Proof of Cross-Level Comparison at Tensor Powers: a Lifted N-Particle Subsolution Lies below N Times a Mean-Field Supersolution with the Tensor-Averaged Cost

Each result cited is universally quantified over the data in its own statement. It is applied to the data named at the point of use, read at the configuration level whenever the dimension named is $dN$. Throughout, $W=W_{2}$; $|s|$ is the absolute value of $s\in\mathbb{R}$; $\tfrac{s}{2}$ is the product of $s\in\mathbb{R}$ with the multiplicative inverse of $2$ (claim 8 of Elementary Order Arithmetic in an Ordered Field), and $s^{-1}$ is the multiplicative inverse of a positive $s$ (claim 7 of that lemma); $I=\{\delta\in\mathbb{R}:0<\delta<1\}$. The number $N$ of particles is read in $\mathbb{R}$, where $1\le N$ and $0<N$ by claims 2 and 3 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field; so $|N|=N$ by Absolute Value in an Ordered Field, and $|Ns|=N|s|$ for $s\in\mathbb{R}$ by claim 4 of Properties of the Absolute Value in an Ordered Field. The letter $\sigma$ is the noise intensity, $\theta$ the control cost and $b$ the bound of $c$; tolerances are written $\vartheta$, and the measure written $\sigma^{*}$ in Intrinsic Test Functions at a Maximiser of the Doubled Difference Linked through Tensor Powers is written $\nu^{*}$ here.

Step 1 (Comparison hypotheses for a Langevin operator in dimension $m$). Let $m$ be one of $d$ and $dN$, let $V'$ be a confining potential on $\mathbb{R}^{m}$, let $\Gamma'\in\mathcal{M}_{p\times m}(\mathbb{R})$, and let $g':\mathcal{P}_{2}(\mathbb{R}^{m})\to\mathbb{R}$ be bounded and uniformly continuous for $W$ and the metric of The Absolute Value Metric on the Real Line. Let $(\mathcal{D}',\mathcal{D}'_{\Sigma},\mathcal{E}',\Sigma')$ be the Langevin free-energy pair with potential $V'$ and noise intensity $\sigma$, with translation Hessian $H_{\mathcal{E}'}$, and let $F'$ be the Langevin Hamilton-Jacobi operator with common noise with potential $V'$, noise intensity $\sigma$, discount $\lambda_{0}$, common-noise matrix $\Gamma'$, control cost $\theta$ and running cost $g'$, everything in dimension $m$ (at the configuration level when $m=dN$). We record four properties.

(L1) By The Langevin Free-Energy Pair is a Wasserstein-Coercive Penalty Pair: Growth Bounds, Continuity of the Translation Hessian, and the First Variation of the Penalty §pair and The Langevin Free-Energy Pair is a Wasserstein-Coercive Penalty Pair: Growth Bounds, Continuity of the Translation Hessian, and the First Variation of the Penalty §coercive, the pair is a Wasserstein-coercive penalty pair and $\mathcal{D}'$ has the map property; by Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §pair and Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §nonempty, $\mathcal{D}'_{\Sigma}\subseteq\mathcal{D}'$ and $\mathcal{D}'_{\Sigma}$ is nonempty.

(L2) By The Langevin Free-Energy Pair is Displacement Convex, with Closed Score Along Couplings and Regular Penalised Maxima §closed, the pair has closed score along couplings.

(L3) By The Hamilton-Jacobi Equation with Common Noise for Controlled Langevin Dynamics in a Confining Potential on the Wasserstein Space §operator, $F'$ is the Hamilton-Jacobi operator with common noise and penalty drift of the pair with discount $\lambda_{0}$, common-noise matrix $\Gamma'$, control cost $\theta$ and running cost $g'$, a second-order equation operator over $\mathcal{D}'_{\Sigma}$, whose $\delta$-shifts relative to the pair are those of The Bundle of Vector Fields over a Set of Measures, Second-Order Equation Operators on the Wasserstein Space, and Their Delta-Shifts §shifted. By The Hamilton-Jacobi Equation with Common Noise for Controlled Langevin Dynamics in a Confining Potential on the Wasserstein Space §equation and The Discounted Hamilton-Jacobi Equation with Common Noise and a Penalty Drift on the Wasserstein Space §equation, a viscosity subsolution, supersolution or solution of the Langevin Hamilton-Jacobi equation with common noise for these data is a function $\mathcal{D}'\to\mathbb{R}$ that is a viscosity subsolution, supersolution or solution of $F'$ relative to the pair, in the sense of Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space §subsolution, Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space §supersolution and Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space §solution.

(L4) $F'$ is locally strictly proper and satisfies the shift-coercivity condition, the shift-semicontinuity condition and the second-order structure condition at uniquely mapped pairs. This is The Hamilton-Jacobi Operator with Common Noise and Penalty Drift Satisfies the Hypotheses of the Comparison Principle for a Displacement Convex Pair §conclusion, applied to the pair (a penalty pair by (L1)), with $\lambda_{0}$, with $\theta$ (which satisfies $0<\theta\le1$), with $p$, $\Gamma'$, $g'$ and $F'$. That lemma names the shift-semicontinuity condition of The Shift-Semicontinuity Condition for an Equation Operator on the Wasserstein Space §semicontinuity; the clauses The Shift-Semicontinuity Condition for an Equation Operator on the Wasserstein Space §converging, The Shift-Semicontinuity Condition for an Equation Operator on the Wasserstein Space §level and The Shift-Semicontinuity Condition for an Equation Operator on the Wasserstein Space §semicontinuity state, word for word, the same three clauses for the same data, so the condition obtained is also the one of The Shift-Semicontinuity Condition for an Equation Operator on the Wasserstein Space §semicontinuity. We discharge the hypotheses of the lemma one by one. (Convexity) is The Langevin Free-Energy Pair is Displacement Convex, with Closed Score Along Couplings and Regular Penalised Maxima §convex. (Semicontinuity) is the first assertion of The Langevin Free-Energy Pair is a Wasserstein-Coercive Penalty Pair: Growth Bounds, Continuity of the Translation Hessian, and the First Variation of the Penalty §growth. (Running cost) is the assumption on $g'$.

The trace as a finite sum of entries. For $k\in[p]$ let $\gamma_{k}\in\mathbb{R}^{m}$ be the $k$th row of $\Gamma'$, the point whose $i$th coordinate is $\gamma_{k,i}=\Gamma'_{ki}$ for $i\in[m]$. Let $\mu\in\mathcal{D}'$; then $H_{\mathcal{E}'}(\mu)\in\mathcal{S}(m)$ by Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §hessian. Apply The Trace as a Sum of Quadratic Forms, its Monotonicity and a Norm Bound §rows with its matrix $A$ taken to be $\Gamma'$ and $X=H_{\mathcal{E}'}(\mu)$; the letters $m$ and $p$ of that lemma are its own dimensions, read here as our $p$ and our $m$ respectively, so that its rows $a_{k}$ are our $\gamma_{k}$ (its hypotheses $1\le p$ and $1\le m$ hold because the matrix sets of Real Matrices, Symmetric Matrices and the Semidefinite Ordering: Standing Notation §matrices, to which $\Gamma'$ belongs, are formed only for dimensions at least $1$). Together with claim 4 of Linearity of the Matrix-Vector Product and the Quadratic Form as a Double Sum, applied with $n=m$, $M=H_{\mathcal{E}'}(\mu)$ and $w=z=\gamma_{k}$ for each $k\in[p]$, and the commutativity and associativity of multiplication in $\mathbb{R}$, it gives

$$\mathrm{tr}\bigl(\Gamma'^{\top}\Gamma'H_{\mathcal{E}'}(\mu)\bigr)=\sum_{k=1}^{p}\gamma_{k}\cdot\bigl(H_{\mathcal{E}'}(\mu)\gamma_{k}\bigr)=\sum_{k=1}^{p}\ \sum_{i=1}^{m}\ \sum_{l=1}^{m}\gamma_{k,i}\gamma_{k,l}\,H_{\mathcal{E}'}(\mu)_{il}.\tag{T}$$

Put $s_{\Gamma'}=\sum_{k=1}^{p}\sum_{i=1}^{m}\sum_{l=1}^{m}|\gamma_{k,i}|\,|\gamma_{k,l}|$, a real number that does not depend on $\mu$ and is nonnegative by claim 5 of Properties of Finite Sums (applied to the innermost sums first), each summand being a product of nonnegative numbers.

