Proof of Cross-Level Comparison through One-Particle Marginals: N Times a Mean-Field Subsolution Lies below a Lifted N-Particle Supersolution under Cost Domination

Each result cited is universally quantified over the data in its own statement, and is applied to the data named at the point of use. Throughout, $|s|$ is the absolute value of $s\in\mathbb{R}$, $\tfrac{s}{2}$ is the product of $s$ with the multiplicative inverse of $2$ (claim 8 of Elementary Order Arithmetic in an Ordered Field), $s^{-1}$ is the multiplicative inverse of a positive $s$ (claim 7 of that lemma), and $I=\{\delta\in\mathbb{R}:0<\delta<1\}$. Commutativity, associativity and distributivity in the field $\mathbb{R}$, and the compatibility of $\le$ with addition, are used without mention. The natural number $N$ is read as a real number through the canonical map of The Canonical Map from the Natural Numbers to a Field; by claims 2 and 3 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field, $1\le N$ and $0<N^{-1}$, and multiplying $1\le N$ by $N^{-1}$ (claim 5 of Elementary Arithmetic in an Ordered Field) gives $N^{-1}\le1$. Moreover, for every $x\in\mathbb{R}$ with $0\le x$, claim 5 of Elementary Arithmetic in an Ordered Field applied to $1\le N$ with the multiplier $x$ gives

The letters $\sigma$ (noise intensity), $\theta$ (control cost), $b$ (cost bound), $c$ (configuration cost), $g$ (running cost) and $p$ (rows of $\Gamma$) keep the meaning of the statement; the measure $\sigma^{*}$ below, named as in Intrinsic Test Functions at a Maximiser of the Doubled Difference Linked through One-Particle Marginals, is unrelated to the noise intensity $\sigma$.

Step 1 (The two levels and the properties of their operators). Let $n$ be one of the two natural numbers $d$ and $dN$. If $n=d$, put $V'=V$, $\Gamma'=\Gamma$ and $g'=g$. If $n=dN$, put $V'=V_{N}$, the $N$-particle potential of $V$, a confining potential on $\mathbb{R}^{dN}$ by The N-Particle Potential of a Confining Potential is a Confining Potential on the Configuration Space §confining; $\Gamma'=\Gamma_{N}$, the $N$-particle common-noise matrix, which lies in $\mathcal{M}_{p\times dN}(\mathbb{R})$ by The N-Particle Hamilton-Jacobi Equation with Individual and Common Noise on the Configuration Space §common-noise; and $g'=G$, where $G(P)=\int_{\mathbb{R}^{dN}}c\,dP$ for $P\in\mathcal{P}_{2}(\mathbb{R}^{dN})$; in this case every result and notion used in this step is read at the configuration level. Since $0\le|c(x)|\le b$ for $x\in\mathbb{R}^{dN}$ (claim 1 of Properties of the Absolute Value in an Ordered Field), $0\le b$ and $c$ is bounded with bound $b$; so 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 $b$, shows that $c$ is Borel and that $G$ is uniformly continuous for $W_{2}$ and the metric of The Absolute Value Metric on the Real Line, with $|G(P)|\le b$ for every $P$. In both cases, therefore, $V'$ is a confining potential on $\mathbb{R}^{n}$, $\Gamma'\in\mathcal{M}_{p\times n}(\mathbb{R})$, and $g':\mathcal{P}_{2}(\mathbb{R}^{n})\to\mathbb{R}$ is uniformly continuous with $|g'|\le b$ (for $n=d$ by hypothesis). Let $(\mathcal{D}',\mathcal{D}'_{\Sigma},\mathcal{E}',\Sigma')$ be the Langevin free-energy pair with potential $V'$ and noise intensity $\sigma$, and $F'$ 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 $g'$. For $n=d$ these are the pair $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ and the operator $F$ of the mean-field equation; for $n=dN$ they are the configuration-level pair $(\mathcal{D}_{N},\mathcal{D}_{N,\Sigma},\mathcal{E}_{N},\Sigma_{N})$ and, by The Lifted N-Particle Hamilton-Jacobi Equation on the Wasserstein Space of the Configuration Space §operator, the lifted $N$-particle operator $F_{N}$ with running cost $c$. 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 $\mathcal{D}'_{\Sigma}$ with $\delta$-shifts relative to the pair.

