Proof of A Comparison Principle for First-Order Equations on the Noise Wasserstein Space Relative to a Noise-Closed Penalty Pair

Each result cited is universally quantified over the data in its own statement. Throughout, $W=W_{a}$, 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; $|s|$ is the absolute value of $s\in\mathbb{R}$, $\alpha^{-1}$ is the multiplicative inverse of a positive $\alpha$, $J=\{\delta\in\mathbb{R}:0<\delta<1\}$, and suprema and infima of nonempty subsets of $\mathbb{R}$ bounded above or below are those of The Real Numbers: Standing Notation and Background §bounds. For $\sigma\in\mathcal{P}^{a}_{\rho}$, $L^{2}(\sigma;X^{a})$ is a real Hilbert space (Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §fields). Elementary order arithmetic and the approximation property Approximation Property of the Supremum and the Infimum in $\mathbb{R}$ are in force by The Real Numbers: Standing Notation and Background §background. The order in which the auxiliary quantities are chosen is: $e_{0}$ and $\mu_{0}$ (Step 1); then, for a given $\theta$, the constants $A$, $R_{0}$, $\lambda$, $(\omega_{1},\omega_{2})$, $\kappa$, $t_{1}$ (Step 2); then $\alpha$ and $t_{2}$ (Step 3); then $\eta_{2}$, $\delta_{1}$ and $\delta_{0}$ (Step 4); then, for a given $\delta$ with $0<\delta<\delta_{0}$, the constants $K$, $B$, $R$, $C$, $\bar{R}$, $\gamma$ and finally $\tau$, after which the perturbed maximiser is taken (Step 5).

Step 1 (The corrected maximum and its monotonicity). By Basic Properties of a Noise-Closed Noise Penalty Pair: Lower Bound, Lower Semicontinuity, Complete Sublevel Sets and Bounded Distances §bounded-below fix $e_{0}\in\mathbb{R}$ with $e_{0}\le\mathcal{E}(\sigma)$ for every $\sigma\in\mathcal{D}$; the set $\mathcal{D}$ contains the nonempty $\mathcal{D}_{\Sigma}$ (Noise Penalty Pairs on the Noise Wasserstein Space: the Penalty, Its Score, and Their Domains §pair, Noise Penalty Pairs on the Noise Wasserstein Space: the Penalty, Its Score, and Their Domains §nonempty), so fix $\mu_{0}\in\mathcal{D}$. For positive $\delta,\alpha$ let $\Psi_{\delta,\alpha}$ and $M(\delta,\alpha)$ be as in Bounds, Monotonicity and Borwein-Preiss Perturbed Maximisers of the Doubled Difference in the Noise Wasserstein Distance for a Noise-Closed Penalty Pair, read with the present pair, $e_{0}$, $u$, $v$, $b$ and $b'$; by Bounds, Monotonicity and Borwein-Preiss Perturbed Maximisers of the Doubled Difference in the Noise Wasserstein Distance for a Noise-Closed Penalty Pair §bounds, $M(\delta,\alpha)$ is a real number at most $b-b'-2\delta e_{0}$. For $\alpha>0$ and $\delta\in J$ put

$$G(\alpha,\delta)=M(\delta,\alpha)+2\delta e_{0},\qquad\text{so that}\qquad G(\alpha,\delta)\le b-b'.\qquad(1\mathrm{a})$$

Lower bound. By Bounds, Monotonicity and Borwein-Preiss Perturbed Maximisers of the Doubled Difference in the Noise Wasserstein Distance for a Noise-Closed Penalty Pair §diagonal at $\mu_{0}$, $u(\mu_{0})-v(\mu_{0})-2\delta\mathcal{E}(\mu_{0})\le M(\delta,\alpha)$; for $\delta\in J$ we have $2\delta\mathcal{E}(\mu_{0})\le2|\mathcal{E}(\mu_{0})|$ and $-2|e_{0}|\le2\delta e_{0}$, so

$$\ell\le G(\alpha,\delta),\qquad\ell=u(\mu_{0})-v(\mu_{0})-2|\mathcal{E}(\mu_{0})|-2|e_{0}|.\qquad(1\mathrm{b})$$

