Proof of The Hamilton-Jacobi Operator with Penalty Drift on the Noise Wasserstein Space Satisfies the Hypotheses of the First-Order Comparison Principle for a Displacement Convex Noise Penalty Pair

Each result cited is universally quantified over the data in its own statement.

Elementary order and arithmetic of real numbers (adding inequalities, multiplying them by nonnegative or positive reals, handling absolute values, the nonnegativity of squares) is carried by The Real Numbers: Standing Notation and Background §background and is used without further mention; so is the fact that for nonnegative reals $x,y$ one has $x<y$, respectively $x\le y$, exactly when $x^{2}<y^{2}$, respectively $x^{2}\le y^{2}$ (claims 1 and 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field). The four assumptions of the statement are referred to as (Convexity), (Lower semicontinuity of the penalty), (Growth) and (Running cost).

Step 0 (Notation and preliminary facts). Fix the data of the statement: the noise penalty pair $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$, the reals $\lambda_{0},\theta$ with $0<\lambda_{0}$ and $0<\theta\le1$, the function $g$, the operator $F$, a real number $C$ as in (Growth), and a real number $M\ge0$ as in (Running cost), so that $|g(\mu)|\le M$ for every $\mu\in\mathcal{D}$.

(0.1) Hilbert spaces and distances. For $\nu\in\mathcal{P}^{a}_{\rho}$ the space $L^{2}(\nu;X^{a})$, with inner product $\langle\cdot,\cdot\rangle_{\nu}$ and norm $\lVert\cdot\rVert_{\nu}$, is a real Hilbert space by Probability Measures on a Hilbert Space Transported in the Noise Norm: Standing Notation §fields, in particular a real inner product space, and $\lVert x\rVert_{\nu}^{2}=\langle x,x\rangle_{\nu}$ by Real Inner Product Space §norm. Inner products are symmetric by condition (a) of Real Inner Product Space §inner-product; bilinearity, the zero vector, homogeneity of the norm and the expansion of $\lVert x\pm y\rVert_{\nu}^{2}$ are Elementary Identities in a Real Inner Product Space §bilinear, Elementary Identities in a Real Inner Product Space §zero, Elementary Identities in a Real Inner Product Space §homogeneity and Elementary Identities in a Real Inner Product Space §expansion; $|\langle x,y\rangle_{\nu}|\le\lVert x\rVert_{\nu}\lVert y\rVert_{\nu}$ by The Cauchy-Schwarz Inequality in a Real Inner Product Space; and the norm satisfies the triangle inequality of The Norm Metric of a Real Inner Product Space: Triangle Inequalities, Limits and Continuity §triangle. The noise space $X^{a}$, with inner product $\langle\cdot,\cdot\rangle_{a}$, is a real Hilbert space and a linear subspace of $X$ carrying the vector operations of $X$, by The Noise Space is a Real Hilbert Space: Orthonormal Basis, Continuous Embedding, Partial Sums, Closed Balls and Borel Measurability §hilbert; its inner product is linear in the first argument by conditions (b) and (c) of Real Inner Product Space §inner-product. For $\nu\in\mathcal{D}_{\Sigma}$ one has $\nu\in\mathcal{D}$, so that $\mathcal{E}(\nu)$ is a real number, and $\Sigma(\nu)\in T^{a}_{\nu}\subseteq L^{2}(\nu;X^{a})$, by Noise Penalty Pairs on the Noise Wasserstein Space: the Penalty, Its Score, and Their Domains §pair. For $\mu,\nu\in\mathcal{P}^{a}_{\rho}$ the ordered pair $(\mu,\nu)$ is noise-connected 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 §connected; $W_{a}(\mu,\nu)$ is a nonnegative real number with $W_{a}(\mu,\nu)^{2}\le I^{a}(\pi)$ for every $\pi\in\Pi^{a}(\mu,\nu)$, by The Noise Wasserstein Distance §distance; $W_{a}(\mu,\nu)=W_{a}(\nu,\mu)$ 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; $W_{a}(\mu,\mu)=0$ 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 §separation; and $W_{a}$ satisfies the triangle inequality 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. Moreover $\rho\in\mathcal{P}^{a}_{\rho}$ 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. Every $\pi\in\Pi^{a}(\mu,\nu)$ belongs to $\Pi(\mu,\nu)$ by Couplings of Finite Noise Cost and Their Noise Cost §couplings, and its noise cost $I^{a}(\pi)$ is a nonnegative real number by Couplings of Finite Noise Cost and Their Noise Cost §cost.

(0.2) Cross pairings and discrepancies. Let $\nu',\nu\in\mathcal{P}(X)$ and $\pi\in\Pi(\nu',\nu)$, and let $\mathcal{K}^{a}$ be the cross pairing of Noise Displacement and Cross Pairings Along Couplings: Bounds, Linearity, Displacement Couplings, Polarisation and a Vanishing Criterion §cross, that clause being applied with $\nu',\nu$ in place of its $\nu,\mu$. (a) For $k\in L^{2}(\nu';X^{a})$ and $w\in L^{2}(\nu;X^{a})$, the discrepancy $D=\int_{X\times X}|k(x)-w(y)|_{a}^{2}\,\pi(dz)$ is a nonnegative real number by Noise Displacement and Cross Pairings Along Couplings: Bounds, Linearity, Displacement Couplings, Polarisation and a Vanishing Criterion §discrepancy, and

$$D=\lVert k\rVert_{\nu'}^{2}-2\,\mathcal{K}^{a}(k,w,\pi)+\lVert w\rVert_{\nu}^{2}$$

by the same clause. (b) For $h,k\in L^{2}(\nu';X^{a})$, $w\in L^{2}(\nu;X^{a})$ and $t\in\mathbb{R}$,

$$\mathcal{K}^{a}(th+k,w,\pi)=t\,\mathcal{K}^{a}(h,w,\pi)+\mathcal{K}^{a}(k,w,\pi).$$

Indeed, fix representatives $h,k,w$. By The Space of Square-Integrable Maps from a Measure Space into a Hilbert Space with an Orthonormal Basis §operations the pointwise map $th+k$ is a representative of the class $th+k$, and by Noise Displacement and Cross Pairings Along Couplings: Bounds, Linearity, Displacement Couplings, Polarisation and a Vanishing Criterion §cross the cross pairing may be computed with any representatives, the functions $z\mapsto\langle h(x),w(y)\rangle_{a}$ and $z\mapsto\langle k(x),w(y)\rangle_{a}$ being integrable against $\pi$. At every $z$, linearity of $\langle\cdot,\cdot\rangle_{a}$ in its first argument (0.1) gives $\langle th(x)+k(x),w(y)\rangle_{a}=t\langle h(x),w(y)\rangle_{a}+\langle k(x),w(y)\rangle_{a}$, and the linearity of the integral, Linearity and Monotonicity of the Lebesgue Integral §integrable, gives the identity.

