N)}(p_{\#}\hat{\mu})|\le\mathcal{E}^{(N)}(p_{\#}\hat{\mu})-e_{0}+|e_{0}|\le\mathcal{E}(\hat{\mu})+c_{0}-e_{0}+|e_{0}|=\mathcal{E}(\hat{\mu})-e_{1}+|e_{0}|$; likewise $|\mathcal{E}^{(N)}(\hat{\nu})|\le\mathcal{E}^{(N)}(\hat{\nu})-e_{0}+|e_{0}|$ by (H3). Multiplying by the positive $\delta$ and using $\delta(2|e_{0}|+1)\le t_{2}/4$ (Step 4),

$$\delta\bigl(|\mathcal{E}^{(N)}(p_{\#}\hat{\mu})|+|\mathcal{E}^{(N)}(\hat{\nu})|+1\bigr)\le\delta\bigl(\mathcal{E}(\hat{\mu})-e_{1}+\mathcal{E}^{(N)}(\hat{\nu})-e_{0}\bigr)+\delta\bigl(2|e_{0}|+1\bigr)<\tfrac{t_{2}}{2}+\tfrac{t_{2}}{4}<t_{2},$$

and the argument is positive; since $\omega_{2}(t,\alpha)\le\epsilon$ for $0\le t\le t_{2}$ (Step 4), $\Omega_{2}\le\epsilon$.

Step 14 (Contradiction; claim 1). By (12b) and Step 13, $\lambda(r_{1}-r_{2})<5\epsilon$. By (6a) and $0<\lambda$, $\lambda(\tfrac{\theta}{2}-\tau)<\lambda(r_{1}-r_{2})$, and $\lambda\tau\le\epsilon$ (Step 5), so $\tfrac{\lambda\theta}{2}<\lambda\tau+5\epsilon\le6\epsilon=\tfrac{3}{8}\lambda\theta$, that is $\tfrac{1}{8}\lambda\theta<0$, contradicting $0<\lambda\theta$. Hence no $\mu_{0}$ as in Step 2 exists: for every $N\ge N_{1}$ and all $u$, $v$ as in claim 1, $u(\mu)\le v(p^{(N)}_{\#}\mu)+\theta$ for every $\mu\in\mathcal{D}$ with $\mathcal{E}(\mu)\le c$. This is claim 1 (fine subsolutions below coarse supersolutions) of the present theorem.

Step 15 (Claim 2). Let $b,b',\theta,c$ be as in claim 2. Steps 1 and 2 are carried out verbatim with these numbers (they involve only $b,b',\theta,c$ and the data; $R_{ij}$ contains both $|F^{-}_{\delta}(\sigma_{0},0,0)|$ and $|F^{+}_{\delta}(\sigma_{0},0,0)|$), giving a grid, constants and $N_{1}$ that depend only on the data and on these $b,b',\theta,c$. Let $N\ge N_{1}$, let $w$ be a viscosity subsolution of $F^{(N)}$ relative to $\mathcal{P}^{(N)}$ with $w\le b$ on $\mathcal{D}^{(N)}$ and $z$ a viscosity supersolution of $F$ relative to $\mathcal{P}$ with $b'\le z$ on $\mathcal{D}$, and suppose that $\mu_{0}\in\mathcal{D}$ satisfies $\mathcal{E}(\mu_{0})\le c$ and $\theta<w(p_{\#}\mu_{0})-z(\mu_{0})$. We list the changes; every step not mentioned is unchanged, and $S$, $S'$, $h=\alpha(\mathrm{id}-S)$, $\varphi_{0}$, $\chi$, $\psi_{0}$, $\chi'$, $q^{*}$, $q'^{*}$ are defined as in Steps 7 and 8.

Step 3. By Basic Properties of the Delta-Envelopes on the Noise Wasserstein Space, and the Envelopes of Bounded Functions for a Noise-Closed Penalty Pair §growth and Basic Properties of the Delta-Envelopes on the Noise Wasserstein Space, and the Envelopes of Bounded Functions for a Noise-Closed Penalty Pair §duality, $z$ has penalty-subordinate growth from below and $-z$ from above relative to $\mathcal{P}$, with $(-z)^{-}_{\delta}=-z^{+}_{\delta}$ on $\mathcal{D}$, and at level N $w$ has penalty-subordinate growth from above. Now $U_{\delta}=-z^{+}_{\delta}$ on $\mathcal{D}$ and $V_{\delta}=w^{-}_{\delta}$ on $\mathcal{D}^{(N)}$, upper semicontinuous by Basic Properties of the Delta-Envelopes on the Noise Wasserstein Space, and the Envelopes of Bounded Functions for a Noise-Closed Penalty Pair §semicontinuity, with $U_{\delta}(\mu)\le-b'-\delta\mathcal{E}(\mu)$ and $V_{\delta}(\nu)\le b-\delta\mathcal{E}^{(N)}(\nu)$ by Basic Properties of the Delta-Envelopes on the Noise Wasserstein Space, and the Envelopes of Bounded Functions for a Noise-Closed Penalty Pair §bounded; the doubling lemma is applied with $b_{1}=-b'$ and $b_{2}=b$, and

$$\Psi_{\delta,\alpha}(\mu,\nu)=w^{-}_{\delta}(\nu)-z^{+}_{\delta}(\mu)-\tfrac{\alpha}{2}W^{(N)}_{a}(p_{\#}\mu,\nu)^{2}.$$