Decreasing the weight. Let $\alpha>0$, $\delta,\delta'\in J$ with $\delta'<\delta$, let $\tau\ge0$ and let $(\mu,\nu)\in\mathcal{D}\times\mathcal{D}$ satisfy $M(\delta,\alpha)-\tau\le\Psi_{\delta,\alpha}(\mu,\nu)$. By Bounds, Monotonicity and Borwein-Preiss Perturbed Maximisers of the Doubled Difference in the Noise Wasserstein Distance for a Noise-Closed Penalty Pair §weight, $M(\delta,\alpha)-\tau+(\delta-\delta')(\mathcal{E}(\mu)+\mathcal{E}(\nu))\le M(\delta',\alpha)$; adding $2(\delta'-\delta)e_{0}$ to both sides,

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

The left side is nonnegative, as $\mathcal{E}(\mu)-e_{0}$ and $\mathcal{E}(\nu)-e_{0}$ are. For every positive $\tau$ a pair $(\mu,\nu)$ as above 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 $G(\alpha,\delta)\le G(\alpha,\delta')$: $G(\alpha,\cdot)$ is nonincreasing on $J$.

Decreasing the strength. Let $0<\alpha'<\alpha$, $\delta\in J$, $\tau\ge0$, and let $(\mu,\nu)\in\mathcal{D}\times\mathcal{D}$ satisfy $M(\delta,\alpha)-\tau\le\Psi_{\delta,\alpha}(\mu,\nu)$. By Bounds, Monotonicity and Borwein-Preiss Perturbed Maximisers of the Doubled Difference in the Noise Wasserstein Distance for a Noise-Closed Penalty Pair §strength,

$$\tfrac{\alpha-\alpha'}{2}\,W(\mu,\nu)^{2}\le M(\delta,\alpha')-M(\delta,\alpha)+\tau=G(\alpha',\delta)-G(\alpha,\delta)+\tau.\qquad(1\mathrm{d})$$

Moreover, since $W(\mu,\nu)^{2}\ge0$, every value $\Psi_{\delta,\alpha}(\mu,\nu)$ is at most $\Psi_{\delta,\alpha'}(\mu,\nu)\le M(\delta,\alpha')$, so $M(\delta,\alpha)\le M(\delta,\alpha')$ as $M(\delta,\alpha)$ is the least upper bound; that is, $G(\alpha,\delta)\le G(\alpha',\delta)$, and $G(\cdot,\delta)$ is nonincreasing on the positive reals.

Step 2 (Claim 1: constants independent of the doubling). Let $\theta$ be positive. Put $A=|b|+|b'|+|e_{0}|+1$ and $R_{0}=2A$, both positive. Since $F$ is locally strictly proper, fix by Locally Strictly Proper First-Order Equation Operators on the Noise Wasserstein Space §strictly-proper, with $Q=\mathcal{D}_{\Sigma}$, a properness constant $\lambda>0$ for $F$ at $R_{0}$; since $F$ satisfies the first-order structure condition, fix by The First-Order Structure Condition at Uniquely Noise-Mapped Pairs §structure a structure pair $(\omega_{1},\omega_{2})$ for $F$ at $R_{0}$. Put $\kappa=\lambda\theta/8$, positive. By clause 2 of Modulus of Continuity fix a positive $t_{1}$ with $\omega_{1}(t)\le\kappa$ whenever $0\le t\le t_{1}$, and put $\beta_{0}=1+2t_{1}^{-1}$ and $\eta_{1}=t_{1}/32$.

Step 3 (Choice of the doubling strength). For $\alpha\ge\beta_{0}$ the set $\{G(\alpha,\delta):\delta\in J\}$ is nonempty and bounded above by $b-b'$ by (1a); let $N(\alpha)$ be its least upper bound. By (1b), $\ell\le N(\alpha)$; by Step 1, $G(\cdot,\delta)$ is nonincreasing, so $G(\alpha,\delta)\le G(\alpha',\delta)\le N(\alpha')$ for $\beta_{0}\le\alpha'<\alpha$ and every $\delta\in J$, whence $N(\alpha)\le N(\alpha')$. The set $\{N(\alpha):\alpha\ge\beta_{0}\}$ is nonempty and bounded below by $\ell$; let $L$ be its greatest lower bound. By Approximation Property of the Supremum and the Infimum in $\mathbb{R}$ §epsilon-below fix $\alpha_{1}\ge\beta_{0}$ with $N(\alpha_{1})<L+\eta_{1}$, and put $\alpha=2\alpha_{1}$, so that $\tfrac{\alpha}{2}=\alpha_{1}$. Then $\alpha\ge\alpha_{1}\ge\beta_{0}$, so $L\le N(\alpha)$ and