(0.3) Displacements of noise-optimal maps. Let $\mu,\nu\in\mathcal{P}^{a}_{\rho}$ and let $S$ be a noise-optimal map from $\mu$ to $\nu$; by Noise-Optimal Maps and Uniquely Noise-Mapped Pairs §map, $S_{\#}\mu=\nu$ and the displacement $S-\mathrm{id}\in L^{2}(\mu;X^{a})$ is the class of the measurable map $x\mapsto S(x)-x$ into $X^{a}$. Apply Noise Displacement and Cross Pairings Along Couplings: Bounds, Linearity, Displacement Couplings, Polarisation and a Vanishing Criterion §displacement with this $\mu$, with $h=S-\mathrm{id}$ and this representative, and with $t=1$. Since the vector operations of $X^{a}$ are those of $X$ (0.1), the map $S_{1}$ there is $x\mapsto x+(S(x)-x)=S(x)$, so $S_{1}=S$, and the coupling $\pi_{S}=(\mathrm{id},S)_{\#}\mu\in\Pi^{a}(\mu,\nu)$ satisfies

$$I^{a}(\pi_{S})=\lVert S-\mathrm{id}\rVert_{\mu}^{2},\qquad\mathcal{J}^{a}(\zeta,\pi_{S})=\langle\zeta,S-\mathrm{id}\rangle_{\mu}\quad\text{for every }\zeta\in L^{2}(\mu;X^{a}),$$

where $\mathcal{J}^{a}$ is the noise displacement pairing of Noise Displacement and Cross Pairings Along Couplings: Bounds, Linearity, Displacement Couplings, Polarisation and a Vanishing Criterion §pairing. The coupling $\pi_{S}$ is noise-optimal by Noise-Optimal Maps and Uniquely Noise-Mapped Pairs §map, so $I^{a}(\pi_{S})=W_{a}(\mu,\nu)^{2}$ by Noise-Optimal Couplings §optimal; as both numbers are nonnegative, $\lVert S-\mathrm{id}\rVert_{\mu}=W_{a}(\mu,\nu)$.

(0.4) The constant $C$ is nonnegative. By Noise Penalty Pairs on the Noise Wasserstein Space: the Penalty, Its Score, and Their Domains §nonempty there is $\mu_{0}\in\mathcal{D}_{\Sigma}\subseteq\mathcal{D}$. Then (Growth) gives $0\le W_{a}(\mu_{0},\rho)^{2}\le C\,(1+|\mathcal{E}(\mu_{0})|)$. If $C<0$, then, as $0<1+|\mathcal{E}(\mu_{0})|$, we would get $C\,(1+|\mathcal{E}(\mu_{0})|)<0$, a contradiction. Hence $0\le C$.

(0.5) Elementary inequalities in $\delta$. Let $\delta\in\mathbb{R}$ with $0<\delta<1$. Then $0<\theta\delta\le\delta<1$, because $0<\theta\le1$. Consequently $0<1+\theta\delta$, $0\le1-\theta\delta\le1$, $\tfrac{\delta}{2}+\tfrac{\theta\delta^{2}}{2}>0$, $\tfrac{\delta}{2}-\tfrac{\theta\delta^{2}}{2}=\tfrac{\delta}{2}(1-\theta\delta)\ge0$, and $\delta-\tfrac{\theta\delta^{2}}{2}=\delta-\tfrac{\delta}{2}\,\theta\delta\ge\tfrac{\delta}{2}>0$. For real $m,s$ one has $2ms\le m^{2}+s^{2}$, since $0\le(m-s)^{2}=m^{2}-2ms+s^{2}$.

(0.6) Expanded form of the shifts. Let $(\nu,q)\in\mathcal{V}^{a}(\mathcal{D}_{\Sigma})$, $r\in\mathbb{R}$ and $\delta>0$, and write $\sigma=\Sigma(\nu)$. 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, the formula of The Discounted Hamilton-Jacobi Equation with a Penalty Drift on the Noise Wasserstein Space §operator, and $\langle\sigma,q\pm\delta\sigma\rangle_{\nu}=\langle\sigma,q\rangle_{\nu}\pm\delta\lVert\sigma\rVert_{\nu}^{2}$ (0.1),

$$F^{-}_{\delta}(\nu,r,q)=\lambda_{0}r+\lambda_{0}\delta\,\mathcal{E}(\nu)+\frac{\theta}{2}\lVert q+\delta\sigma\rVert_{\nu}^{2}+\langle\sigma,q\rangle_{\nu}+\delta\lVert\sigma\rVert_{\nu}^{2}-g(\nu),\tag{0a}$$ $$F^{+}_{\delta}(\nu,r,q)=\lambda_{0}r-\lambda_{0}\delta\,\mathcal{E}(\nu)+\frac{\theta}{2}\lVert q-\delta\sigma\rVert_{\nu}^{2}+\langle\sigma,q\rangle_{\nu}-\delta\lVert\sigma\rVert_{\nu}^{2}-g(\nu).\tag{0c}$$

Expanding $\lVert q\pm\delta\sigma\rVert_{\nu}^{2}=\lVert q\rVert_{\nu}^{2}\pm2\delta\langle\sigma,q\rangle_{\nu}+\delta^{2}\lVert\sigma\rVert_{\nu}^{2}$ by (0.1) gives

$$F^{-}_{\delta}(\nu,r,q)=\lambda_{0}r+\lambda_{0}\delta\,\mathcal{E}(\nu)+\frac{\theta}{2}\lVert q\rVert_{\nu}^{2}+(1+\theta\delta)\langle\sigma,q\rangle_{\nu}+\Bigl(\delta+\frac{\theta\delta^{2}}{2}\Bigr)\lVert\sigma\rVert_{\nu}^{2}-g(\nu),\tag{0b}$$ $$F^{+}_{\delta}(\nu,r,q)=\lambda_{0}r-\lambda_{0}\delta\,\mathcal{E}(\nu)+\frac{\theta}{2}\lVert q\rVert_{\nu}^{2}+(1-\theta\delta)\langle\sigma,q\rangle_{\nu}-\Bigl(\delta-\frac{\theta\delta^{2}}{2}\Bigr)\lVert\sigma\rVert_{\nu}^{2}-g(\nu).\tag{0d}$$