(Growth). Let $C$ be the constant of The Langevin Free-Energy Pair is a Wasserstein-Coercive Penalty Pair: Growth Bounds, Continuity of the Translation Hessian, and the First Variation of the Penalty §growth, so that $M_{2}(\mu)\le C(1+|\mathcal{E}'(\mu)|)$ and $|\mathrm{tr}\,H_{\mathcal{E}'}(\mu)|\le C(1+|\mathcal{E}'(\mu)|)$ for every $\mu\in\mathcal{D}'$. Let $\mu\in\mathcal{D}'$ and write $t_{\mu}=C(1+|\mathcal{E}'(\mu)|)$. For $i,l\in[m]$, The Langevin Free-Energy Pair is a Wasserstein-Coercive Penalty Pair: Growth Bounds, Continuity of the Translation Hessian, and the First Variation of the Penalty §entries gives $|H_{\mathcal{E}'}(\mu)_{il}|\le\mathrm{tr}\,H_{\mathcal{E}'}(\mu)\le|\mathrm{tr}\,H_{\mathcal{E}'}(\mu)|\le t_{\mu}$ (claim 3 of Properties of the Absolute Value in an Ordered Field). By claim 4 of Properties of the Absolute Value in an Ordered Field, used twice, and claim 5 of Elementary Arithmetic in an Ordered Field with the nonnegative multiplier $|\gamma_{k,i}|\,|\gamma_{k,l}|$, every summand of (T) satisfies

$$\bigl|\gamma_{k,i}\gamma_{k,l}\,H_{\mathcal{E}'}(\mu)_{il}\bigr|=|\gamma_{k,i}|\,|\gamma_{k,l}|\,|H_{\mathcal{E}'}(\mu)_{il}|\le t_{\mu}\,|\gamma_{k,i}|\,|\gamma_{k,l}| .$$

Claim 2 of Comparison and Absolute Value Bounds for Finite Sums of Real Numbers bounds the absolute value of each of the three nested sums in (T) by the sum of the absolute values of its summands, claim 1 of that lemma carries these bounds through the enclosing sums, and claim 3 of Properties of Finite Sums (applied to each of the three sums) takes out the factor $t_{\mu}$; so

$$\bigl|\mathrm{tr}\bigl(\Gamma'^{\top}\Gamma'H_{\mathcal{E}'}(\mu)\bigr)\bigr|\le\sum_{k=1}^{p}\sum_{i=1}^{m}\sum_{l=1}^{m}t_{\mu}\,|\gamma_{k,i}|\,|\gamma_{k,l}|=s_{\Gamma'}\,C\bigl(1+|\mathcal{E}'(\mu)|\bigr).$$

Put $C'=|C|(1+s_{\Gamma'})$. Since $0\le1+|\mathcal{E}'(\mu)|$, $C\le|C|$, $s_{\Gamma'}C\le s_{\Gamma'}|C|$, $|C|\le C'$ and $s_{\Gamma'}|C|\le C'$ (as $0\le s_{\Gamma'}$ and $0\le|C|$; claims 2, 3 and 5 of Elementary Arithmetic in an Ordered Field), we obtain $M_{2}(\mu)\le C'(1+|\mathcal{E}'(\mu)|)$ and $|\mathrm{tr}(\Gamma'^{\top}\Gamma'H_{\mathcal{E}'}(\mu))|\le C'(1+|\mathcal{E}'(\mu)|)$ for every $\mu\in\mathcal{D}'$, which is (Growth) with the constant $C'$.

(Hessian continuity). Let $R'$ be positive and $S_{R'}=\{\mu\in\mathcal{D}':|\mathcal{E}'(\mu)|\le R'\}$. For $i,l\in[m]$ the restriction of $\mu\mapsto H_{\mathcal{E}'}(\mu)_{il}$ to $S_{R'}$ is continuous by The Langevin Free-Energy Pair is a Wasserstein-Coercive Penalty Pair: Growth Bounds, Continuity of the Translation Hessian, and the First Variation of the Penalty §entries, so for $k\in[p]$ the restriction of $\mu\mapsto\gamma_{k,i}\gamma_{k,l}\,H_{\mathcal{E}'}(\mu)_{il}$ to $S_{R'}$ is continuous by claim 5 of Continuity of Sums and Products of Real-Valued Functions on a Metric Space (the multiple $cf$ with $c=\gamma_{k,i}\gamma_{k,l}$), taken in the metric space $(\mathcal{P}_{2}(\mathbb{R}^{m}),W)$ with the subset $S_{R'}$. A finite sum of functions continuous on $S_{R'}$ is continuous on $S_{R'}$, by induction on the number of summands along the recursion of claim 1 of Properties of Finite Sums, each step being claim 5 of Continuity of Sums and Products of Real-Valued Functions on a Metric Space for $f+g$. Applied to the three nested sums of (T), this shows that the restriction of $\mu\mapsto\mathrm{tr}(\Gamma'^{\top}\Gamma'H_{\mathcal{E}'}(\mu))$ to $S_{R'}$ is continuous, which is (Hessian continuity).

Step 2 (The two levels). Particle level. By The Tensor-Averaged Cost of a Bounded Uniformly Continuous Cost is Bounded and Uniformly Continuous §bound, $|\tilde{c}(\mu)|\le\tfrac{b}{N}\le|\tfrac{b}{N}|$ for every $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ (claim 3 of Properties of the Absolute Value in an Ordered Field), and $0\le|\tfrac{b}{N}|$ (claim 1 of that lemma), so $\tilde{c}$ is bounded with bound $|\tfrac{b}{N}|$; it is uniformly continuous by The Tensor-Averaged Cost of a Bounded Uniformly Continuous Cost is Bounded and Uniformly Continuous §uniform. So Step 1 applies with $m=d$, $V'=V$, $\Gamma'=\Gamma$ and $g'=\tilde{c}$: its pair is the Langevin free-energy pair $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ with potential $V$ and noise intensity $\sigma$ named in the mean-field equation, and its operator is the operator $F$ of that equation. We write $F^{-}_{\delta}$, $F^{+}_{\delta}$ for the $\delta$-shifts of $F$ relative to this pair. By (L3), $v$ is a viscosity supersolution of $F$ relative to $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$.