$$N(\tfrac{\alpha}{2})-N(\alpha)<\eta_{1}.\qquad(3\mathrm{a})$$

Moreover $1<\beta_{0}\le\alpha$, and from $2t_{1}^{-1}<\alpha$ we get $\alpha^{-1}<t_{1}/2$. By The First-Order Structure Condition at Uniquely Noise-Mapped Pairs §pair the function with value $\omega_{2}(t,\alpha)$ at $t\ge0$ is a modulus of continuity; by clause 2 of Modulus of Continuity fix a positive $t_{2}$ with $\omega_{2}(t,\alpha)\le\kappa$ whenever $0\le t\le t_{2}$.

Step 4 (Choice of the threshold). Let $\eta_{2}$ be the least of $\eta_{1}$ and $t_{2}/8$. By Approximation Property of the Supremum and the Infimum in $\mathbb{R}$ §epsilon-above fix $\delta_{1}\in J$ with $N(\alpha)-\eta_{2}<G(\alpha,\delta_{1})$, and let $\delta_{0}$ be the least of $\delta_{1}$ and $t_{2}\bigl(8|e_{0}|+4\bigr)^{-1}$, a positive number. Let $\delta\in\mathbb{R}$ satisfy $0<\delta<\delta_{0}$. Then $\delta\in J$, $\tfrac{\delta}{2}\in J$, $\delta(2|e_{0}|+1)<t_{2}/4$ and $\delta<\delta_{1}$, so, $G(\alpha,\cdot)$ being nonincreasing (Step 1),

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

Step 5 (Claim 1: the bound). With $\delta$ as in Step 4 we show $M(\delta,\alpha)\le\theta$. Suppose instead $\theta<M(\delta,\alpha)$. Write $\Psi=\Psi_{\delta,\alpha}$ and $M=M(\delta,\alpha)$.

Step 5a (Constants depending on $\delta$ and $\alpha$). By Bounds, Monotonicity and Borwein-Preiss Perturbed Maximisers of the Doubled Difference in the Noise Wasserstein Distance for a Noise-Closed Penalty Pair §radius at $\delta$ fix $K,B\in\mathbb{R}$ with $0\le B$, not depending on the strength or on the tolerance, with $W(\sigma,\rho)\le B$ for every $\sigma\in\mathcal{D}$ with $\mathcal{E}(\sigma)\le K$. Put

$$R=(2\alpha+5)B+|e_{0}|+|K|+A+1,$$

a positive real. Since $F$ satisfies the shift-coercivity condition and $0<\delta<1$, fix by The Shift-Coercivity Condition for a First-Order Equation Operator on the Noise Wasserstein Space §coercivity a score bound $C\ge0$ for $F$ at $(\delta,R)$, and put $\bar{R}=R+C$. Since $F$ has momentum-continuous shifts, fix, applied with $\delta$ and $\bar{R}$, the positive constant that First-Order Equation Operators on the Noise Wasserstein Space with Momentum-Continuous Shifts §momentum provides for $\eta=\kappa$, which we call $\gamma$. Finally let $\tau$ be the least of $\tfrac{1}{2}$, $\tfrac{\theta}{8}$, $\eta_{1}$, $\tfrac{t_{2}}{8}$ and $\gamma\bigl(8B+2\bigr)^{-1}$. Then $0<\tau<1$, $\tau<\theta$, $\lambda\tau\le\kappa$, $\tau\le\eta_{1}$, $\tau\le t_{2}/8$ and $4\tau B<\gamma$ (as $0\le4B<8B+2$).