Step 1 (Local strict properness, claim 1). Let $R>0$, $(\nu,q)\in\mathcal{V}^{a}(\mathcal{D}_{\Sigma})$ and $-R\le s\le r\le R$. By the formula of The Discounted Hamilton-Jacobi Equation with a Penalty Drift on the Noise Wasserstein Space §operator all terms other than $\lambda_{0}r$ and $\lambda_{0}s$ cancel, so $F(\nu,r,q)-F(\nu,s,q)=\lambda_{0}(r-s)$, and in particular $\lambda_{0}(r-s)\le F(\nu,r,q)-F(\nu,s,q)$. Hence the positive real $\lambda_{0}$ is a properness constant for $F$ at $R$, for every $R>0$, and $F$ is locally strictly proper. None of the four assumptions is used here.

Step 2 (Shift-coercivity, claim 2). Let $\delta,R\in\mathbb{R}$ with $0<\delta<1$ and $0<R$. Put

$$A=2\lambda_{0}R+M+\frac{R^{2}}{2},\qquad B=R+2A+\frac{R^{2}}{2\delta},\qquad C_{\delta,R}=1+\frac{4(R+B)}{\delta};$$

these are nonnegative reals, as $0\le M$. We show that $C_{\delta,R}$ is a score bound for $F$ at $(\delta,R)$.

Let $\xi=(\nu,r,q)$ and $\eta=(\nu',r',q')$ be $R$-bounded test data with $F^{-}_{\delta}(\xi)-F^{+}_{\delta}(\eta)<R$. By Test Data for a First-Order Equation Operator on the Noise Wasserstein Space and the Admissible Sets §admissible every member of $S^{-}_{\delta,R}$ is such a $\xi$ for some $\eta$, and every member of $S^{+}_{\delta,R}$ is such an $\eta$ for some $\xi$; so it suffices to show $s\le C_{\delta,R}$ and $s'\le C_{\delta,R}$, where $s=\lVert\Sigma(\nu)\rVert_{\nu}$ and $s'=\lVert\Sigma(\nu')\rVert_{\nu'}$. By Test Data for a First-Order Equation Operator on the Noise Wasserstein Space and the Admissible Sets §bounded, $|r|,|r'|<R$, $|\mathcal{E}(\nu)|,|\mathcal{E}(\nu')|<R$ and $\lVert q\rVert_{\nu},\lVert q'\rVert_{\nu'}<R$.

Lower bound for $F^{-}_{\delta}(\xi)$. In (0a) we have $\lambda_{0}r\ge-\lambda_{0}R$; $\lambda_{0}\delta\mathcal{E}(\nu)\ge-\lambda_{0}\delta|\mathcal{E}(\nu)|\ge-\lambda_{0}R$ as $\delta<1$; $\tfrac{\theta}{2}\lVert q+\delta\Sigma(\nu)\rVert_{\nu}^{2}\ge0$; $\langle\Sigma(\nu),q\rangle_{\nu}\ge-sR$ by the Cauchy-Schwarz inequality of (0.1); $\delta\lVert\Sigma(\nu)\rVert_{\nu}^{2}=\delta s^{2}\ge\tfrac{\delta}{2}s^{2}$; and $-g(\nu)\ge-M$ by the choice of $M$ in Step 0, as $\nu\in\mathcal{D}$. Hence, since $\tfrac{R^{2}}{2}\ge0$,

$$F^{-}_{\delta}(\xi)\ \ge\ \frac{\delta}{2}s^{2}-Rs-A=\varphi(s)-A,\qquad\text{where }\varphi(x)=\frac{\delta}{2}x^{2}-Rx\ \ (x\in\mathbb{R}).$$

Upper bound for $F^{+}_{\delta}(\eta)$. In (0d) we have $\lambda_{0}r'\le\lambda_{0}R$; $-\lambda_{0}\delta\mathcal{E}(\nu')\le\lambda_{0}R$; $\tfrac{\theta}{2}\lVert q'\rVert_{\nu'}^{2}\le\tfrac{R^{2}}{2}$ as $\theta\le1$; $(1-\theta\delta)\langle\Sigma(\nu'),q'\rangle_{\nu'}\le(1-\theta\delta)s'R\le Rs'$ by Cauchy-Schwarz and $0\le1-\theta\delta\le1$ (0.5); $-(\delta-\tfrac{\theta\delta^{2}}{2})s'^{2}\le-\tfrac{\delta}{2}s'^{2}$ by (0.5); and $-g(\nu')\le M$. Hence $F^{+}_{\delta}(\eta)\le A-\varphi(s')$.

Conclusion. For every real $x$, $0\le\tfrac{\delta}{2}(x-\tfrac{R}{\delta})^{2}=\varphi(x)+\tfrac{R^{2}}{2\delta}$, so $\varphi(x)\ge-\tfrac{R^{2}}{2\delta}$. From the two bounds, $\varphi(s)+\varphi(s')-2A\le F^{-}_{\delta}(\xi)-F^{+}_{\delta}(\eta)<R$, hence $\varphi(s)<R+2A-\varphi(s')\le B$ and likewise $\varphi(s')<B$. Now let $x\ge0$ with $\varphi(x)<B$, and suppose $x>C_{\delta,R}$. Then $x>1$, $x>\tfrac{4R}{\delta}$ and $x>\tfrac{4B}{\delta}$, all three numbers being at most $C_{\delta,R}$. The second gives $Rx<\tfrac{\delta}{4}x^{2}$, so $\varphi(x)>\tfrac{\delta}{4}x^{2}$; the first gives $x^{2}>x$, so $\tfrac{\delta}{4}x^{2}>\tfrac{\delta}{4}x$; and the third gives $\tfrac{\delta}{4}x>B$. Thus $\varphi(x)>B$, a contradiction. Hence $x\le C_{\delta,R}$; applied to $x=s$ and $x=s'$ this proves the claim. As $\delta,R$ were arbitrary, $F$ satisfies the shift-coercivity condition. Of the assumptions only the boundedness in (Running cost) and $\theta\le1$ were used.