(Q) 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 mean-field equation is a function $\mathcal{D}\to\mathbb{R}$ that is a viscosity subsolution, supersolution or solution of $F$ relative to $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ in the sense of Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space; by The Lifted N-Particle Hamilton-Jacobi Equation on the Wasserstein Space of the Configuration Space §equation and the same two clauses read at the configuration level with the data $V_{N}$, $\sigma$, $\lambda_{0}$, $\Gamma_{N}$, $\theta$, $G$, a viscosity subsolution, supersolution or solution of the $N$-particle equation is a function $\mathcal{D}_{N}\to\mathbb{R}$ that is one of $F_{N}$ relative to $(\mathcal{D}_{N},\mathcal{D}_{N,\Sigma},\mathcal{E}_{N},\Sigma_{N})$.

(P1) 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, the pair is a penalty pair; in particular $\mathcal{D}'_{\Sigma}\subseteq\mathcal{D}'$ and $\mathcal{D}'_{\Sigma}$ is nonempty (Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §pair, Penalty Pairs on the Wasserstein Space: the Penalty, Its Score, and Their Domains §nonempty).

(P2) 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 §coercive, the pair is Wasserstein-coercive and $\mathcal{D}'$ has the map property.

(P3) 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.

(P4) 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 §growth, $\mathcal{E}'$ is lower semicontinuous on $\mathcal{D}'$.

(P5) $F'$ is locally strictly proper and satisfies the shift-coercivity condition, the shift-semicontinuity condition of The Shift-Semicontinuity Condition for an Equation Operator on the Wasserstein Space §semicontinuity 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 (P1)), with $\lambda_{0}$, with $\theta$ (which satisfies $0<\theta\le1$), with $p$, $\Gamma'$, $g'$ and $F'$; we discharge its hypotheses. (Convexity) is The Langevin Free-Energy Pair is Displacement Convex, with Closed Score Along Couplings and Regular Penalised Maxima §convex; (Semicontinuity) is (P4); (Running cost) holds because $|g'(\nu)|\le b\le|b|$ and $0\le|b|$, so that $g'$ is bounded with bound $|b|$, and $g'$ is uniformly continuous as shown above.

The trace as a finite sum of entries. For $k\in[p]$ let $\gamma_{k}\in\mathbb{R}^{n}$ be the $k$th row of $\Gamma'$, with $i$th coordinate $\gamma_{k,i}=\Gamma'_{ki}$ for $i\in[n]$. Let $\mu\in\mathcal{D}'$; then $H_{\mathcal{E}'}(\mu)\in\mathcal{S}(n)$ 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 $n$, so that its rows $a_{k}$ are our $\gamma_{k}$ (its hypotheses $1\le m$ and $1\le p$ 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 its $n$ equal to our $n$, $M=H_{\mathcal{E}'}(\mu)$ and $w=z=\gamma_{k}$ for each $k\in[p]$, it gives

Put $s_{\Gamma'}=\sum_{k=1}^{p}\sum_{i=1}^{n}\sum_{l=1}^{n}|\gamma_{k,i}|\,|\gamma_{k,l}|$, a real number independent of $\mu$ and nonnegative by claim 5 of Properties of Finite Sums (applied to the innermost sums first), each summand being a product of nonnegative numbers.

For (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 $\mu\in\mathcal{D}'$. Let $\mu\in\mathcal{D}'$ and $t_{\mu}=C(1+|\mathcal{E}'(\mu)|)$. For $i,l\in[n]$, 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 and claim 3 of Properties of the Absolute Value in an Ordered Field give $|H_{\mathcal{E}'}(\mu)_{il}|\le\mathrm{tr}\,H_{\mathcal{E}'}(\mu)\le|\mathrm{tr}\,H_{\mathcal{E}'}(\mu)|\le t_{\mu}$. 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 $|\gamma_{k,i}\gamma_{k,l}H_{\mathcal{E}'}(\mu)_{il}|\le t_{\mu}|\gamma_{k,i}|\,|\gamma_{k,l}|$. 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 takes out the factor $t_{\mu}$; so $|\mathrm{tr}(\Gamma'^{\top}\Gamma'H_{\mathcal{E}'}(\mu))|\le s_{\Gamma'}C(1+|\mathcal{E}'(\mu)|)$. 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 $0\le|C|$), 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 the hypothesis with the constant $C'$.