Step 5b (The perturbed maximiser). By Bounds, Monotonicity and Borwein-Preiss Perturbed Maximisers of the Doubled Difference in the Noise Wasserstein Distance for a Noise-Closed Penalty Pair §perturbed, with $\alpha$ and $\tau$, there are $(\hat{\mu},\hat{\nu})\in\mathcal{D}\times\mathcal{D}$, sequences $(\mu_{k})_{k}$ and $(\nu_{k})_{k}$ in $\mathcal{D}$ and positive reals $c_{k}$ whose series converges with $\sum_{k=1}^{\infty}c_{k}\le\tau$, such that the following hold: by Bounds, Monotonicity and Borwein-Preiss Perturbed Maximisers of the Doubled Difference in the Noise Wasserstein Distance for a Noise-Closed Penalty Pair §localised, the measures $\hat{\mu}$, $\hat{\nu}$, $\mu_{k}$, $\nu_{k}$ lie in $\mathcal{D}_{K}=\{\sigma\in\mathcal{D}:\mathcal{E}(\sigma)\le K\}$, so their $W$-distances to $\rho$ are at most $B$; by Bounds, Monotonicity and Borwein-Preiss Perturbed Maximisers of the Doubled Difference in the Noise Wasserstein Distance for a Noise-Closed Penalty Pair §near-maximiser, $M-\tau\le\Psi(\hat{\mu},\hat{\nu})$; and by Bounds, Monotonicity and Borwein-Preiss Perturbed Maximisers of the Doubled Difference in the Noise Wasserstein Distance for a Noise-Closed Penalty Pair §strict-maximum, with $\Phi$ as there, $\Phi(\mu,\nu)<\Phi(\hat{\mu},\hat{\nu})$ for every $(\mu,\nu)\in\mathcal{D}\times\mathcal{D}$ other than $(\hat{\mu},\hat{\nu})$. 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 $B$, the distance between any two of these measures is at most $2B$.

Step 5c (Test functions). Since $\mathcal{D}$ has the noise map property, $\hat{\mu},\hat{\nu}\in\mathcal{D}$ and $\hat{\mu},\hat{\nu},\mu_{k},\nu_{k}\in\mathcal{P}^{a}_{\rho}$, the ordered pairs $(\hat{\mu},\hat{\nu})$, $(\hat{\nu},\hat{\mu})$, $(\hat{\mu},\mu_{k})$ and $(\hat{\nu},\nu_{k})$ are uniquely noise-mapped (The Noise Map Property of a Set of Probability Measures §map-property), so by Noise-Optimal Maps and Uniquely Noise-Mapped Pairs §uniquely-mapped fix a noise-optimal map $S$ from $\hat{\mu}$ to $\hat{\nu}$ and a noise-optimal map $S'$ from $\hat{\nu}$ to $\hat{\mu}$, and, by Axiom of Countable Choice, for each $k$ noise-optimal maps $S_{k}$ from $\hat{\mu}$ to $\mu_{k}$ and $S'_{k}$ from $\hat{\nu}$ to $\nu_{k}$. Let $\varphi_{0},\psi_{0},\chi,\chi':\mathcal{P}^{a}_{\rho}\to\mathbb{R}$ be

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

the series converging by Squared Noise Wasserstein Distances, Their Convergent Series and Linear Combinations are Noise Intrinsic Test Functions §series-convergence (with $Q=\mathcal{D}$, which has the noise map property, the nonnegative bound $B$, the centres $\mu_{k}$, respectively $\nu_{k}$, whose $W$-distances to $\rho$ are at most $B$ by Step 5b, and $\beta_{k}=c_{k}$). By Squared Noise Wasserstein Distances, Their Convergent Series and Linear Combinations are Noise Intrinsic Test Functions §distance, with $\hat{\nu}$, respectively $\hat{\mu}$, as its fixed measure, $\varphi_{0}$ and $\psi_{0}$ are noise intrinsic test functions on $\mathcal{D}$ with $\nabla\varphi_{0}(\hat{\mu})=2(\mathrm{id}-S)$ and $\nabla\psi_{0}(\hat{\nu})=2(\mathrm{id}-S')$; by Squared Noise Wasserstein Distances, Their Convergent Series and Linear Combinations are Noise Intrinsic Test Functions §series-test, $\chi$ and $\chi'$ are noise intrinsic test functions 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 $S_{k}$, respectively $S'_{k}$, the gradient $q^{*}=\nabla\chi(\hat{\mu})$ is the sum in $L^{2}(\hat{\mu};X^{a})$ of the convergent series $\sum_{k=1}^{\infty}2c_{k}(\mathrm{id}-S_{k})$, the gradient $q'^{*}=\nabla\chi'(\hat{\nu})$ that of $\sum_{k=1}^{\infty}2c_{k}(\mathrm{id}-S'_{k})$, and

$$\lVert q^{*}\rVert_{\hat{\mu}}\le2\sum_{k=1}^{\infty}c_{k}W(\hat{\mu},\mu_{k})\le4B\sum_{k=1}^{\infty}c_{k}\le4\tau B,$$