As $b_{1}+b_{2}=b-b'$, (3a) and (3b) hold verbatim. In (3c), Basic Properties of the Delta-Envelopes on the Noise Wasserstein Space, and the Envelopes of Bounded Functions for a Noise-Closed Penalty Pair §semicontinuity gives $w(p_{\#}\mu_{0})-\delta\mathcal{E}^{(N)}(p_{\#}\mu_{0})\le w^{-}_{\delta}(p_{\#}\mu_{0})$ and $z^{+}_{\delta}(\mu_{0})\le z(\mu_{0})+\delta\mathcal{E}(\mu_{0})$, so

$$M(\delta,\alpha)\ge w^{-}_{\delta}(p_{\#}\mu_{0})-z^{+}_{\delta}(\mu_{0})\ge w(p_{\#}\mu_{0})-z(\mu_{0})-\delta\bigl(2|c|+c_{0}\bigr)>\tfrac{\theta}{2}.\qquad(3\mathrm{c}')$$

In (3g) the hypotheses of Bounds, Monotonicity and Borwein-Preiss Perturbed Maximisers of the Cross-Level Doubled Function in the Coarse Noise Wasserstein Distance for Compatible Noise-Closed Penalty Pairs §weight come from Basic Properties of the Delta-Envelopes on the Noise Wasserstein Space, and the Envelopes of Bounded Functions for a Noise-Closed Penalty Pair §monotone: $z^{+}_{\delta'}(\mu)\le z^{+}_{\delta}(\mu)-(\delta-\delta')\mathcal{E}(\mu)$, that is $U_{\delta}(\mu)+(\delta-\delta')\mathcal{E}(\mu)\le U_{\delta'}(\mu)$, and $w^{-}_{\delta}(\nu)+(\delta-\delta')\mathcal{E}^{(N)}(\nu)\le w^{-}_{\delta'}(\nu)$. Steps 4 and 5 are unchanged.

Step 6. Put $r_{1}=z^{+}_{\delta}(\hat{\mu})$ and $r_{2}=w^{-}_{\delta}(\hat{\nu})$. Now $\Psi(\hat{\mu},\hat{\nu})=r_{2}-r_{1}-\tfrac{\alpha}{2}W^{(N)}_{a}(p_{\#}\hat{\mu},\hat{\nu})^{2}$, so

$$r_{2}-r_{1}>\tfrac{\theta}{2}-\tau>0.\qquad(6\mathrm{a}')$$

(6b), (6c) and (6d) hold verbatim, as they use only (3a), $\Psi(\hat{\mu},\hat{\nu})>0$, (1a) and (H3). By Basic Properties of the Delta-Envelopes on the Noise Wasserstein Space, and the Envelopes of Bounded Functions for a Noise-Closed Penalty Pair §bounded, $r_{2}\le b-\delta\mathcal{E}^{(N)}(\hat{\nu})\le|b|+|e_{0}|$ and $r_{1}\ge b'+\delta\mathcal{E}(\hat{\mu})\ge-|b'|-|e_{1}|$, so with (6a$'$)

$$-A<r_{1}<r_{2}<A.\qquad(6\mathrm{e}')$$

Step 8. Put $\varphi=\bigl(-\tfrac{\alpha}{2}\bigr)\varphi_{0}+(-1)\chi$, a noise intrinsic test function on $\mathcal{D}$ by Squared Noise Wasserstein Distances, Their Convergent Series and Linear Combinations are Noise Intrinsic Test Functions §linear, with, by Properties of a Mode-Restriction Link: Isometric Embedding, Contraction of Noise Norms and Noise Wasserstein Distances, and Pull-Back of Cylindrical Functions and Tangent Fields §fields-linear,