Step 3 (An estimate along a coupling). Claim. Let $\nu',\nu\in\mathcal{P}(X)$, $\pi\in\Pi(\nu',\nu)$, $h,k\in L^{2}(\nu';X^{a})$, $w\in L^{2}(\nu;X^{a})$, and let $\ell\in\mathbb{R}$ be positive with $\lVert h\rVert_{\nu'}\le\ell$. Let $D=\int_{X\times X}|k(x)-w(y)|_{a}^{2}\,\pi(dz)$ be the discrepancy of $k$ and $w$ along $\pi$. Then

$$\bigl(\langle h,k\rangle_{\nu'}-\mathcal{K}^{a}(h,w,\pi)\bigr)^{2}\le\ell^{2}D .$$

Proof. Put $\beta=\langle h,k\rangle_{\nu'}-\mathcal{K}^{a}(h,w,\pi)$ and let $t\in\mathbb{R}$. The field $th+k$ lies in $L^{2}(\nu';X^{a})$, and by (0.2)(a) its discrepancy with $w$ along $\pi$ is nonnegative and equals $\lVert th+k\rVert_{\nu'}^{2}-2\mathcal{K}^{a}(th+k,w,\pi)+\lVert w\rVert_{\nu}^{2}$. By (0.1), $\lVert th+k\rVert_{\nu'}^{2}=t^{2}\lVert h\rVert_{\nu'}^{2}+2t\langle h,k\rangle_{\nu'}+\lVert k\rVert_{\nu'}^{2}$; by (0.2)(b), $\mathcal{K}^{a}(th+k,w,\pi)=t\mathcal{K}^{a}(h,w,\pi)+\mathcal{K}^{a}(k,w,\pi)$; and by (0.2)(a) again, $\lVert k\rVert_{\nu'}^{2}-2\mathcal{K}^{a}(k,w,\pi)+\lVert w\rVert_{\nu}^{2}=D$. Therefore

$$0\le t^{2}\lVert h\rVert_{\nu'}^{2}+2t\beta+D\le t^{2}\ell^{2}+2t\beta+D\qquad\text{for every }t\in\mathbb{R}.$$

With $t=-\beta\ell^{-2}$ this reads $0\le\beta^{2}\ell^{-2}-2\beta^{2}\ell^{-2}+D=D-\beta^{2}\ell^{-2}$, and multiplying by $\ell^{2}>0$ gives the claim.

Consequence. Let $\nu\in\mathcal{P}(X)$, $w\in L^{2}(\nu;X^{a})$ and $\ell>0$, and for every $n\in\mathbb{N}$ let $\nu'_{n}\in\mathcal{P}(X)$, $\pi_{n}\in\Pi(\nu'_{n},\nu)$ and $h_{n},k_{n}\in L^{2}(\nu'_{n};X^{a})$ with $\lVert h_{n}\rVert_{\nu'_{n}}\le\ell$, such that the discrepancies $D_{n}$ of $k_{n}$ and $w$ along $\pi_{n}$ converge to $0$. Then $\beta_{n}=\langle h_{n},k_{n}\rangle_{\nu'_{n}}-\mathcal{K}^{a}(h_{n},w,\pi_{n})$ converges to $0$: given $\varepsilon>0$, choose $N$ with $D_{n}<\varepsilon^{2}\ell^{-2}$ for $n\ge N$ (the $D_{n}$ being nonnegative); then $|\beta_{n}|^{2}=\beta_{n}^{2}\le\ell^{2}D_{n}<\varepsilon^{2}$ by the claim, so $|\beta_{n}|<\varepsilon$.

Step 4 (Shift-semicontinuity, claim 3). Let $\delta,R\in\mathbb{R}$ with $0<\delta<1$ and $0<R$, let $\xi_{n}=(\nu_{n},r_{n},q_{n})$ ($n\in\mathbb{N}$) and $\xi=(\nu,r,q)$ be test data, and let $\pi_{n}\in\Pi^{a}(\nu_{n},\nu)$ be couplings such that $(\xi_{n})$ converges to $\xi$ along $(\pi_{n})$ with score bounded by $R$. Write $\sigma_{n}=\Sigma(\nu_{n})$, $\sigma=\Sigma(\nu)$ and $\mathcal{K}_{n}(k,w)=\mathcal{K}^{a}(k,w,\pi_{n})$, with $\pi_{n}\in\Pi(\nu_{n},\nu)$ by (0.1). By that clause: every $\xi_{n}$ is $R$-bounded, so $|\mathcal{E}(\nu_{n})|<R$ and $\lVert q_{n}\rVert_{\nu_{n}}<R$; $\lVert\sigma_{n}\rVert_{\nu_{n}}\le R$; $(\pi_{n})$ is a sequence of couplings of vanishing noise cost, so $\lim_{n}I^{a}(\pi_{n})=0$; $(q_{n})$ converges strongly to $q$, so the discrepancies $D_{n}$ of $q_{n}$ and $q$ along $\pi_{n}$ converge to $0$; $(\sigma_{n})$ converges weakly to $\sigma$, so $\lim_{n}\mathcal{K}_{n}(\sigma_{n},w)=\langle\sigma,w\rangle_{\nu}$ for every $w\in L^{2}(\nu;X^{a})$; and $r_{n}\to r$. The measures $\nu_{n},\nu$ lie in $\mathcal{D}_{\Sigma}\subseteq\mathcal{D}$ by Test Data for a First-Order Equation Operator on the Noise Wasserstein Space and the Admissible Sets §data and Noise Penalty Pairs on the Noise Wasserstein Space: the Penalty, Its Score, and Their Domains §pair. Limits of sums, differences and scalar multiples of convergent real sequences are computed by Arithmetic of Limits of Real Sequences §sums and Arithmetic of Limits of Real Sequences §scalar.

(4.1) $W_{a}(\nu,\nu_{n})\to0$. Given $\varepsilon>0$, choose $N$ with $I^{a}(\pi_{n})<\varepsilon^{2}$ for $n\ge N$, by Limit of a Sequence of Real Numbers. Then $W_{a}(\nu_{n},\nu)^{2}\le I^{a}(\pi_{n})<\varepsilon^{2}$ by (0.1), so $W_{a}(\nu_{n},\nu)<\varepsilon$, and $W_{a}(\nu,\nu_{n})<\varepsilon$ by symmetry (0.1), for $n\ge N$.