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 $W(\hat{\mu},\mu_{k})\le2B$ (Step 5b) and the comparison and linearity of convergent series, and the last step uses $\sum_{k}c_{k}\le\tau$ and $0\le B$; likewise $\lVert q'^{*}\rVert_{\hat{\nu}}\le4\tau B$. Put

$$\varphi=\tfrac{\alpha}{2}\varphi_{0}+\chi,\qquad\psi=\bigl(-\tfrac{\alpha}{2}\bigr)\psi_{0}+(-1)\chi'.$$

By Squared Noise Wasserstein Distances, Their Convergent Series and Linear Combinations are Noise Intrinsic Test Functions §linear these are noise intrinsic test functions on $\mathcal{D}$, with

$$q_{1}:=\nabla\varphi(\hat{\mu})=\alpha(\mathrm{id}-S)+q^{*},\qquad q_{2}:=\nabla\psi(\hat{\nu})=\alpha(S'-\mathrm{id})-q'^{*},$$

using $-\alpha(\mathrm{id}-S')=\alpha(S'-\mathrm{id})$ in $L^{2}(\hat{\nu};X^{a})$.

Step 5d (Touching). For $(\mu,\nu)\in\mathcal{D}\times\mathcal{D}$ the series $\sum_{k}c_{k}\bigl(W(\mu,\mu_{k})^{2}+W(\nu,\nu_{k})^{2}\bigr)$ has sum $\chi(\mu)+\chi'(\nu)$ by Elementary Properties of Series of Real Numbers §linearity, so

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

Taking $\nu=\hat{\nu}$, Step 5b 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 $1$). Taking $\mu=\hat{\mu}$ and using $W(\hat{\mu},\nu)=W(\nu,\hat{\mu})$ (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), $\Phi(\hat{\mu},\nu)\le\Phi(\hat{\mu},\hat{\nu})$ for every $\nu\in\mathcal{D}$ 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 $\mathcal{D}$.

Step 5e (Exact inequalities). 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 and $v$ from below; the pair is noise-closed with closed score along noise couplings, $F$ satisfies the shift-coercivity and shift-semicontinuity conditions, $0<\delta<1$, $u$ is a viscosity subsolution and $v$ a viscosity supersolution. So Exact Inequalities at a Touching Point of a Penalised Viscosity Subsolution or Supersolution on the Noise Wasserstein Space §subsolution with $\varphi$ and Exact Inequalities at a Touching Point of a Penalised Viscosity Subsolution or Supersolution on the Noise Wasserstein Space §supersolution with $\psi$ give $\hat{\mu},\hat{\nu}\in\mathcal{D}_{\Sigma}$ and, with $r_{1}=u^{-}_{\delta}(\hat{\mu})$ and $r_{2}=v^{+}_{\delta}(\hat{\nu})$,

$$F^{-}_{\delta}(\hat{\mu},r_{1},q_{1})\le0\le F^{+}_{\delta}(\hat{\nu},r_{2},q_{2}).\qquad(5\mathrm{a})$$

Step 5f (Bounds). As $\Psi(\hat{\mu},\hat{\nu})=r_{1}-r_{2}-\tfrac{\alpha}{2}W(\hat{\mu},\hat{\nu})^{2}$, Step 5b and $\theta<M$ give

$$r_{1}-r_{2}\ge\Psi(\hat{\mu},\hat{\nu})\ge M-\tau>\theta-\tau>0.\qquad(5\mathrm{b})$$