For (Hessian continuity), let $R_{0}$ be positive and $S_{R_{0}}=\{\mu\in\mathcal{D}':|\mathcal{E}'(\mu)|\le R_{0}\}$. For $i,l\in[n]$ the restriction of $\mu\mapsto H_{\mathcal{E}'}(\mu)_{il}$ to $S_{R_{0}}$ 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 $S_{R_{0}}$ 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 $(\mathcal{P}_{2}(\mathbb{R}^{n}),W_{2})$ with the subset $S_{R_{0}}$. A finite sum of functions continuous on $S_{R_{0}}$ is continuous on $S_{R_{0}}$, 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 $S_{R_{0}}$ is continuous.

(P6) The statement of The Shift-Semicontinuity Condition for an Equation Operator on the Wasserstein Space coincides word for word with that of The Shift-Semicontinuity Condition for an Equation Operator on the Wasserstein Space: the same test data, the same clause (converging), the same two implications in clause (level) and the same clause (semicontinuity). Hence, by (P5), $F'$ satisfies the shift-semicontinuity condition of The Shift-Semicontinuity Condition for an Equation Operator on the Wasserstein Space §semicontinuity.

Applied with $n=d$ and with $n=dN$, Step 1 gives (Q) and (P1)-(P6) for $F$ on the particle-level pair and for $F_{N}$ on the configuration-level pair.

Step 2 (The doubled difference across the levels). Let $v$ and $U$ be as in clause 1, and fix $b_{v},b_{U}\in\mathbb{R}$ with $v(\mu)\le b_{v}$ for $\mu\in\mathcal{D}$ and $b_{U}\le U(P)$ for $P\in\mathcal{D}_{N}$. By (Q), $v$ is a viscosity subsolution of $F$ relative to the particle-level pair and $U$ a viscosity supersolution of $F_{N}$ relative to the configuration-level pair. Both pairs are Wasserstein-coercive by (P2), so by The Delta-Envelopes of Bounded Functions and Their Monotonicity in the Weight, for a Wasserstein-Coercive Penalty Pair §growth $v$ has penalty-subordinate growth from above and $U$ from below, and for positive $\delta$ the $\delta$-envelopes $v^{-}_{\delta}$ on $\mathcal{D}$ (relative to the particle-level pair) and $U^{+}_{\delta}$ on $\mathcal{D}_{N}$ (relative to the configuration-level pair) are defined. By Basic Properties of a Wasserstein-Coercive Penalty Pair §bounded-below, applied to each pair (at the configuration level for the second), and claim 9 of Elementary Order Arithmetic in an Ordered Field, fix $e_{0}\in\mathbb{R}$ with $e_{0}\le\mathcal{E}(\mu)$ for $\mu\in\mathcal{D}$ and $e_{0}\le\mathcal{E}_{N}(P)$ for $P\in\mathcal{D}_{N}$. For $P\in\mathcal{P}_{2}(\mathbb{R}^{dN})$, $P^{[1]}\in\mathcal{P}_{2}(\mathbb{R}^{d})$ by Tensor Powers and One-Particle Marginals: Particle Laws, Product Integrals, Push-Forwards, Moments, Product Maps and Diagonal Shifts §moments; for $P\in\mathcal{D}_{N}$, $P^{[1]}\in\mathcal{D}$ and $N\mathcal{E}(P^{[1]})\le\mathcal{E}_{N}(P)$ by The Langevin Free-Energy Pair across the Particle and Configuration Levels: Tensor Powers, One-Particle Marginals and Diagonal Translation Hessians §marginal.

Energy of the marginal. Let $P\in\mathcal{D}_{N}$. Then $e_{0}\le\mathcal{E}(P^{[1]})$, and