(4.2) Convergent terms. (a) $g(\nu_{n})\to g(\nu)$: given $\varepsilon>0$, (Running cost) gives $\gamma>0$ with $|g(\mu)-g(\mu')|<\varepsilon$ for $\mu,\mu'\in\mathcal{D}$ with $W_{a}(\mu,\mu')<\gamma$, and by (4.1) this applies to $\mu=\nu_{n}$, $\mu'=\nu$ for all large $n$. (b) $\lVert q_{n}\rVert_{\nu_{n}}^{2}\to\lVert q\rVert_{\nu}^{2}$: by the consequence in Step 3 with $\ell=R$, $h_{n}=k_{n}=q_{n}$ and $w=q$, the numbers $\beta_{n}=\lVert q_{n}\rVert_{\nu_{n}}^{2}-\mathcal{K}_{n}(q_{n},q)$ tend to $0$; by (0.2)(a), $D_{n}=\lVert q_{n}\rVert_{\nu_{n}}^{2}-2\mathcal{K}_{n}(q_{n},q)+\lVert q\rVert_{\nu}^{2}=2\beta_{n}-\lVert q_{n}\rVert_{\nu_{n}}^{2}+\lVert q\rVert_{\nu}^{2}$, so $\lVert q_{n}\rVert_{\nu_{n}}^{2}=2\beta_{n}-D_{n}+\lVert q\rVert_{\nu}^{2}\to\lVert q\rVert_{\nu}^{2}$. (c) $\langle\sigma_{n},q_{n}\rangle_{\nu_{n}}\to\langle\sigma,q\rangle_{\nu}$: by the consequence in Step 3 with $\ell=R$, $h_{n}=\sigma_{n}$, $k_{n}=q_{n}$ and $w=q$, the difference $\langle\sigma_{n},q_{n}\rangle_{\nu_{n}}-\mathcal{K}_{n}(\sigma_{n},q)$ tends to $0$, and $\mathcal{K}_{n}(\sigma_{n},q)\to\langle\sigma,q\rangle_{\nu}$ by weak convergence with $w=q$.

(4.3) Lower semicontinuous terms. (a) For every $\varepsilon>0$ there is $N$ with $\mathcal{E}(\nu_{n})>\mathcal{E}(\nu)-\varepsilon$ for $n\ge N$: by (Lower semicontinuity of the penalty) and Lower Semicontinuous Function on a Subset of a Metric Space, $\mathcal{E}$ is lower semicontinuous at $\nu\in\mathcal{D}$ relative to $\mathcal{D}$, which gives $\gamma>0$ with $\mathcal{E}(\nu)-\varepsilon<\mathcal{E}(\mu)$ for $\mu\in\mathcal{D}$ with $W_{a}(\nu,\mu)<\gamma$; apply (4.1), as $\nu_{n}\in\mathcal{D}$. (b) For every $\varepsilon>0$ there is $N$ with $\lVert\sigma_{n}\rVert_{\nu_{n}}^{2}>\lVert\sigma\rVert_{\nu}^{2}-\varepsilon$ for $n\ge N$: by (0.2)(a), $0\le\lVert\sigma_{n}\rVert_{\nu_{n}}^{2}-2\mathcal{K}_{n}(\sigma_{n},\sigma)+\lVert\sigma\rVert_{\nu}^{2}$, so $\lVert\sigma_{n}\rVert_{\nu_{n}}^{2}\ge2\mathcal{K}_{n}(\sigma_{n},\sigma)-\lVert\sigma\rVert_{\nu}^{2}$; the right side converges to $2\lVert\sigma\rVert_{\nu}^{2}-\lVert\sigma\rVert_{\nu}^{2}=\lVert\sigma\rVert_{\nu}^{2}$ by weak convergence with $w=\sigma$, so it exceeds $\lVert\sigma\rVert_{\nu}^{2}-\varepsilon$ for large $n$.

(4.4) The lower shift. By (0b), $F^{-}_{\delta}(\xi_{n})=u_{n}+v_{n}$ and $F^{-}_{\delta}(\xi)=u+v$, where

$$u_{n}=\lambda_{0}r_{n}+\frac{\theta}{2}\lVert q_{n}\rVert_{\nu_{n}}^{2}+(1+\theta\delta)\langle\sigma_{n},q_{n}\rangle_{\nu_{n}}-g(\nu_{n}),\qquad v_{n}=\lambda_{0}\delta\,\mathcal{E}(\nu_{n})+\Bigl(\delta+\frac{\theta\delta^{2}}{2}\Bigr)\lVert\sigma_{n}\rVert_{\nu_{n}}^{2},$$

and $u,v$ are the same expressions at $\xi$. By (4.2) and the limit laws, $u_{n}\to u$. Let $k=\lambda_{0}\delta+\delta+\tfrac{\theta\delta^{2}}{2}+1>0$. Given $\varepsilon>0$, (4.3) applied with $\varepsilon k^{-1}$ gives, for large $n$, $v_{n}\ge v-(\lambda_{0}\delta+\delta+\tfrac{\theta\delta^{2}}{2})\varepsilon k^{-1}\ge v-\varepsilon$, the coefficients being nonnegative by (0.5). Now let $c\in\mathbb{R}$ be such that for every $\varepsilon>0$ there is $N$ with $F^{-}_{\delta}(\xi_{n})\le c+\varepsilon$ for $n\ge N$. Fix $\varepsilon>0$ and choose $n$ so large that $F^{-}_{\delta}(\xi_{n})\le c+\tfrac{\varepsilon}{3}$, $|u_{n}-u|<\tfrac{\varepsilon}{3}$ and $v_{n}\ge v-\tfrac{\varepsilon}{3}$ (the largest of three thresholds, each obtained with the positive number $\tfrac{\varepsilon}{3}$ in place of $\varepsilon$). Then

$$F^{-}_{\delta}(\xi)=u+v<u_{n}+\tfrac{\varepsilon}{3}+v_{n}+\tfrac{\varepsilon}{3}=F^{-}_{\delta}(\xi_{n})+\tfrac{2\varepsilon}{3}\le c+\varepsilon .$$

As $\varepsilon>0$ is arbitrary, Comparison of Real Numbers with Arbitrary Positive Slack §slack-above gives $F^{-}_{\delta}(\xi)\le c$.