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, $r_{1}\le b-\delta\mathcal{E}(\hat{\mu})\le b-\delta e_{0}\le|b|+|e_{0}|$ and $r_{2}\ge b'+\delta\mathcal{E}(\hat{\nu})\ge-|b'|-|e_{0}|$, using $\delta\in J$; with $r_{2}<r_{1}$ this gives $-A<r_{2}<r_{1}<A$. By Bounds, Monotonicity and Borwein-Preiss Perturbed Maximisers of the Doubled Difference in the Noise Wasserstein Distance for a Noise-Closed Penalty Pair §bounds and (5b), $0<\Psi(\hat{\mu},\hat{\nu})\le b-b'-\delta(\mathcal{E}(\hat{\mu})+\mathcal{E}(\hat{\nu}))$, so $\delta\mathcal{E}(\hat{\mu})<b-b'-\delta\mathcal{E}(\hat{\nu})\le b-b'-\delta e_{0}<A$, while $\delta\mathcal{E}(\hat{\mu})\ge\delta e_{0}\ge-|e_{0}|>-A$; thus $\delta|\mathcal{E}(\hat{\mu})|<A$ and likewise $\delta|\mathcal{E}(\hat{\nu})|<A$. Also $|\mathcal{E}(\hat{\mu})|\le|e_{0}|+|K|$ and $|\mathcal{E}(\hat{\nu})|\le|e_{0}|+|K|$, since $e_{0}\le\mathcal{E}\le K$ at both points. By The Noise-Optimal Map: Transport Cost, Stability Along Nearly Optimal Couplings and Stability Under Perturbation of the Source §cost, $\lVert S-\mathrm{id}\rVert_{\hat{\mu}}^{2}=W(\hat{\mu},\hat{\nu})^{2}$; by the scaling identity of The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §metric in the real Hilbert space $L^{2}(\hat{\mu};X^{a})$, $\lVert\alpha(\mathrm{id}-S)\rVert_{\hat{\mu}}=\alpha\lVert S-\mathrm{id}\rVert_{\hat{\mu}}$, and claim 3 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives $\lVert S-\mathrm{id}\rVert_{\hat{\mu}}=W(\hat{\mu},\hat{\nu})$, so

$$\lVert\alpha(\mathrm{id}-S)\rVert_{\hat{\mu}}=\alpha W(\hat{\mu},\hat{\nu})\le2\alpha B,$$

and likewise $\lVert\alpha(S'-\mathrm{id})\rVert_{\hat{\nu}}=\alpha W(\hat{\nu},\hat{\mu})\le2\alpha B$. By The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §triangle, Step 5c and $\tau<1$, $\lVert q_{1}\rVert_{\hat{\mu}}\le(2\alpha+4)B<R$ and $\lVert q_{2}\rVert_{\hat{\nu}}<R$, while $\lVert q_{1}-\alpha(\mathrm{id}-S)\rVert_{\hat{\mu}}=\lVert q^{*}\rVert_{\hat{\mu}}\le4\tau B<\gamma$ and $\lVert q_{2}-\alpha(S'-\mathrm{id})\rVert_{\hat{\nu}}=\lVert q'^{*}\rVert_{\hat{\nu}}<\gamma$.

Step 5g (Score bounds). The triples $\xi=(\hat{\mu},r_{1},q_{1})$ and $\zeta=(\hat{\nu},r_{2},q_{2})$ are test data for $F$, as $\hat{\mu},\hat{\nu}\in\mathcal{D}_{\Sigma}$. By Steps 5b and 5f each is $R$-bounded: the $W$-distances to $\rho$ are at most $B<R$, the penalties have absolute value at most $|e_{0}|+|K|<R$, $|r_{1}|,|r_{2}|<A<R$, and the fields have norm below $R$. By (5a), $F^{-}_{\delta}(\xi)-F^{+}_{\delta}(\zeta)\le0<R$, so $\xi\in S^{-}_{\delta,R}$ and $\zeta\in S^{+}_{\delta,R}$ (Test Data for a First-Order Equation Operator on the Noise Wasserstein Space and the Admissible Sets §admissible), each witnessing the other. As $C$ is a score bound at $(\delta,R)$ (The Shift-Coercivity Condition for a First-Order Equation Operator on the Noise Wasserstein Space §bound), $\lVert\Sigma(\hat{\mu})\rVert_{\hat{\mu}}\le C\le\bar{R}$ and $\lVert\Sigma(\hat{\nu})\rVert_{\hat{\nu}}\le\bar{R}$.