Indeed, $0\le\mathcal{E}_{N}(P)-e_{0}$ (claim 3 of Elementary Arithmetic in an Ordered Field). If $0\le\mathcal{E}(P^{[1]})$, then $|\mathcal{E}(P^{[1]})|=\mathcal{E}(P^{[1]})\le N\mathcal{E}(P^{[1]})\le\mathcal{E}_{N}(P)=(\mathcal{E}_{N}(P)-e_{0})+e_{0}$ by (0), and $e_{0}\le|e_{0}|$ (claim 3 of Properties of the Absolute Value in an Ordered Field). Otherwise $\mathcal{E}(P^{[1]})<0$, $|\mathcal{E}(P^{[1]})|=-\mathcal{E}(P^{[1]})\le-e_{0}\le|e_{0}|$ (Absolute Value in an Ordered Field, claim 4 of Elementary Order Arithmetic in an Ordered Field and claim 3 of Properties of the Absolute Value in an Ordered Field), and adding the nonnegative $\mathcal{E}_{N}(P)-e_{0}$ keeps the bound. The second inequality of (2a) follows from $\mathcal{E}_{N}(P)\le|\mathcal{E}_{N}(P)|$ and $-e_{0}\le|e_{0}|$ (claims 2 and 3 of Properties of the Absolute Value in an Ordered Field).

The link. Let $L:\mathcal{P}_{2}(\mathbb{R}^{d})\times\mathcal{P}_{2}(\mathbb{R}^{dN})\to\mathbb{R}$ have the value $L(\mu,P)=\tfrac{N}{2}W_{2}(\mu,P^{[1]})^{2}$; it is nonnegative, as $0\le W_{2}$ and $0<\tfrac{N}{2}$ (claims 5 and 8 of Elementary Order Arithmetic in an Ordered Field). It is continuous in the sense of Existence, Penalty Bounds and Monotonicity of the Maximum of a Two-Space Doubled Difference with a Continuous Link: let $W_{2}(\mu_{j},\mu)\to0$ and $W_{2}(P_{j},P)\to0$. 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} §marginal (with $q=d$) and (0), $W_{2}(P_{j}^{[1]},P^{[1]})^{2}\le NW_{2}(P_{j}^{[1]},P^{[1]})^{2}\le W_{2}(P_{j},P)^{2}$, so $W_{2}(P_{j}^{[1]},P^{[1]})\le W_{2}(P_{j},P)$ by claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field. By the triangle inequality and symmetry of $W_{2}$ (The Quadratic Wasserstein Distance is a Metric on the Wasserstein Space §triangle, The Quadratic Wasserstein Distance is a Metric on the Wasserstein Space §symmetry), used on each side, and claim 6 of Properties of the Absolute Value in an Ordered Field,

and the right side converges to $0$ (claim 1 of Arithmetic of Limits of Real Sequences); so, by Limit of a Sequence of Real Numbers, $W_{2}(\mu_{j},P_{j}^{[1]})\to W_{2}(\mu,P^{[1]})$, and by claims 2 and 3 of Arithmetic of Limits of Real Sequences, $L(\mu_{j},P_{j})\to L(\mu,P)$.

The doubled difference. We apply Existence, Penalty Bounds and Monotonicity of the Maximum of a Two-Space Doubled Difference with a Continuous Link with $n_{1}=d$, $n_{2}=dN$, the particle-level pair as its first pair and the configuration-level pair as its second (both Wasserstein-coercive by (P2)), the present $e_{0}$, its $u$ taken to be our $v$ and its $v$ our $U$, $b=b_{v}$, $b'=b_{U}$, $\kappa_{1}=N$, $\kappa_{2}=1$ and the link $L$. For positive $\delta,\alpha$ let $\Psi_{\delta,\alpha}:\mathcal{D}\times\mathcal{D}_{N}\to\mathbb{R}$ and $M(\delta,\alpha)$ be the function and the supremum named there:

since $\alpha L(\mu,P)=\tfrac{N\alpha}{2}W_{2}(\mu,P^{[1]})^{2}$. This is also the function $\Psi$ of Intrinsic Test Functions at a Maximiser of the Doubled Difference Linked through One-Particle Marginals for the data $v$, $U$, $b_{v}$, $b_{U}$, $\delta$, $\alpha$. By Existence, Penalty Bounds and Monotonicity of the Maximum of a Two-Space Doubled Difference with a Continuous Link §maximiser, $M(\delta,\alpha)$ is a real number, $M(\delta,\alpha)\le Nb_{v}-b_{U}-(N+1)\delta e_{0}$, and $\Psi_{\delta,\alpha}$ has a maximising pair.