Configuration level. By The N-Particle Potential of a Confining Potential is a Confining Potential on the Configuration Space §confining, the $N$-particle potential $V_{N}$ is a confining potential on $\mathbb{R}^{dN}$, and by The N-Particle Hamilton-Jacobi Equation with Individual and Common Noise on the Configuration Space §common-noise, $\Gamma_{N}\in\mathcal{M}_{p\times dN}(\mathbb{R})$. Since $0\le|c(x)|\le b$ for any $x\in\mathbb{R}^{dN}$ (claim 1 of Properties of the Absolute Value in an Ordered Field), $0\le b$, so $c$ is bounded with bound $b$, and it is uniformly continuous by hypothesis. By Running Costs for N-Particle Systems: Integrals of Bounded Uniformly Continuous Functions and Costs of the Empirical Measure §integral, applied with $m=dN$, this $c$ and this $b$, $c$ is Borel and integrable with respect to every $P\in\mathcal{P}_{2}(\mathbb{R}^{dN})$, and the function $\bar{c}:\mathcal{P}_{2}(\mathbb{R}^{dN})\to\mathbb{R}$, $\bar{c}(P)=\int_{\mathbb{R}^{dN}}c\,dP$, is uniformly continuous with $|\bar{c}(P)|\le b$ for every $P$, hence bounded with bound $b$. By The Lifted N-Particle Hamilton-Jacobi Equation on the Wasserstein Space of the Configuration Space §operator, the lifted operator $F_{N}$ is the Langevin Hamilton-Jacobi operator with common noise, at the configuration level, with potential $V_{N}$, noise intensity $\sigma$, discount $\lambda_{0}$, common-noise matrix $\Gamma_{N}$, control cost $\theta$ and running cost $\bar{c}$, over the score domain of the Langevin free-energy pair $(\mathcal{D}_{N},\mathcal{D}_{N,\Sigma},\mathcal{E}_{N},\Sigma_{N})$ with potential $V_{N}$ and noise intensity $\sigma$ formed at the configuration level. So Step 1 applies with $m=dN$, $V'=V_{N}$, $\Gamma'=\Gamma_{N}$ and $g'=\bar{c}$. We write $F^{-}_{N,\delta}$, $F^{+}_{N,\delta}$ for the $\delta$-shifts of $F_{N}$ relative to this pair, and $H_{\mathcal{E}_{N}}$ for its translation Hessian. By The Lifted N-Particle Hamilton-Jacobi Equation on the Wasserstein Space of the Configuration Space §equation and (L3), $U$ is a viscosity subsolution of $F_{N}$ relative to $(\mathcal{D}_{N},\mathcal{D}_{N,\Sigma},\mathcal{E}_{N},\Sigma_{N})$.

Across the levels. The data $V$, $\lambda_{0}$, $\sigma$, $\theta$, $p$, $\Gamma$ and the bounded Borel cost $c$ are data of Cross-Level Inequalities between the Shifted Lifted N-Particle and Mean-Field Langevin Operators at Tensor Powers and through One-Particle Marginals with $g=\tilde{c}$, and of The Langevin Free-Energy Pair across the Particle and Configuration Levels: Tensor Powers, One-Particle Marginals and Diagonal Translation Hessians; the pairs and the operators $F_{N}$ and $F$ named there are the present ones. By The Langevin Free-Energy Pair across the Particle and Configuration Levels: Tensor Powers, One-Particle Marginals and Diagonal Translation Hessians §tensor,

$$\mu^{\otimes N}\in\mathcal{D}_{N}\ \text{and}\ \mathcal{E}_{N}(\mu^{\otimes N})=N\,\mathcal{E}(\mu)\ \text{for}\ \mu\in\mathcal{D};\qquad\mu^{\otimes N}\in\mathcal{D}_{N,\Sigma}\ \text{for}\ \mu\in\mathcal{D}_{\Sigma}.\qquad(2\mathrm{a})$$

Step 3 (Fixed constants). By hypothesis fix $b_{U},b_{v}\in\mathbb{R}$ with $U(P)\le b_{U}$ for every $P\in\mathcal{D}_{N}$ and $b_{v}\le v(\mu)$ for every $\mu\in\mathcal{D}$. By The Delta-Envelopes of Bounded Functions and Their Monotonicity in the Weight, for a Wasserstein-Coercive Penalty Pair §growth, applied to the two pairs (Wasserstein-coercive penalty pairs by (L1)), $U$ has penalty-subordinate growth from above relative to the configuration-level pair and $v$ penalty-subordinate growth from below relative to the particle-level pair; so for positive $\delta$ the $\delta$-envelopes $U^{-}_{\delta}:\mathcal{D}_{N}\to\mathbb{R}$ and $v^{+}_{\delta}:\mathcal{D}\to\mathbb{R}$ are defined. By Basic Properties of a Wasserstein-Coercive Penalty Pair §bounded-below for each pair and claim 9 of Elementary Order Arithmetic in an Ordered Field, fix $e_{0}\in\mathbb{R}$ with $e_{0}\le\mathcal{E}_{N}(P)$ for every $P\in\mathcal{D}_{N}$ and $e_{0}\le\mathcal{E}(\mu)$ for every $\mu\in\mathcal{D}$. Put

$$B'=|b_{U}|+N|b_{v}|+(1+N)|e_{0}|,\qquad B=B'+1,\qquad R=2B .$$

$B'$ is a sum of products of nonnegative numbers, so $0\le B'$ (claims 2 and 5 of Elementary Arithmetic in an Ordered Field); hence $0<B$ and $0<R$ (claims 1, 5 and 6 of Elementary Order Arithmetic in an Ordered Field). By (L4) for $F_{N}$, fix a properness constant $\lambda>0$ for $F_{N}$ at $R$ and a second-order structure pair $(\omega_{1},\omega_{2})$ for $F_{N}$ at $R$, both at the configuration level. None of $B'$, $B$, $R$, $\lambda$, $\omega_{1}$, $\omega_{2}$ depends on $\delta$ or $\alpha$ below.

Step 4 (The doubled difference and its maximum). For $P\in\mathcal{P}_{2}(\mathbb{R}^{dN})$ and $\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ we have $\mu^{\otimes N}\in\mathcal{P}_{2}(\mathbb{R}^{dN})$ by Tensor Powers and One-Particle Marginals: Particle Laws, Product Integrals, Push-Forwards, Moments, Product Maps and Diagonal Shifts §moments; put $L(P,\mu)=\tfrac{1}{2}W(P,\mu^{\otimes N})^{2}$, which is nonnegative since $0<\tfrac{1}{2}$ (claim 8 of Elementary Order Arithmetic in an Ordered Field) and by claim 5 of Elementary Arithmetic in an Ordered Field.

Continuity of $L$. Let $P_{j},P\in\mathcal{P}_{2}(\mathbb{R}^{dN})$ and $\mu_{j},\mu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ ($j\in\mathbb{N}$) be such that $(W(P_{j},P))_{j}$ and $(W(\mu_{j},\mu))_{j}$ converge to $0$. Put $a_{j}=W(P_{j},\mu_{j}^{\otimes N})$, $a=W(P,\mu^{\otimes N})$ and $w_{j}=W(\mu_{j}^{\otimes N},\mu^{\otimes N})$, all nonnegative (The Quadratic Wasserstein Distance is a Metric on the Wasserstein Space §metric). The triangle inequality and symmetry (The Quadratic Wasserstein Distance is a Metric on the Wasserstein Space §triangle, The Quadratic Wasserstein Distance is a Metric on the Wasserstein Space §symmetry) give $a_{j}\le W(P_{j},P)+a+w_{j}$ and $a\le W(P_{j},P)+a_{j}+w_{j}$, so $|a_{j}-a|\le W(P_{j},P)+w_{j}$ (claim 6 of Properties of the Absolute Value in an Ordered Field). By Tensor Powers Scale the Wasserstein Distance by the Square Root of N, and the One-Particle Marginal is Lipschitz with Constant N^{-1/2} §tensor, $w_{j}^{2}=N\,W(\mu_{j},\mu)^{2}$; since $N\le N\cdot N$ (claim 5 of Elementary Arithmetic in an Ordered Field with $1\le N$ and $0\le N$) and $0\le W(\mu_{j},\mu)^{2}$, $w_{j}^{2}\le\bigl(N\,W(\mu_{j},\mu)\bigr)^{2}$, so $w_{j}\le N\,W(\mu_{j},\mu)$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, both sides being nonnegative. Hence $|a_{j}-a|\le e_{j}$ with $e_{j}=W(P_{j},P)+N\,W(\mu_{j},\mu)$, and $(e_{j})_{j}$ converges to $0$ by claims 1 and 3 of Arithmetic of Limits of Real Sequences. Given positive $\varepsilon$, Limit of a Sequence of Real Numbers gives $J$ with $e_{j}=|e_{j}-0|<\varepsilon$ for $j\ge J$, so $|a_{j}-a|<\varepsilon$ for $j\ge J$ (claim 2 of Elementary Order Arithmetic in an Ordered Field); thus $(a_{j})_{j}$ converges to $a$, and $L(P_{j},\mu_{j})=\tfrac{1}{2}a_{j}a_{j}$ converges to $\tfrac{1}{2}a\,a=L(P,\mu)$ by claims 2 and 3 of Arithmetic of Limits of Real Sequences.