Step 5h (Removing the perturbation of the momentum). By the choice of $\gamma$ in Step 5a (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 $q'=\alpha(\mathrm{id}-S)$, all of which satisfy the bounds there by Steps 5f and 5g ($|r_{1}|<A\le\bar{R}$, both fields of norm below $R\le\bar{R}$, their difference of norm below $\gamma$, $\lVert\Sigma(\hat{\mu})\rVert_{\hat{\mu}}\le\bar{R}$), and then at $\hat{\nu}$ with $r_{2}$, $q=q_{2}$ and $q'=\alpha(S'-\mathrm{id})$, (5a) gives

$$F^{-}_{\delta}\bigl(\hat{\mu},r_{1},\alpha(\mathrm{id}-S)\bigr)<\kappa,\qquad-\kappa<F^{+}_{\delta}\bigl(\hat{\nu},r_{2},\alpha(S'-\mathrm{id})\bigr).\qquad(5\mathrm{c})$$

Step 5i (Properness). 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, $F^{-}_{\delta}(\hat{\mu},r,\alpha(\mathrm{id}-S))$ is $F$ evaluated at $\hat{\mu}$, $r+\delta\mathcal{E}(\hat{\mu})$ and $\alpha(\mathrm{id}-S)+\delta\Sigma(\hat{\mu})$ for every $r\in\mathbb{R}$, the first and last arguments not depending on $r$, and $(\hat{\mu},\alpha(\mathrm{id}-S)+\delta\Sigma(\hat{\mu}))\in\mathcal{V}^{a}(\mathcal{D}_{\Sigma})$. By (5b) and Step 5f, $r_{2}+\delta\mathcal{E}(\hat{\mu})\le r_{1}+\delta\mathcal{E}(\hat{\mu})$ and both have absolute value below $A+A=R_{0}$, so the properness constant $\lambda$ at $R_{0}$ (Locally Strictly Proper First-Order Equation Operators on the Noise Wasserstein Space §constant) gives

$$\lambda(r_{1}-r_{2})\le F^{-}_{\delta}\bigl(\hat{\mu},r_{1},\alpha(\mathrm{id}-S)\bigr)-F^{-}_{\delta}\bigl(\hat{\mu},r_{2},\alpha(\mathrm{id}-S)\bigr).\qquad(5\mathrm{d})$$

Step 5j (The structure condition). We have $1<\alpha$, $0<\delta<1$, $\hat{\mu},\hat{\nu}\in\mathcal{D}_{\Sigma}$ with both ordered pairs $(\hat{\mu},\hat{\nu})$ and $(\hat{\nu},\hat{\mu})$ uniquely noise-mapped (Step 5c), $S$ and $S'$ noise-optimal maps between them, $\delta(|\mathcal{E}(\hat{\mu})|+|\mathcal{E}(\hat{\nu})|)<2A=R_{0}$ and $-R_{0}\le r_{2}\le R_{0}$ (Step 5f). So The First-Order Structure Condition at Uniquely Noise-Mapped Pairs §pair, for the pair $(\omega_{1},\omega_{2})$ at $R_{0}$ with the value slot $r_{2}$, gives, writing $\Omega_{1}=\omega_{1}\bigl(\alpha W(\hat{\mu},\hat{\nu})^{2}+\alpha^{-1}\bigr)$ and $\Omega_{2}=\omega_{2}\bigl(\delta(|\mathcal{E}(\hat{\mu})|+|\mathcal{E}(\hat{\nu})|+1),\alpha\bigr)$,

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

Adding this to (5d) and using (5c),

$$\lambda(r_{1}-r_{2})\le F^{-}_{\delta}\bigl(\hat{\mu},r_{1},\alpha(\mathrm{id}-S)\bigr)-F^{+}_{\delta}\bigl(\hat{\nu},r_{2},\alpha(S'-\mathrm{id})\bigr)+\Omega_{1}+\Omega_{2}<2\kappa+\Omega_{1}+\Omega_{2}.\qquad(5\mathrm{e})$$

Step 5k (The first modulus). By (1d) with $\alpha'=\tfrac{\alpha}{2}$, whose coefficient is $\tfrac{\alpha}{4}$, applied to $(\hat{\mu},\hat{\nu})$ (admissible by Step 5b), then $G(\tfrac{\alpha}{2},\delta)\le N(\tfrac{\alpha}{2})$, (4a), $\eta_{2}\le\eta_{1}$, (3a) and $\tau\le\eta_{1}$,

$$\tfrac{\alpha}{4}W(\hat{\mu},\hat{\nu})^{2}\le G(\tfrac{\alpha}{2},\delta)-G(\alpha,\delta)+\tau<N(\tfrac{\alpha}{2})-N(\alpha)+\eta_{2}+\tau<3\eta_{1},$$

so $\alpha W(\hat{\mu},\hat{\nu})^{2}<12\eta_{1}<t_{1}/2$, and with $\alpha^{-1}<t_{1}/2$ (Step 3) the argument of $\omega_{1}$ lies in $[0,t_{1}]$; hence $\Omega_{1}\le\kappa$.