(4.5) The upper shift. By (0d), $F^{+}_{\delta}(\xi_{n})=u'_{n}-v'_{n}$ and $F^{+}_{\delta}(\xi)=u'-v'$, where

$$u'_{n}=\lambda_{0}r_{n}+\frac{\theta}{2}\lVert q_{n}\rVert_{\nu_{n}}^{2}+(1-\theta\delta)\langle\sigma_{n},q_{n}\rangle_{\nu_{n}}-g(\nu_{n}),\qquad v'_{n}=\lambda_{0}\delta\,\mathcal{E}(\nu_{n})+\Bigl(\delta-\frac{\theta\delta^{2}}{2}\Bigr)\lVert\sigma_{n}\rVert_{\nu_{n}}^{2},$$

and $u',v'$ are the same expressions at $\xi$. As in (4.4), $u'_{n}\to u'$, and, the coefficients $\lambda_{0}\delta$ and $\delta-\tfrac{\theta\delta^{2}}{2}$ being nonnegative by (0.5), for every $\varepsilon>0$ we have $v'_{n}\ge v'-\varepsilon$ for large $n$. Let $c\in\mathbb{R}$ be such that for every $\varepsilon>0$ there is $N$ with $c-\varepsilon\le F^{+}_{\delta}(\xi_{n})$ for $n\ge N$. Fix $\varepsilon>0$ and choose $n$ so large that $c-\tfrac{\varepsilon}{3}\le F^{+}_{\delta}(\xi_{n})$, $|u'_{n}-u'|<\tfrac{\varepsilon}{3}$ and $v'_{n}\ge v'-\tfrac{\varepsilon}{3}$ (each threshold obtained with $\tfrac{\varepsilon}{3}$ in place of $\varepsilon$). Then

$$c-\tfrac{\varepsilon}{3}\le u'_{n}-v'_{n}<u'+\tfrac{\varepsilon}{3}-v'+\tfrac{\varepsilon}{3}=F^{+}_{\delta}(\xi)+\tfrac{2\varepsilon}{3},$$

so $c-\varepsilon\le F^{+}_{\delta}(\xi)$; as $\varepsilon>0$ is arbitrary, Comparison of Real Numbers with Arbitrary Positive Slack §slack-below gives $c\le F^{+}_{\delta}(\xi)$.

By (4.4) and (4.5), $F$ is shift-semicontinuous at $(\delta,R)$; as $\delta,R$ were arbitrary, $F$ satisfies the shift-semicontinuity condition.

Step 5 (First-order structure at uniquely noise-mapped pairs, claim 4). Let $T=\{t\in\mathbb{R}:0\le t\}$. The pair $(\omega_{1},\omega_{2})$ below is chosen first; it depends only on $g,\lambda_{0},\theta,C$, and we show it is a structure pair for $F$ at $R$ for every positive $R$.

(5.1) The modulus $\omega_{1}$. For $s\in T$ let $G(s)=\{|g(\mu')-g(\nu')|:\mu',\nu'\in\mathcal{D},\ W_{a}(\mu',\nu')^{2}\le s\}$. It contains $0=|g(\mu_{0})-g(\mu_{0})|$, with $\mu_{0}$ from (0.4), since $W_{a}(\mu_{0},\mu_{0})=0$ by (0.1); and it is bounded above by $2M$. Hence $\omega_{1}(s)=\sup G(s)$ exists by The Real Numbers: Standing Notation and Background §bounds, and $0\le\omega_{1}(s)$ because $\omega_{1}(s)$ is an upper bound of $G(s)\ni0$ (Upper Bound and Least Upper Bound). Given $\varepsilon>0$, (Running cost) gives $\gamma>0$ with $|g(\mu')-g(\nu')|<\varepsilon$ whenever $\mu',\nu'\in\mathcal{D}$ and $W_{a}(\mu',\nu')<\gamma$. Put $\gamma_{1}=\gamma^{2}/4>0$. If $t\in T$ and $t\le\gamma_{1}$, every element of $G(t)$ comes from $\mu',\nu'\in\mathcal{D}$ with $W_{a}(\mu',\nu')^{2}\le(\gamma/2)^{2}$, so $W_{a}(\mu',\nu')\le\gamma/2<\gamma$ and the element is $<\varepsilon$; thus $\varepsilon$ is an upper bound of $G(t)$ and $\omega_{1}(t)\le\varepsilon$, the supremum being the least upper bound (Upper Bound and Least Upper Bound). So $\omega_{1}$ is a modulus of continuity, and by construction $|g(\mu')-g(\nu')|\le\omega_{1}(s)$ whenever $\mu',\nu'\in\mathcal{D}$, $s\in T$ and $W_{a}(\mu',\nu')^{2}\le s$.

(5.2) The function $\omega_{2}$. For $t\in T$ and real $\alpha>1$ put $\omega_{2}(t,\alpha)=(\lambda_{0}+4C\theta^{2}\alpha^{2})\,t$. For each $\alpha>1$ the coefficient is nonnegative by (0.4), so $t\mapsto\omega_{2}(t,\alpha)$ is a modulus of continuity by Linear Moduli of Continuity §modulus.

(5.3) The inequality. Let $R>0$, and let $\alpha,\delta,\mu,\nu,S,S',r$ be as in The First-Order Structure Condition at Uniquely Noise-Mapped Pairs §pair: $1<\alpha$, $0<\delta<1$, $\mu,\nu\in\mathcal{D}_{\Sigma}$ with both ordered pairs $(\mu,\nu)$ and $(\nu,\mu)$ uniquely noise-mapped, $S$ a noise-optimal map from $\mu$ to $\nu$ and $S'$ one from $\nu$ to $\mu$, and $-R\le r\le R$. (Neither the unique mapping nor the condition $\delta(|\mathcal{E}(\mu)|+|\mathcal{E}(\nu)|)\le R$ will be needed.) Write $W=W_{a}(\mu,\nu)$, $\sigma=\Sigma(\mu)\in L^{2}(\mu;X^{a})$, $\tau=\Sigma(\nu)\in L^{2}(\nu;X^{a})$, $p=\alpha(\mathrm{id}-S)=-\alpha(S-\mathrm{id})\in L^{2}(\mu;X^{a})$, $p'=\alpha(S'-\mathrm{id})\in L^{2}(\nu;X^{a})$, $e=|\mathcal{E}(\mu)|+|\mathcal{E}(\nu)|$ and $t=\delta(e+1)$, and let $\Delta=F^{-}_{\delta}(\mu,r,p)-F^{+}_{\delta}(\nu,r,p')$ be the difference to be bounded below.

Norms of the displacements. By (0.3), $\lVert S-\mathrm{id}\rVert_{\mu}=W$ and $\lVert S'-\mathrm{id}\rVert_{\nu}=W_{a}(\nu,\mu)=W$ (symmetry, (0.1)); hence, by homogeneity (0.1) with $|\alpha|=\alpha$ and $|-\alpha|=\alpha$, $\lVert p\rVert_{\mu}=\lVert p'\rVert_{\nu}=\alpha W$.