The maximum. Apply Existence, Penalty Bounds and Monotonicity of the Maximum of a Two-Space Doubled Difference with a Continuous Link with $n_{1}=dN$ and the configuration-level pair, $n_{2}=d$ and the particle-level pair (Wasserstein-coercive penalty pairs by (L1)), the present $e_{0}$, $u=U$, $v=v$, $b=b_{U}$, $b'=b_{v}$, $\kappa_{1}=1$, $\kappa_{2}=N$ and the link $L$ just shown to be nonnegative and continuous. For positive $\delta,\alpha$ its function $\Psi_{\delta,\alpha}:\mathcal{D}_{N}\times\mathcal{D}\to\mathbb{R}$ has the value

$$\Psi_{\delta,\alpha}(P,\mu)=U^{-}_{\delta}(P)-N\,v^{+}_{\delta}(\mu)-\tfrac{\alpha}{2}\,W(P,\mu^{\otimes N})^{2},$$

since $1\cdot x=x$ and $\alpha\cdot(\tfrac{1}{2}x)=\tfrac{\alpha}{2}x$; let $M(\delta,\alpha)$ be the supremum of its values. Its clauses read as follows, for all positive $\delta,\alpha$.

(4a) By Existence, Penalty Bounds and Monotonicity of the Maximum of a Two-Space Doubled Difference with a Continuous Link §maximiser, every value of $\Psi_{\delta,\alpha}$ is at most $b_{U}-Nb_{v}-(1+N)\delta e_{0}$, so $M(\delta,\alpha)$, the least upper bound, is a real number at most $b_{U}-Nb_{v}-(1+N)\delta e_{0}$; and $\Psi_{\delta,\alpha}$ has a maximising pair.

(4b) By Existence, Penalty Bounds and Monotonicity of the Maximum of a Two-Space Doubled Difference with a Continuous Link §penalty, if $0\le\Psi_{\delta,\alpha}(P,\mu)$ then $\delta|\mathcal{E}_{N}(P)|\le B_{\delta}$ and $N\delta|\mathcal{E}(\mu)|\le B_{\delta}$, where $B_{\delta}=|b_{U}|+N|b_{v}|+(1+N)\delta|e_{0}|$.

(4c) By Existence, Penalty Bounds and Monotonicity of the Maximum of a Two-Space Doubled Difference with a Continuous Link §weight, if $0<\delta'<\delta$ and $(\hat{P},\hat{\mu})$ is a maximising pair of $\Psi_{\delta,\alpha}$, then $M(\delta,\alpha)+(\delta-\delta')\bigl(\mathcal{E}_{N}(\hat{P})+N\,\mathcal{E}(\hat{\mu})\bigr)\le M(\delta',\alpha)$.

(4d) By Existence, Penalty Bounds and Monotonicity of the Maximum of a Two-Space Doubled Difference with a Continuous Link §strength, if $0<\alpha'<\alpha$ and $(\hat{P},\hat{\mu})$ is a maximising pair of $\Psi_{\delta,\alpha}$, then $M(\delta,\alpha)+(\alpha-\alpha')\,\tfrac{1}{2}W(\hat{P},\hat{\mu}^{\otimes N})^{2}\le M(\delta,\alpha')$.

Step 5 (The estimate at a maximiser). Let $\delta\in I$ and $\alpha\in\mathbb{R}$ with $1<\alpha$ satisfy $0\le M(\delta,\alpha)$; write $M=M(\delta,\alpha)$ and $\Psi=\Psi_{\delta,\alpha}$. We show that there is a maximising pair $(\rho^{*},\nu^{*})$ of $\Psi$ such that, with $Q^{*}=(\nu^{*})^{\otimes N}$,

$$\lambda M\le\omega_{1}\bigl(\alpha\,W(\rho^{*},Q^{*})^{2}+\alpha^{-1}\bigr)+\omega_{2}\bigl(\delta\,(|\mathcal{E}_{N}(\rho^{*})|+|\mathcal{E}_{N}(Q^{*})|+1),\ \alpha\bigr).\qquad(\mathrm{E})$$

(5a) Test data. By (4a) fix a maximising pair $(\hat{P},\hat{\mu})$ of $\Psi$; as $M$ is an upper bound of the values of $\Psi$, $\Psi(P,\mu)\le\Psi(\hat{P},\hat{\mu})$ for all $(P,\mu)\in\mathcal{D}_{N}\times\mathcal{D}$. Apply Intrinsic Test Functions at a Maximiser of the Doubled Difference Linked through Tensor Powers to the particle-level and configuration-level pairs (Wasserstein-coercive penalty pairs whose penalty domains have the map property, by (L1)), for which $\mu^{\otimes N}\in\mathcal{D}_{N}$ for $\mu\in\mathcal{D}$ by (2a), with $U$, $v$, $b_{U}$, $b_{v}$, $\delta$, $\alpha$ and $(\hat{P},\hat{\mu})$; its function $\Psi$ is ours by Step 4. It provides $\rho^{*}\in\mathcal{D}_{N}$, $\nu^{*}\in\mathcal{D}$, $\mathbb{X}\in\mathcal{S}(dN)$ and $\mathbb{Y}\in\mathcal{S}(d)$; with $Q^{*}=(\nu^{*})^{\otimes N}\in\mathcal{D}_{N}$, both ordered pairs $(\rho^{*},Q^{*})$ and $(Q^{*},\rho^{*})$ are uniquely mapped, and we let $S$ and $S'$ be the optimal maps named there. Put

$$V_{*}=\alpha(\mathrm{id}-S)\in L^{2}(\rho^{*};\mathbb{R}^{dN}),\quad V'_{*}=\alpha(S'-\mathrm{id})\in L^{2}(Q^{*};\mathbb{R}^{dN}),\quad s_{*}=U^{-}_{\delta}(\rho^{*}),\quad t_{*}=v^{+}_{\delta}(\nu^{*}),$$

scalar multiples in the vector spaces of The Intrinsic Calculus on the Wasserstein Space: Standing Notation of the classes named in the lemma. By Intrinsic Test Functions at a Maximiser of the Doubled Difference Linked through Tensor Powers §maximiser, $\Psi(\rho^{*},\nu^{*})=\Psi(\hat{P},\hat{\mu})=M$, so $(\rho^{*},\nu^{*})$ is a maximising pair of $\Psi$. By Intrinsic Test Functions at a Maximiser of the Doubled Difference Linked through Tensor Powers §admitted, fix $\mathbb{Y}_{N}\in\mathcal{S}(dN)$ such that $(\mathbb{X},\mathbb{Y}_{N})$ is admitted at $\alpha$ at the configuration level and $a^{\oplus}\cdot(\mathbb{Y}_{N}a^{\oplus})=N\,a\cdot(\mathbb{Y}a)$ for every $a\in\mathbb{R}^{d}$.