$$q_{1}:=\nabla\varphi(\hat{\mu})=j\circ(-h)\circ p-q^{*}.$$

Now $\Phi(\mu,\nu)=w^{-}_{\delta}(\nu)-z^{+}_{\delta}(\mu)-\tfrac{\alpha}{2}W^{(N)}_{a}(p_{\#}\mu,\nu)^{2}-\chi(\mu)-\chi'(\nu)$, and taking $\nu=\hat{\nu}$ gives $z^{+}_{\delta}(\hat{\mu})-\varphi(\hat{\mu})\le z^{+}_{\delta}(\mu)-\varphi(\mu)$ for $\mu\in\mathcal{D}$: a local minimum at $\hat{\mu}$ relative to $\mathcal{D}$. Exact Inequalities at a Touching Point of a Penalised Viscosity Subsolution or Supersolution on the Noise Wasserstein Space §supersolution at the fine level ($z$ a viscosity supersolution with penalty-subordinate growth from below) gives $\hat{\mu}\in\mathcal{D}_{\Sigma}$ and

$$0\le F^{+}_{\delta}(\hat{\mu},r_{1},q_{1}).\qquad(8\mathrm{a}')$$

(8b) holds with $j\circ(-h)\circ p$ in place of $j\circ h\circ p$, whose norm is $\lVert h\rVert^{(N)}_{p_{\#}\hat{\mu}}$ by Properties of a Mode-Restriction Link: Isometric Embedding, Contraction of Noise Norms and Noise Wasserstein Distances, and Pull-Back of Cylindrical Functions and Tangent Fields §fields-isometry, as $\lVert-h\rVert^{(N)}_{p_{\#}\hat{\mu}}=\lVert h\rVert^{(N)}_{p_{\#}\hat{\mu}}$.

Step 9. With $\xi=(\hat{\mu},r_{1},q_{1})$ and the same partner $\zeta_{0}=(\sigma_{0},0,0)$, (8a$'$) gives $F^{-}_{\delta}(\zeta_{0})-F^{+}_{\delta}(\xi)\le F^{-}_{\delta}(\sigma_{0},0,0)\le|F^{-}_{\delta}(\sigma_{0},0,0)|<R$; so $\xi\in S^{+}_{\delta,R}$ by Test Data for a First-Order Equation Operator on the Noise Wasserstein Space and the Admissible Sets §admissible, and (9a) $\lVert\Sigma(\hat{\mu})\rVert_{\hat{\mu}}\le C^{f}$ holds.

Step 10. Momentum continuity at $\hat{\mu}$ with $r_{1}$, $q=q_{1}$ and $q'=j\circ(-h)\circ p$, and (8a$'$), give

$$-\epsilon<F^{+}_{\delta}\bigl(\hat{\mu},r_{1},j\circ(-h)\circ p\bigr).\qquad(10\mathrm{a}')$$

The cross-level datum is $(\hat{\mu},r_{1},-h)$, with $-h\in T^{a,(N)}_{p_{\#}\hat{\mu}}$ (Step 7), $\bar{R}$-bounded as before; the $\epsilon$-super-consistency at $(\delta,\bar{R})$ (Cross-Level Data and Eta-Consistency of a Fine and a Coarse Level at (delta, R) §super-consistent) and (10a$'$) give

$$-2\epsilon<F^{+}_{\delta}\bigl(\hat{\mu},r_{1},j\circ(-h)\circ p\bigr)-\epsilon\le F^{(N),+}_{\delta}\bigl(p_{\#}\hat{\mu},r_{1},\alpha(S-\mathrm{id})\bigr).\qquad(10\mathrm{b}')$$

Step 11. Put $\psi=\tfrac{\alpha}{2}\psi_{0}+\chi'$, with $q_{2}:=\nabla\psi(\hat{\nu})=\alpha(\mathrm{id}-S')+q'^{*}$ by Squared Noise Wasserstein Distances, Their Convergent Series and Linear Combinations are Noise Intrinsic Test Functions §linear at level N. Taking $\mu=\hat{\mu}$ gives $w^{-}_{\delta}(\nu)-\psi(\nu)\le w^{-}_{\delta}(\hat{\nu})-\psi(\hat{\nu})$ for $\nu\in\mathcal{D}^{(N)}$: a local maximum at $\hat{\nu}$ relative to $\mathcal{D}^{(N)}$. Exact Inequalities at a Touching Point of a Penalised Viscosity Subsolution or Supersolution on the Noise Wasserstein Space §subsolution at level N ($w$ a viscosity subsolution with penalty-subordinate growth from above; $F^{(N)}$ shift-coercive by (H5) and shift-semicontinuous by (H2)) gives $\hat{\nu}\in\mathcal{D}^{(N)}_{\Sigma}$ and