A bound on $W^{2}$. The measures $\mu,\nu,\rho$ lie in $\mathcal{P}^{a}_{\rho}$, so by the triangle inequality and symmetry (0.1), $0\le W\le W_{a}(\mu,\rho)+W_{a}(\nu,\rho)$. Hence $W^{2}\le(W_{a}(\mu,\rho)+W_{a}(\nu,\rho))^{2}\le2W_{a}(\mu,\rho)^{2}+2W_{a}(\nu,\rho)^{2}$, the last step by (0.5) with $m=W_{a}(\mu,\rho)$, $s=W_{a}(\nu,\rho)$. As $\mu,\nu\in\mathcal{D}$, (Growth) yields $W^{2}\le2C(1+|\mathcal{E}(\mu)|)+2C(1+|\mathcal{E}(\nu)|)=2C(2+e)$. Since $0\le C$ (0.4) and $2+e\le2(1+e)$, we get $\delta W^{2}\le2C\delta(2+e)\le4C\delta(1+e)=4Ct$.

Expansion of $\Delta$. Subtracting (0c) at $(\nu,r,p')$ from (0a) at $(\mu,r,p)$, the terms $\lambda_{0}r$ cancel and

$$\Delta=\lambda_{0}\delta\bigl(\mathcal{E}(\mu)+\mathcal{E}(\nu)\bigr)+\frac{\theta}{2}\Bigl(\lVert p+\delta\sigma\rVert_{\mu}^{2}-\lVert p'-\delta\tau\rVert_{\nu}^{2}\Bigr)+\bigl(\langle\sigma,p\rangle_{\mu}-\langle\tau,p'\rangle_{\nu}\bigr)+\delta\lVert\sigma\rVert_{\mu}^{2}+\delta\lVert\tau\rVert_{\nu}^{2}+g(\nu)-g(\mu).$$

By (0.1), $\lVert p+\delta\sigma\rVert_{\mu}^{2}=\lVert p\rVert_{\mu}^{2}+2\delta\langle p,\sigma\rangle_{\mu}+\delta^{2}\lVert\sigma\rVert_{\mu}^{2}$ and $\lVert p'-\delta\tau\rVert_{\nu}^{2}=\lVert p'\rVert_{\nu}^{2}-2\delta\langle p',\tau\rangle_{\nu}+\delta^{2}\lVert\tau\rVert_{\nu}^{2}$; as $\lVert p\rVert_{\mu}^{2}=\lVert p'\rVert_{\nu}^{2}=\alpha^{2}W^{2}$,

$$\frac{\theta}{2}\Bigl(\lVert p+\delta\sigma\rVert_{\mu}^{2}-\lVert p'-\delta\tau\rVert_{\nu}^{2}\Bigr)=\theta\delta\langle p,\sigma\rangle_{\mu}+\theta\delta\langle p',\tau\rangle_{\nu}+\frac{\theta\delta^{2}}{2}\lVert\sigma\rVert_{\mu}^{2}-\frac{\theta\delta^{2}}{2}\lVert\tau\rVert_{\nu}^{2}.$$

Bounds for the individual terms. (i) Monotonicity of the score. The pair is displacement convex by (Convexity), that is, $0$-displacement convex in the sense of Lambda-Displacement Convexity of a Noise Penalty Pair §convex. By (0.3) the couplings $\pi_{S}=(\mathrm{id},S)_{\#}\mu\in\Pi^{a}(\mu,\nu)$ and $\pi_{S'}=(\mathrm{id},S')_{\#}\nu\in\Pi^{a}(\nu,\mu)$ are noise-optimal, with $\mathcal{J}^{a}(\sigma,\pi_{S})=\langle\sigma,S-\mathrm{id}\rangle_{\mu}$ and $\mathcal{J}^{a}(\tau,\pi_{S'})=\langle\tau,S'-\mathrm{id}\rangle_{\nu}$. Since $\mu,\nu\in\mathcal{D}_{\Sigma}\subseteq\mathcal{D}$, the convexity inequality of Lambda-Displacement Convexity of a Noise Penalty Pair §convex with $\lambda=0$ applies to $\mu\in\mathcal{D}_{\Sigma}$, $\nu\in\mathcal{D}$ and $\pi_{S}$, and to $\nu\in\mathcal{D}_{\Sigma}$, $\mu\in\mathcal{D}$ and $\pi_{S'}$; the cost terms carry the factor $\tfrac{0}{2}=0$, so

$$\mathcal{E}(\mu)+\langle\sigma,S-\mathrm{id}\rangle_{\mu}\le\mathcal{E}(\nu),\qquad\mathcal{E}(\nu)+\langle\tau,S'-\mathrm{id}\rangle_{\nu}\le\mathcal{E}(\mu).$$

Adding, the real numbers $\mathcal{E}(\mu)$ and $\mathcal{E}(\nu)$ cancel and $\langle\sigma,S-\mathrm{id}\rangle_{\mu}+\langle\tau,S'-\mathrm{id}\rangle_{\nu}\le0$. By bilinearity (0.1), $\langle\sigma,p\rangle_{\mu}-\langle\tau,p'\rangle_{\nu}=-\alpha\bigl(\langle\sigma,S-\mathrm{id}\rangle_{\mu}+\langle\tau,S'-\mathrm{id}\rangle_{\nu}\bigr)\ge0$. (ii) Cross terms: by Cauchy-Schwarz (0.1) and (0.5) with $m=\lVert\sigma\rVert_{\mu}$, $s=\theta\alpha W$,

$$\theta\delta\langle p,\sigma\rangle_{\mu}\ge-\theta\delta\,\alpha W\lVert\sigma\rVert_{\mu}=-\frac{\delta}{2}\,2ms\ge-\frac{\delta}{2}\lVert\sigma\rVert_{\mu}^{2}-\frac{\delta}{2}\theta^{2}\alpha^{2}W^{2},$$