Step 5l (The second modulus). By (1c) with $\delta'=\tfrac{\delta}{2}\in J$, applied to $(\hat{\mu},\hat{\nu})$, then $G(\alpha,\tfrac{\delta}{2})\le N(\alpha)$ and (4a),

$$\tfrac{\delta}{2}\bigl(\mathcal{E}(\hat{\mu})-e_{0}+\mathcal{E}(\hat{\nu})-e_{0}\bigr)\le G(\alpha,\tfrac{\delta}{2})-G(\alpha,\delta)+\tau<\eta_{2}+\tau\le\tfrac{t_{2}}{8}+\tfrac{t_{2}}{8}.$$

As $\mathcal{E}(\hat{\mu})-e_{0}\ge0$, the triangle inequality for $|\cdot|$ applied to $\mathcal{E}(\hat{\mu})=(\mathcal{E}(\hat{\mu})-e_{0})+e_{0}$ gives $|\mathcal{E}(\hat{\mu})|\le\mathcal{E}(\hat{\mu})-e_{0}+|e_{0}|$, and likewise for $\hat{\nu}$. Multiplying by the positive $\delta$ and using $\delta(2|e_{0}|+1)<t_{2}/4$ (Step 4),

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

and the argument is positive; hence $\Omega_{2}\le\kappa$.

Step 5m (Contradiction). By (5e), Steps 5k and 5l, $\lambda(r_{1}-r_{2})<4\kappa$. By (5b) and $0<\lambda$, $\lambda(\theta-\tau)<\lambda(r_{1}-r_{2})$, and $\lambda\tau\le\kappa$ (Step 5a), so $\lambda\theta<\lambda\tau+4\kappa\le5\kappa=\tfrac{5}{8}\lambda\theta$, that is $\tfrac{3}{8}\lambda\theta<0$, contradicting $0<\lambda\theta$. Therefore $M(\delta,\alpha)\le\theta$. For $\mu\in\mathcal{D}$, since $W(\mu,\mu)=0$ (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 §separation),

$$u^{-}_{\delta}(\mu)-v^{+}_{\delta}(\mu)=\Psi_{\delta,\alpha}(\mu,\mu)\le M(\delta,\alpha)\le\theta .$$

As $\delta$ with $0<\delta<\delta_{0}$ was arbitrary, this is claim 1 (comparison of the envelopes) of the present theorem.

Step 6 (Claim 2). Let $\mu\in\mathcal{D}$ and let $\theta$ be positive; let $\delta_{0}$ be as in claim 1 for $\theta$, and let $\delta$ be half the least of $\delta_{0}$ and $\theta\bigl(2|\mathcal{E}(\mu)|+1\bigr)^{-1}$, so that $0<\delta<\delta_{0}$ and $2\delta|\mathcal{E}(\mu)|\le\theta$. Since $u$ has penalty-subordinate growth from above and $v$ from below (Step 5e), 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 $u(\mu)-\delta\mathcal{E}(\mu)\le u^{-}_{\delta}(\mu)$ and $v^{+}_{\delta}(\mu)\le v(\mu)+\delta\mathcal{E}(\mu)$, so by claim 1 (comparison of the envelopes) of the present theorem

$$u(\mu)-v(\mu)=\bigl(u(\mu)-\delta\mathcal{E}(\mu)\bigr)-\bigl(v(\mu)+\delta\mathcal{E}(\mu)\bigr)+2\delta\mathcal{E}(\mu)\le u^{-}_{\delta}(\mu)-v^{+}_{\delta}(\mu)+2\delta|\mathcal{E}(\mu)|\le2\theta .$$

As $\theta$ was arbitrary, $u(\mu)-v(\mu)\le0$ by Comparison of Real Numbers with Arbitrary Positive Slack §slack-above, that is $u(\mu)\le v(\mu)$; this is claim 2 (comparison) of the present theorem.