$$F^{(N),-}_{\delta}(\hat{\nu},r_{2},q_{2})\le0.\qquad(11\mathrm{a}')$$

The bounds on $q_{2}$ hold as before with $\alpha(\mathrm{id}-S')$ in place of $\alpha(S'-\mathrm{id})$. The datum $\zeta=(\hat{\nu},r_{2},q_{2})$ is $R$-bounded as before ($|r_{2}|<A$ by (6e$'$)), the partner is $\xi'=(p_{\#}\hat{\mu},r_{1},-h)$, $R$-bounded as before, and by (10b$'$) and (11a$'$), $F^{(N),-}_{\delta}(\zeta)-F^{(N),+}_{\delta}(\xi')<2\epsilon<R$; so $\zeta\in S^{-}_{\delta,R}$ at level N and $\lVert\Sigma^{(N)}(\hat{\nu})\rVert^{(N)}_{\hat{\nu}}\le C^{c}$. Momentum continuity at level N ((H7)) at $\hat{\nu}$ with $r_{2}$, $q=q_{2}$ and $q'=\alpha(\mathrm{id}-S')$, and (11a$'$), give

$$F^{(N),-}_{\delta}\bigl(\hat{\nu},r_{2},\alpha(\mathrm{id}-S')\bigr)<\epsilon.\qquad(11\mathrm{b}')$$

Step 12. Properness is used at $\hat{\nu}$: by The Bundle of Noise Fields over a Set of Measures, First-Order Equation Operators on the Noise Wasserstein Space, and Their Delta-Shifts §shifted at level N, $F^{(N),-}_{\delta}(\hat{\nu},r,\alpha(\mathrm{id}-S'))$ is $F^{(N)}$ at $\hat{\nu}$, $r+\delta\mathcal{E}^{(N)}(\hat{\nu})$ and $\alpha(\mathrm{id}-S')+\delta\Sigma^{(N)}(\hat{\nu})$; by (6a$'$), (6b) and (6e$'$), $r_{1}+\delta\mathcal{E}^{(N)}(\hat{\nu})\le r_{2}+\delta\mathcal{E}^{(N)}(\hat{\nu})$, both of absolute value below $R_{0}$, so

$$\lambda(r_{2}-r_{1})\le F^{(N),-}_{\delta}\bigl(\hat{\nu},r_{2},\alpha(\mathrm{id}-S')\bigr)-F^{(N),-}_{\delta}\bigl(\hat{\nu},r_{1},\alpha(\mathrm{id}-S')\bigr).$$

The First-Order Structure Condition at Uniquely Noise-Mapped Pairs §pair at level N is applied with its measures $\mu$ and $\nu$ being $\hat{\nu}$ and $p_{\#}\hat{\mu}$, its map $S$ being $S'$ (noise-optimal from $\hat{\nu}$ to $p_{\#}\hat{\mu}$) and its map $S'$ being $S$ (noise-optimal from $p_{\#}\hat{\mu}$ to $\hat{\nu}$), and the value slot $r_{1}$ ($-R_{0}\le r_{1}\le R_{0}$ by (6e$'$)); the arguments of $\omega_{1}$ and $\omega_{2}$ are those of Step 12, by The Noise Wasserstein Distance is a Metric on the Measures Noise-Connected to the Reference Measure: Existence of Noise-Optimal Couplings, Comparison with the Quadratic Wasserstein Distance and Lower Semicontinuity §symmetry at level N and commutativity of addition, so

$$-\Omega_{1}-\Omega_{2}\le F^{(N),-}_{\delta}\bigl(\hat{\nu},r_{1},\alpha(\mathrm{id}-S')\bigr)-F^{(N),+}_{\delta}\bigl(p_{\#}\hat{\mu},r_{1},\alpha(S-\mathrm{id})\bigr).$$

Adding, and using (11b$'$) and (10b$'$),

$$\lambda(r_{2}-r_{1})\le F^{(N),-}_{\delta}\bigl(\hat{\nu},r_{2},\alpha(\mathrm{id}-S')\bigr)-F^{(N),+}_{\delta}\bigl(p_{\#}\hat{\mu},r_{1},\alpha(S-\mathrm{id})\bigr)+\Omega_{1}+\Omega_{2}<3\epsilon+\Omega_{1}+\Omega_{2}.$$

Steps 13 and 14. Step 13 is unchanged (it uses only (3f), (3g), (4a), (1a), (H3) and Step 5), so $\Omega_{1},\Omega_{2}\le\epsilon$ and $\lambda(r_{2}-r_{1})<5\epsilon$; with (6a$'$) and $\lambda\tau\le\epsilon$, $\tfrac{\lambda\theta}{2}<6\epsilon=\tfrac{3}{8}\lambda\theta$, a contradiction as in Step 14. Hence for every $N\ge N_{1}$ and all $w$, $z$ as in claim 2, $w(p^{(N)}_{\#}\mu)\le z(\mu)+\theta$ for every $\mu\in\mathcal{D}$ with $\mathcal{E}(\mu)\le c$. This is claim 2 (coarse subsolutions below fine supersolutions) of the present theorem.