and in the same way $\theta\delta\langle p',\tau\rangle_{\nu}\ge-\tfrac{\delta}{2}\lVert\tau\rVert_{\nu}^{2}-\tfrac{\delta}{2}\theta^{2}\alpha^{2}W^{2}$. (iii) Collecting the score terms: the coefficient of $\lVert\sigma\rVert_{\mu}^{2}$ becomes $\delta+\tfrac{\theta\delta^{2}}{2}-\tfrac{\delta}{2}=\tfrac{\delta}{2}+\tfrac{\theta\delta^{2}}{2}$ and that of $\lVert\tau\rVert_{\nu}^{2}$ becomes $\delta-\tfrac{\theta\delta^{2}}{2}-\tfrac{\delta}{2}=\tfrac{\delta}{2}(1-\theta\delta)$, both nonnegative by (0.5) (the second one is where $\theta\le1$ enters), so these terms are $\ge0$; the remaining contribution is $-\theta^{2}\alpha^{2}\delta W^{2}\ge-4C\theta^{2}\alpha^{2}t$. (iv) Penalty terms: $\lambda_{0}\delta(\mathcal{E}(\mu)+\mathcal{E}(\nu))\ge-\lambda_{0}\delta e\ge-\lambda_{0}t$. (v) Running cost: $W^{2}\le\alpha W^{2}\le\alpha W^{2}+\alpha^{-1}$, as $1<\alpha$, $0\le W^{2}$ and $0<\alpha^{-1}$; so (5.1) with $s=\alpha W^{2}+\alpha^{-1}$, applied to $\mu,\nu\in\mathcal{D}$, gives $g(\nu)-g(\mu)\ge-|g(\mu)-g(\nu)|\ge-\omega_{1}(\alpha W^{2}+\alpha^{-1})$.

Adding (i)-(v) to the expansion of $\Delta$,

$$-\omega_{1}\bigl(\alpha W_{a}(\mu,\nu)^{2}+\alpha^{-1}\bigr)-\omega_{2}\bigl(\delta(|\mathcal{E}(\mu)|+|\mathcal{E}(\nu)|+1),\alpha\bigr)\le F^{-}_{\delta}\bigl(\mu,r,\alpha(\mathrm{id}-S)\bigr)-F^{+}_{\delta}\bigl(\nu,r,\alpha(S'-\mathrm{id})\bigr).$$

Hence $(\omega_{1},\omega_{2})$ is a structure pair for $F$ at every $R>0$, and $F$ satisfies the first-order structure condition at uniquely noise-mapped pairs.

Step 6 (Momentum-continuous shifts, claim 5). A Lipschitz bound. Let $\delta,R\in\mathbb{R}$ with $0<\delta<1$ and $0<R$, let $\nu\in\mathcal{D}_{\Sigma}$ with $\lVert\Sigma(\nu)\rVert_{\nu}\le R$, let $r\in\mathbb{R}$, and let $q,q'\in L^{2}(\nu;X^{a})$ with $\lVert q\rVert_{\nu}\le R$ and $\lVert q'\rVert_{\nu}\le R$; write $\sigma=\Sigma(\nu)$. For $w,w'\in L^{2}(\nu;X^{a})$, bilinearity and symmetry (0.1) give $\langle w-w',w+w'\rangle_{\nu}=\lVert w\rVert_{\nu}^{2}-\lVert w'\rVert_{\nu}^{2}$. With $w=q+\delta\sigma$ and $w'=q'+\delta\sigma$, so that $w-w'=q-q'$ and $w+w'=q+q'+2\delta\sigma$, the formula (0a) shows that the terms $\lambda_{0}r+\lambda_{0}\delta\mathcal{E}(\nu)$, $\delta\lVert\sigma\rVert_{\nu}^{2}$ and $-g(\nu)$ cancel in the difference, and

$$F^{-}_{\delta}(\nu,r,q)-F^{-}_{\delta}(\nu,r,q')=\frac{\theta}{2}\bigl(\lVert w\rVert_{\nu}^{2}-\lVert w'\rVert_{\nu}^{2}\bigr)+\langle\sigma,q-q'\rangle_{\nu}=\frac{\theta}{2}\,\langle q-q',\,q+q'+2\delta\sigma\rangle_{\nu}+\langle\sigma,q-q'\rangle_{\nu}.$$

By the triangle inequality and homogeneity (0.1), $\lVert q+q'+2\delta\sigma\rVert_{\nu}\le2R+2\delta R=2(1+\delta)R$. Taking absolute values, with the triangle inequality for the absolute value and Cauchy-Schwarz (0.1) for each inner product, and using $\theta>0$ and $\lVert\sigma\rVert_{\nu}\le R$,

$$\bigl|F^{-}_{\delta}(\nu,r,q)-F^{-}_{\delta}(\nu,r,q')\bigr|\le\bigl(\theta(1+\delta)+1\bigr)R\,\lVert q-q'\rVert_{\nu}\le(2\theta+1)R\,\lVert q-q'\rVert_{\nu},$$

since $\delta<1$. In the same way, from (0c) with $w=q-\delta\sigma$ and $w'=q'-\delta\sigma$, $F^{+}_{\delta}(\nu,r,q)-F^{+}_{\delta}(\nu,r,q')=\tfrac{\theta}{2}\langle q-q',q+q'-2\delta\sigma\rangle_{\nu}+\langle\sigma,q-q'\rangle_{\nu}$, $\lVert q+q'-2\delta\sigma\rVert_{\nu}\le2(1+\delta)R$, and $|F^{+}_{\delta}(\nu,r,q)-F^{+}_{\delta}(\nu,r,q')|\le(2\theta+1)R\,\lVert q-q'\rVert_{\nu}$.

The condition. Let $\delta,R,\eta\in\mathbb{R}$ with $0<\delta<1$, $0<R$ and $0<\eta$ be given; then $(2\theta+1)R$ is positive, and we choose $\kappa=\eta\,\bigl((2\theta+1)R\bigr)^{-1}>0$. For $\nu\in\mathcal{D}_{\Sigma}$ with $\lVert\Sigma(\nu)\rVert_{\nu}\le R$, $r\in\mathbb{R}$ with $|r|\le R$, and $q,q'\in L^{2}(\nu;X^{a})$ with $\lVert q\rVert_{\nu}\le R$, $\lVert q'\rVert_{\nu}\le R$ and $\lVert q-q'\rVert_{\nu}<\kappa$, the Lipschitz bound gives $|F^{\mp}_{\delta}(\nu,r,q)-F^{\mp}_{\delta}(\nu,r,q')|\le(2\theta+1)R\lVert q-q'\rVert_{\nu}<(2\theta+1)R\,\kappa=\eta$ for both shifts. Hence $F$ has momentum-continuous shifts relative to the pair. None of the four assumptions, nor $\theta\le1$, is used here.

Step 7 (Claim 6). Steps 1, 2, 4, 5 and 6 prove claims 1 to 5, and claim 6 collects them. $\blacksquare$