(5b) Closure at the configuration level. Apply Closure of Approximate Test Data for Viscosity Sub- and Supersolutions on the Wasserstein Space §subsolution at the configuration level to the configuration-level pair (Wasserstein-coercive with closed score along couplings, by (L1) and (L2)), to $F_{N}$ (a second-order equation operator over $\mathcal{D}_{N,\Sigma}$ satisfying the shift-coercivity and shift-semicontinuity conditions, by (L3) and (L4)), to $\delta\in I$, to $u=U$ (bounded above by $b_{U}$ and a viscosity subsolution of $F_{N}$, by Step 2), and to $\rho^{*}$, $V_{*}$ and $\mathbb{X}$. Its hypothesis for each positive $\varepsilon$ is Intrinsic Test Functions at a Maximiser of the Doubled Difference Linked through Tensor Powers §subsolution, the map $y\mapsto\alpha(y-S(y))$ being a representative of $V_{*}$. Hence $\rho^{*}\in\mathcal{D}_{N,\Sigma}$ and

$$F^{-}_{N,\delta}\bigl(\rho^{*},s_{*},V_{*},\mathbb{X}\bigr)\le0.\qquad(5\mathrm{b})$$

(5c) Closure at the particle level. Since $(Q^{*})^{[1]}=\nu^{*}$ (Tensor Powers and One-Particle Marginals: Particle Laws, Product Integrals, Push-Forwards, Moments, Product Maps and Diagonal Shifts §marginal-of-tensor), the one-particle projection $\Pi_{Q^{*}}$ takes values in $T_{\nu^{*}}\subseteq L^{2}(\nu^{*};\mathbb{R}^{d})$; put $W_{*}=\alpha\,\Pi_{Q^{*}}(S'-\mathrm{id})\in L^{2}(\nu^{*};\mathbb{R}^{d})$. Apply Closure of Approximate Test Data for Viscosity Sub- and Supersolutions on the Wasserstein Space §supersolution to the particle-level pair (Wasserstein-coercive with closed score along couplings, by (L1) and (L2)), to $F$ (by (L3) and (L4)), to $\delta\in I$, to $v$ (bounded below by $b_{v}$ and a viscosity supersolution of $F$, by Step 2), and to $\nu^{*}$, $W_{*}$ and $\mathbb{Y}$; its hypothesis for each positive $\varepsilon$ is Intrinsic Test Functions at a Maximiser of the Doubled Difference Linked through Tensor Powers §supersolution. Hence $\nu^{*}\in\mathcal{D}_{\Sigma}$ and $0\le F^{+}_{\delta}(\nu^{*},t_{*},W_{*},\mathbb{Y})$.

(5d) Across the levels. By (2a), $Q^{*}\in\mathcal{D}_{N,\Sigma}$. By The Projection onto One-Particle Tangent Fields: Pairing with Product Fields, Contraction, and Product Fields of Tangent Fields §contraction, $\Pi_{Q^{*}}$ is linear, so $\Pi_{Q^{*}}(V'_{*})=W_{*}$. Apply Cross-Level Inequalities between the Shifted Lifted N-Particle and Mean-Field Langevin Operators at Tensor Powers and through One-Particle Marginals §tensor with $\delta\in I$, the matrices $\mathbb{Y}_{N}$ and $\mathbb{Y}$ of (5a), $\mu=\nu^{*}\in\mathcal{D}_{\Sigma}$, $r=t_{*}$ and its field $G$ taken to be $V'_{*}\in L^{2}(Q^{*};\mathbb{R}^{dN})$:

$$N\,F^{+}_{\delta}\bigl(\nu^{*},t_{*},W_{*},\mathbb{Y}\bigr)\le F^{+}_{N,\delta}\bigl(Q^{*},Nt_{*},V'_{*},\mathbb{Y}_{N}\bigr).$$

As $0\le F^{+}_{\delta}(\nu^{*},t_{*},W_{*},\mathbb{Y})$ and $0\le N$, claim 5 of Elementary Arithmetic in an Ordered Field gives $0=N\cdot0\le N\,F^{+}_{\delta}(\nu^{*},t_{*},W_{*},\mathbb{Y})$, whence

$$0\le F^{+}_{N,\delta}\bigl(Q^{*},Nt_{*},V'_{*},\mathbb{Y}_{N}\bigr).\qquad(5\mathrm{d})$$

(5e) Bounds. Since $W(\rho^{*},Q^{*})^{2}\ge0$ and $0<\tfrac{\alpha}{2}$, and $s_{*}-Nt_{*}-\tfrac{\alpha}{2}W(\rho^{*},Q^{*})^{2}=\Psi(\rho^{*},\nu^{*})=M\ge0$, we get $M\le s_{*}-Nt_{*}$ and $0\le s_{*}-Nt_{*}$, that is $Nt_{*}\le s_{*}$ (claim 3 of Elementary Arithmetic in an Ordered Field). As $0\le M=\Psi(\rho^{*},\nu^{*})$, (4b) gives $\delta|\mathcal{E}_{N}(\rho^{*})|\le B_{\delta}$ and $N\delta|\mathcal{E}(\nu^{*})|\le B_{\delta}$; since $\delta<1$ and $0\le|e_{0}|$, $\delta|e_{0}|\le|e_{0}|$ (claim 5 of Elementary Arithmetic in an Ordered Field), so $B_{\delta}\le B'\le B$. By (2a), $\mathcal{E}_{N}(Q^{*})=N\,\mathcal{E}(\nu^{*})$, so $\delta|\mathcal{E}_{N}(Q^{*})|=N\delta|\mathcal{E}(\nu^{*})|\le B$. Therefore

$$\delta\bigl(|\mathcal{E}_{N}(\rho^{*})|+|\mathcal{E}_{N}(Q^{*})|\bigr)\le2B=R.$$

By The Delta-Envelopes of Bounded Functions and Their Monotonicity in the Weight, for a Wasserstein-Coercive Penalty Pair §bounded for the configuration-level pair and the bound $b_{U}$, $s_{*}\le b_{U}-\delta\mathcal{E}_{N}(\rho^{*})$; as $e_{0}\le\mathcal{E}_{N}(\rho^{*})$ and $0<\delta$, $\delta e_{0}\le\delta\mathcal{E}_{N}(\rho^{*})$ (claim 5 of Elementary Arithmetic in an Ordered Field), so, using $b_{U}\le|b_{U}|$ and $-\delta e_{0}\le|\delta e_{0}|=\delta|e_{0}|\le|e_{0}|$ (claims 3 and 4 of Properties of the Absolute Value in an Ordered Field and claim 4 of Elementary Order Arithmetic in an Ordered Field), $s_{*}\le|b_{U}|+|e_{0}|$. Likewise, by the same clause for the particle-level pair and the bound $b_{v}$, $t_{*}\ge b_{v}+\delta\mathcal{E}(\nu^{*})\ge b_{v}+\delta e_{0}\ge-|b_{v}|-|e_{0}|$, and multiplying by $0<N$, $Nt_{*}\ge-N|b_{v}|-N|e_{0}|$. Since $0\le|b_{U}|$, $0\le N|b_{v}|$ and $0\le|e_{0}|\le N|e_{0}|$,

$$-B'\le-N|b_{v}|-N|e_{0}|\le Nt_{*}\le s_{*}\le|b_{U}|+|e_{0}|\le B',$$

so $|Nt_{*}|\le B'\le R$ and $|s_{*}|\le B'$ (claim 6 of Properties of the Absolute Value in an Ordered Field); in particular $-R\le Nt_{*}\le R$. As $|\delta\mathcal{E}_{N}(\rho^{*})|=\delta|\mathcal{E}_{N}(\rho^{*})|\le B$, the triangle inequality (claim 5 of that lemma) gives $|s_{*}+\delta\mathcal{E}_{N}(\rho^{*})|\le B'+B\le R$ and $|Nt_{*}+\delta\mathcal{E}_{N}(\rho^{*})|\le R$, and $Nt_{*}+\delta\mathcal{E}_{N}(\rho^{*})\le s_{*}+\delta\mathcal{E}_{N}(\rho^{*})$.

(5f) Structure pair and properness. The measures $\rho^{*}$ and $Q^{*}$ lie in $\mathcal{D}_{N,\Sigma}$ by (5b) and (5d); both ordered pairs $(\rho^{*},Q^{*})$ and $(Q^{*},\rho^{*})$ are uniquely mapped, with optimal maps $S$ and $S'$ (5a); $1<\alpha$ and $\delta\in I$; $\delta(|\mathcal{E}_{N}(\rho^{*})|+|\mathcal{E}_{N}(Q^{*})|)\le R$ and $-R\le Nt_{*}\le R$ by (5e); and $(\mathbb{X},\mathbb{Y}_{N})$ is admitted at $\alpha$ (5a). So The Second-Order Structure Condition at Uniquely Mapped Pairs on the Wasserstein Space §pair, at the configuration level, for the pair $(\omega_{1},\omega_{2})$ at $R$ with $\mu=\rho^{*}$, $\nu=Q^{*}$ and the value slot $r=Nt_{*}$, gives

$$-\omega_{1}\bigl(\alpha W(\rho^{*},Q^{*})^{2}+\alpha^{-1}\bigr)-\omega_{2}\bigl(\delta(|\mathcal{E}_{N}(\rho^{*})|+|\mathcal{E}_{N}(Q^{*})|+1),\alpha\bigr)\le F^{-}_{N,\delta}(\rho^{*},Nt_{*},V_{*},\mathbb{X})-F^{+}_{N,\delta}(Q^{*},Nt_{*},V'_{*},\mathbb{Y}_{N}).$$

By The Bundle of Vector Fields over a Set of Measures, Second-Order Equation Operators on the Wasserstein Space, and Their Delta-Shifts §shifted, $F^{-}_{N,\delta}(\rho^{*},r,V_{*},\mathbb{X})=F_{N}\bigl(\rho^{*},r+\delta\mathcal{E}_{N}(\rho^{*}),V_{*}+\delta\Sigma_{N}(\rho^{*}),\mathbb{X}+\delta H_{\mathcal{E}_{N}}(\rho^{*})\bigr)$ for every $r\in\mathbb{R}$, where $(\rho^{*},V_{*}+\delta\Sigma_{N}(\rho^{*}))\in\mathcal{V}(\mathcal{D}_{N,\Sigma})$ and the last two arguments do not depend on $r$. By (5e), the two values $Nt_{*}+\delta\mathcal{E}_{N}(\rho^{*})\le s_{*}+\delta\mathcal{E}_{N}(\rho^{*})$ lie in $[-R,R]$, so the properness constant $\lambda$ at $R$ (Locally Strictly Proper Second-Order Equation Operator on the Wasserstein Space §constant, at the configuration level with $Q=\mathcal{D}_{N,\Sigma}$) gives

$$\lambda(s_{*}-Nt_{*})\le F^{-}_{N,\delta}(\rho^{*},s_{*},V_{*},\mathbb{X})-F^{-}_{N,\delta}(\rho^{*},Nt_{*},V_{*},\mathbb{X}).$$

Adding the two displays, and then using (5b) and (5d),

$$\lambda(s_{*}-Nt_{*})\le\omega_{1}\bigl(\alpha W(\rho^{*},Q^{*})^{2}+\alpha^{-1}\bigr)+\omega_{2}\bigl(\delta(|\mathcal{E}_{N}(\rho^{*})|+|\mathcal{E}_{N}(Q^{*})|+1),\alpha\bigr)+F^{-}_{N,\delta}(\rho^{*},s_{*},V_{*},\mathbb{X})-F^{+}_{N,\delta}(Q^{*},Nt_{*},V'_{*},\mathbb{Y}_{N}),$$

whose last two terms have a sum at most $0$. Finally $M\le s_{*}-Nt_{*}$ (5e) and $0<\lambda$ give $\lambda M\le\lambda(s_{*}-Nt_{*})$ (claim 5 of Elementary Arithmetic in an Ordered Field). This is (E) for the maximising pair $(\rho^{*},\nu^{*})$.

Step 6 (The corrected maximum and its monotonicity). For positive $\alpha$ and $\delta\in I$ put $K(\alpha,\delta)=M(\delta,\alpha)+(1+N)\delta e_{0}$.

(6a) By (4a), $K(\alpha,\delta)\le b_{U}-Nb_{v}$.

(6b) Lower bound. By (L1) fix $\mu_{0}\in\mathcal{D}_{\Sigma}\subseteq\mathcal{D}$; then $\mu_{0}^{\otimes N}\in\mathcal{D}_{N}$ and $\mathcal{E}_{N}(\mu_{0}^{\otimes N})=N\,\mathcal{E}(\mu_{0})$ by (2a), and $W(\mu_{0}^{\otimes N},\mu_{0}^{\otimes N})=0$ by The Quadratic Wasserstein Distance is a Metric on the Wasserstein Space §separation. By Basic Properties of the Delta-Envelopes on the Wasserstein Space §semicontinuity, applied to $U$ and the configuration-level pair and to $v$ and the particle-level pair (with the growth of Step 3), $U(P)-\delta\mathcal{E}_{N}(P)\le U^{-}_{\delta}(P)$ for $P\in\mathcal{D}_{N}$ and $v^{+}_{\delta}(\mu)\le v(\mu)+\delta\mathcal{E}(\mu)$ for $\mu\in\mathcal{D}$. So, $M(\delta,\alpha)$ being an upper bound of the values of $\Psi_{\delta,\alpha}$ and $N$ positive (claims 4 of Elementary Order Arithmetic in an Ordered Field and 5 of Elementary Arithmetic in an Ordered Field),

$$M(\delta,\alpha)\ge\Psi_{\delta,\alpha}(\mu_{0}^{\otimes N},\mu_{0})\ge U(\mu_{0}^{\otimes N})-N\,v(\mu_{0})-2N\delta\,\mathcal{E}(\mu_{0}).$$

For $\delta\in I$, $N\delta\mathcal{E}(\mu_{0})\le N\delta|\mathcal{E}(\mu_{0})|\le N|\mathcal{E}(\mu_{0})|$ and $-(1+N)|e_{0}|\le-(1+N)\delta|e_{0}|\le(1+N)\delta e_{0}$ (claims 3 and 4 of Properties of the Absolute Value in an Ordered Field, claim 5 of Elementary Arithmetic in an Ordered Field and claim 4 of Elementary Order Arithmetic in an Ordered Field); hence

$$\ell\le K(\alpha,\delta),\qquad\ell=U(\mu_{0}^{\otimes N})-N\,v(\mu_{0})-2N|\mathcal{E}(\mu_{0})|-(1+N)|e_{0}|.$$

(6c) Decreasing the weight. Let $\alpha>0$, $\delta,\delta'\in I$ with $\delta'<\delta$, and let $(\hat{P},\hat{\mu})$ be a maximising pair of $\Psi_{\delta,\alpha}$. By (4c), $M(\delta',\alpha)-M(\delta,\alpha)\ge(\delta-\delta')(\mathcal{E}_{N}(\hat{P})+N\mathcal{E}(\hat{\mu}))$; adding $(1+N)(\delta'-\delta)e_{0}$,

$$K(\alpha,\delta')-K(\alpha,\delta)\ge(\delta-\delta')\bigl(\mathcal{E}_{N}(\hat{P})-e_{0}+N(\mathcal{E}(\hat{\mu})-e_{0})\bigr)\ge0,$$

the last because $\mathcal{E}_{N}(\hat{P})-e_{0}$ and $\mathcal{E}(\hat{\mu})-e_{0}$ are nonnegative (claim 3 of Elementary Arithmetic in an Ordered Field) and $0<N$, $0<\delta-\delta'$ (claims 2 and 5 of that lemma). Since a maximising pair exists (4a), $K(\alpha,\cdot)$ is nonincreasing on $I$.

(6d) Decreasing the strength. Let $0<\alpha'<\alpha$, $\delta\in I$, and let $(\hat{P},\hat{\mu})$ be a maximising pair of $\Psi_{\delta,\alpha}$. By (4d),

$$K(\alpha',\delta)-K(\alpha,\delta)=M(\delta,\alpha')-M(\delta,\alpha)\ge(\alpha-\alpha')\,\tfrac{1}{2}W(\hat{P},\hat{\mu}^{\otimes N})^{2}\ge0,$$

so $K(\cdot,\delta)$ is nonincreasing on the positive reals.

Step 7 (Constants for a tolerance). Let $\vartheta\in\mathbb{R}$ be positive, and put $\zeta=\lambda\vartheta/4$, positive by claims 5 and 8 of Elementary Order Arithmetic in an Ordered Field. By clause 2 of Modulus of Continuity fix a positive $\tau_{1}$ with $\omega_{1}(t)\le\zeta$ whenever $0\le t\le\tau_{1}$, and put $\beta_{0}=1+2\tau_{1}^{-1}$ and $\eta_{1}=\tau_{1}/16$, both positive. The quantities chosen below are chosen in the order $\tau_{1}$, $\alpha$, $\tau_{2}$, $\eta_{2}$, $\delta_{1}$, $\delta_{0}$, each depending only on $\vartheta$ and those before it.

Step 8 (Choice of the strength). For $\alpha\ge\beta_{0}$ the set $\{K(\alpha,\delta):\delta\in I\}$ is nonempty (it contains $K(\alpha,\tfrac{1}{2})$, as $\tfrac{1}{2}\in I$ by claim 8 of Elementary Order Arithmetic in an Ordered Field) and bounded above by $b_{U}-Nb_{v}$ by (6a); let $\Lambda(\alpha)$ be its least upper bound (The Real Numbers: Standing Notation and Background §bounds). By (6b), $\ell\le\Lambda(\alpha)$. By (6d), $\Lambda$ is nonincreasing on $\{\alpha:\alpha\ge\beta_{0}\}$: for $\beta_{0}\le\alpha'<\alpha$ and every $\delta\in I$, $K(\alpha,\delta)\le K(\alpha',\delta)\le\Lambda(\alpha')$. The set $\{\Lambda(\alpha):\alpha\ge\beta_{0}\}$ is nonempty and bounded below by $\ell$; let $\Lambda_{*}$ be its greatest lower bound (The Real Numbers: Standing Notation and Background §bounds). By claim 4 of Approximation Property of the Supremum and the Infimum in $\mathbb{R}$ fix $\alpha_{1}\ge\beta_{0}$ with $\Lambda(\alpha_{1})<\Lambda_{*}+\eta_{1}$, and put $\alpha=2\alpha_{1}$, so that $\tfrac{\alpha}{2}=\alpha_{1}$. Then $\alpha\ge\alpha_{1}\ge\beta_{0}$, so $\Lambda_{*}\le\Lambda(\alpha)$ and

$$\Lambda(\tfrac{\alpha}{2})-\Lambda(\alpha)<\eta_{1}.\qquad(8\mathrm{a})$$

Moreover $1<\beta_{0}\le\alpha$, and from $2\tau_{1}^{-1}<\beta_{0}\le\alpha$, multiplying by the positive $\alpha^{-1}\tau_{1}/2$ (claim 10 of Elementary Order Arithmetic in an Ordered Field), $\alpha^{-1}<\tau_{1}/2$. By The Second-Order Structure Condition at Uniquely Mapped Pairs on the Wasserstein Space §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 $\tau_{2}$ with $\omega_{2}(t,\alpha)\le\zeta$ whenever $0\le t\le\tau_{2}$.

Step 9 (Choice of the threshold). Let $\eta_{2}$ be the least of $\eta_{1}$ and $\tau_{2}/4$ (claim 9 of Elementary Order Arithmetic in an Ordered Field). By claim 3 of Approximation Property of the Supremum and the Infimum in $\mathbb{R}$ fix $\delta_{1}\in I$ with $\Lambda(\alpha)-\eta_{2}<K(\alpha,\delta_{1})$, and let $\delta_{0}$ be the least of $\delta_{1}$ and $\tau_{2}\bigl(2(1+N)|e_{0}|+2\bigr)^{-1}$ (claims 7 and 9 of Elementary Order Arithmetic in an Ordered Field), a positive number. Let $\delta\in\mathbb{R}$ satisfy $0<\delta<\delta_{0}$. Then $\delta\in I$ and $\delta<\delta_{1}$, so, $K(\alpha,\cdot)$ being nonincreasing (6c),

$$\Lambda(\alpha)-\eta_{2}<K(\alpha,\delta_{1})\le K(\alpha,\delta);\qquad(9\mathrm{a})$$

and $\delta\bigl(2(1+N)|e_{0}|+2\bigr)<\tau_{2}$ (claim 10 of Elementary Order Arithmetic in an Ordered Field), so $\delta\bigl((1+N)|e_{0}|+1\bigr)<\tau_{2}/2$.

Step 10 (The maximum is at most the tolerance). With $\alpha$ and $\delta$ as in Step 9 we show $M(\delta,\alpha)\le\vartheta$. Suppose instead $\vartheta<M(\delta,\alpha)$; then $0\le M(\delta,\alpha)$, and as $\delta\in I$ and $1<\alpha$, Step 5 provides a maximising pair $(\rho^{*},\nu^{*})$ of $\Psi_{\delta,\alpha}$ with $Q^{*}=(\nu^{*})^{\otimes N}$ and (E).

The first modulus. By (6d) with $\alpha'=\tfrac{\alpha}{2}$ and the maximising pair $(\rho^{*},\nu^{*})$, whose coefficient is $(\alpha-\tfrac{\alpha}{2})\tfrac{1}{2}=\tfrac{\alpha}{4}$, then $K(\tfrac{\alpha}{2},\delta)\le\Lambda(\tfrac{\alpha}{2})$ (as $\tfrac{\alpha}{2}=\alpha_{1}\ge\beta_{0}$), (9a), (8a) and $\eta_{2}\le\eta_{1}$,

$$\tfrac{\alpha}{4}W(\rho^{*},Q^{*})^{2}\le K(\tfrac{\alpha}{2},\delta)-K(\alpha,\delta)<\Lambda(\tfrac{\alpha}{2})-\Lambda(\alpha)+\eta_{2}<2\eta_{1},$$

so $\alpha W(\rho^{*},Q^{*})^{2}<8\eta_{1}=\tau_{1}/2$; with $\alpha^{-1}<\tau_{1}/2$, the first argument in (E), nonnegative by The Second-Order Structure Condition at Uniquely Mapped Pairs on the Wasserstein Space §pair, lies in $[0,\tau_{1}]$, and the first modulus in (E) is at most $\zeta$.

The second modulus. By (6c) with $\delta'=\tfrac{\delta}{2}\in I$ and the maximising pair $(\rho^{*},\nu^{*})$, then $K(\alpha,\tfrac{\delta}{2})\le\Lambda(\alpha)$ and (9a),

$$\tfrac{\delta}{2}\bigl(\mathcal{E}_{N}(\rho^{*})-e_{0}+N(\mathcal{E}(\nu^{*})-e_{0})\bigr)\le K(\alpha,\tfrac{\delta}{2})-K(\alpha,\delta)<\eta_{2}\le\tfrac{\tau_{2}}{4}.$$

As $\mathcal{E}_{N}(\rho^{*})-e_{0}\ge0$, the triangle inequality (claims 1 and 5 of Properties of the Absolute Value in an Ordered Field) applied to $\mathcal{E}_{N}(\rho^{*})=(\mathcal{E}_{N}(\rho^{*})-e_{0})+e_{0}$ gives $|\mathcal{E}_{N}(\rho^{*})|\le\mathcal{E}_{N}(\rho^{*})-e_{0}+|e_{0}|$, and likewise $|\mathcal{E}_{N}(Q^{*})|=N|\mathcal{E}(\nu^{*})|\le N(\mathcal{E}(\nu^{*})-e_{0})+N|e_{0}|$, using (2a). Multiplying by the positive $\delta$ and using Step 9,

$$\delta\bigl(|\mathcal{E}_{N}(\rho^{*})|+|\mathcal{E}_{N}(Q^{*})|+1\bigr)\le\delta\bigl(\mathcal{E}_{N}(\rho^{*})-e_{0}+N(\mathcal{E}(\nu^{*})-e_{0})\bigr)+\delta\bigl((1+N)|e_{0}|+1\bigr)<\tfrac{\tau_{2}}{2}+\tfrac{\tau_{2}}{2}=\tau_{2},$$

and the argument is positive; hence the second modulus in (E) is at most $\zeta$.

So (E) gives $\lambda M(\delta,\alpha)\le2\zeta=\lambda\vartheta/2$. But $\vartheta<M(\delta,\alpha)$ and $0<\lambda$ give $\lambda\vartheta<\lambda M(\delta,\alpha)$ (claim 10 of Elementary Order Arithmetic in an Ordered Field), so $\lambda\vartheta<\lambda\vartheta/2$ (claim 2 of that lemma), that is $\lambda\vartheta/2<0$ (claims 1 and 8 of that lemma), contradicting $0<\lambda\vartheta/2$ (claims 5 and 8 of that lemma). Therefore $M(\delta,\alpha)\le\vartheta$. We have shown: for every positive $\vartheta$ there are $\alpha>1$ and a positive $\delta_{0}$ with

$$M(\delta,\alpha)\le\vartheta\qquad\text{for every }\delta\in\mathbb{R}\text{ with }0<\delta<\delta_{0}.\qquad(10\mathrm{a})$$

Step 11 (Clause 1). Let $\mu\in\mathcal{D}$; then $\mu^{\otimes N}\in\mathcal{D}_{N}$ and $\mathcal{E}_{N}(\mu^{\otimes N})=N\,\mathcal{E}(\mu)$ by (2a). Let $\varepsilon\in\mathbb{R}$ be positive, put $\vartheta=\tfrac{\varepsilon}{2}$, positive by claim 8 of Elementary Order Arithmetic in an Ordered Field, and let $\alpha$ and $\delta_{0}$ be as in (10a) for this $\vartheta$. Let $\delta_{2}$ be the least of $\delta_{0}$ and $\vartheta\bigl(2N|\mathcal{E}(\mu)|+1\bigr)^{-1}$ (claims 7 and 9 of that lemma), and $\delta=\tfrac{\delta_{2}}{2}$; then $0<\delta<\delta_{2}\le\delta_{0}$, and $\delta\bigl(2N|\mathcal{E}(\mu)|+1\bigr)\le\vartheta$ (claim 5 of Elementary Arithmetic in an Ordered Field), so $2N\delta|\mathcal{E}(\mu)|\le\vartheta$ as $0<\delta$. By Basic Properties of the Delta-Envelopes on the Wasserstein Space §semicontinuity, as in (6b), $U(\mu^{\otimes N})-\delta\mathcal{E}_{N}(\mu^{\otimes N})\le U^{-}_{\delta}(\mu^{\otimes N})$ and $v^{+}_{\delta}(\mu)\le v(\mu)+\delta\mathcal{E}(\mu)$; with $W(\mu^{\otimes N},\mu^{\otimes N})=0$ (The Quadratic Wasserstein Distance is a Metric on the Wasserstein Space §separation), (10a), and $\mathcal{E}(\mu)\le|\mathcal{E}(\mu)|$ (claim 3 of Properties of the Absolute Value in an Ordered Field),

$$\begin{aligned} U(\mu^{\otimes N})-N\,v(\mu)&=\bigl(U(\mu^{\otimes N})-\delta\mathcal{E}_{N}(\mu^{\otimes N})\bigr)-N\bigl(v(\mu)+\delta\mathcal{E}(\mu)\bigr)+2N\delta\,\mathcal{E}(\mu)\\ &\le U^{-}_{\delta}(\mu^{\otimes N})-N\,v^{+}_{\delta}(\mu)-\tfrac{\alpha}{2}W(\mu^{\otimes N},\mu^{\otimes N})^{2}+2N\delta|\mathcal{E}(\mu)|\\ &=\Psi_{\delta,\alpha}(\mu^{\otimes N},\mu)+2N\delta|\mathcal{E}(\mu)|\le M(\delta,\alpha)+\vartheta\le2\vartheta=\varepsilon , \end{aligned}$$

using claim 4 of Elementary Order Arithmetic in an Ordered Field and claim 5 of Elementary Arithmetic in an Ordered Field for the term multiplied by $-N$, and that $M(\delta,\alpha)$ is an upper bound of the values of $\Psi_{\delta,\alpha}$. As $\varepsilon$ was an arbitrary positive number, Comparison of Real Numbers with Arbitrary Positive Slack §slack-above with $a=U(\mu^{\otimes N})-N\,v(\mu)$ and $b=0$ gives $U(\mu^{\otimes N})-N\,v(\mu)\le0$, that is $U(\mu^{\otimes N})\le N\,v(\mu)$ (claim 3 of Elementary Arithmetic in an Ordered Field). This is clause 1.

Step 12 (Clause 2). The data of the statement are data of Well-Posedness of the Lifted N-Particle Hamilton-Jacobi Equation: Comparison, Existence and Uniqueness of a Bounded Viscosity Solution with the cost $c$ and its bound $b$, and, by Step 2, of Well-Posedness of the Hamilton-Jacobi Equation with Common Noise for Controlled Langevin Dynamics: Existence and Uniqueness of a Bounded Viscosity Solution with the running cost $g=\tilde{c}$, uniformly continuous with $|\tilde{c}(\nu)|\le\tfrac{b}{N}$ for every $\nu$, and the bound $\tfrac{b}{N}$ in place of $b$. Let $U_{N}$ and $\bar{u}_{N}$ be the bounded viscosity solutions of the $N$-particle and of the mean-field equation given by Well-Posedness of the Lifted N-Particle Hamilton-Jacobi Equation: Comparison, Existence and Uniqueness of a Bounded Viscosity Solution §existence and Well-Posedness of the Hamilton-Jacobi Equation with Common Noise for Controlled Langevin Dynamics: Existence and Uniqueness of a Bounded Viscosity Solution §existence, unique by Well-Posedness of the Lifted N-Particle Hamilton-Jacobi Equation: Comparison, Existence and Uniqueness of a Bounded Viscosity Solution §uniqueness and Well-Posedness of the Hamilton-Jacobi Equation with Common Noise for Controlled Langevin Dynamics: Existence and Uniqueness of a Bounded Viscosity Solution §uniqueness. By The Lifted N-Particle Hamilton-Jacobi Equation on the Wasserstein Space of the Configuration Space §equation and (L3) for the configuration level, $U_{N}$ is a viscosity solution of $F_{N}$ relative to the configuration-level pair, hence by Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space §solution a viscosity subsolution of $F_{N}$, that is (by the same two references) a viscosity subsolution of the $N$-particle equation; likewise, by (L3) for the particle level and Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space §solution, $\bar{u}_{N}$ is a viscosity supersolution of the mean-field equation. Being bounded, with bounds $k_{U}$ and $k_{v}$ say, they satisfy $U_{N}(P)\le k_{U}$ for $P\in\mathcal{D}_{N}$ and $-k_{v}\le\bar{u}_{N}(\mu)$ for $\mu\in\mathcal{D}$ (claim 6 of Properties of the Absolute Value in an Ordered Field). Clause 1, applied with $U=U_{N}$ and $v=\bar{u}_{N}$, gives $U_{N}(\mu^{\otimes N})\le N\,\bar{u}_{N}(\mu)$ for every $\mu\in\mathcal{D}$.