Step 3 (The corrected maximum and its monotonicity). For $\alpha>0$ and $\delta\in I$ put

by Step 2. Lower bound. By (P1) at the configuration level fix $P_{0}\in\mathcal{D}_{N}$, and put $\mu_{0}=P_{0}^{[1]}\in\mathcal{D}$. By Basic Properties of the Delta-Envelopes on the Wasserstein Space §semicontinuity (for each pair, $v$ and $U$ having the growth recorded in Step 2), $v(\mu)-\delta\mathcal{E}(\mu)\le v^{-}_{\delta}(\mu)$ for $\mu\in\mathcal{D}$ and $U^{+}_{\delta}(P)\le U(P)+\delta\mathcal{E}_{N}(P)$ for $P\in\mathcal{D}_{N}$. As $W_{2}(\mu_{0},\mu_{0})=0$ (The Quadratic Wasserstein Distance is a Metric on the Wasserstein Space §separation),

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

Decreasing the weight. Let $\alpha>0$, $\delta,\delta'\in I$ with $\delta'<\delta$, and let $(\hat{\mu},\hat{P})$ be a maximising pair of $\Psi_{\delta,\alpha}$. By Existence, Penalty Bounds and Monotonicity of the Maximum of a Two-Space Doubled Difference with a Continuous Link §weight, $M(\delta',\alpha)-M(\delta,\alpha)\ge(\delta-\delta')(N\mathcal{E}(\hat{\mu})+\mathcal{E}_{N}(\hat{P}))$; adding $(N+1)(\delta'-\delta)e_{0}$,

the last because $\mathcal{E}(\hat{\mu})-e_{0}$ and $\mathcal{E}_{N}(\hat{P})-e_{0}$ 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, $K(\alpha,\cdot)$ is nonincreasing on $I$.

Decreasing the strength. Let $0<\alpha'<\alpha$, $\delta\in I$, and let $(\hat{\mu},\hat{P})$ be a maximising pair of $\Psi_{\delta,\alpha}$. By Existence, Penalty Bounds and Monotonicity of the Maximum of a Two-Space Doubled Difference with a Continuous Link §strength,

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

Step 4 (The structure estimate at a maximising pair). Put $B=N|b_{v}|+|b_{U}|+(N+1)|e_{0}|+1$ and $R=3B$; $B$ does not depend on $\delta$ or $\alpha$, $1\le B$ and $0<R$. By (P5) for $n=d$, fix a properness constant $\lambda>0$ for $F$ at $R$ and a second-order structure pair $(\omega_{1},\omega_{2})$ for $F$ at $R$. We show: if $\delta\in I$, $1<\alpha$ and $0\le M(\delta,\alpha)$, then there is a maximising pair $(\rho^{*},\sigma^{*})$ of $\Psi_{\delta,\alpha}$ such that, with $\nu^{*}=(\sigma^{*})^{[1]}$,

Fix such $\delta,\alpha$ and write $M=M(\delta,\alpha)$, $\Psi=\Psi_{\delta,\alpha}$.

(4a) The maximising pair. By Step 2 let $(\hat{\mu},\hat{P})$ be a maximising pair of $\Psi$. We apply Intrinsic Test Functions at a Maximiser of the Doubled Difference Linked through One-Particle Marginals to the particle-level and configuration-level pairs, which are Wasserstein-coercive with penalty domains having the map property by (P2), with $P^{[1]}\in\mathcal{D}$ for $P\in\mathcal{D}_{N}$ (Step 2), to its $u$ taken to be $v$ and its $U$ our $U$, to $b_{v}$, $b_{U}$, $\delta$, $\alpha$ and $(\hat{\mu},\hat{P})$. It provides $\rho^{*}\in\mathcal{D}$, $\sigma^{*}\in\mathcal{D}_{N}$, $\mathbb{X}\in\mathcal{S}(d)$ and $\mathbb{Y}_{N}\in\mathcal{S}(dN)$; with $\nu^{*}=(\sigma^{*})^{[1]}\in\mathcal{D}$, both ordered pairs $(\rho^{*},\nu^{*})$ and $(\nu^{*},\rho^{*})$ are uniquely mapped, and we let $S$ and $S'$ be the optimal maps named there, from $\rho^{*}$ to $\nu^{*}$ and from $\nu^{*}$ to $\rho^{*}$. By Intrinsic Test Functions at a Maximiser of the Doubled Difference Linked through One-Particle Marginals §maximiser, $\Psi(\rho^{*},\sigma^{*})=\Psi(\hat{\mu},\hat{P})=M$, so $(\rho^{*},\sigma^{*})$ is a maximising pair of $\Psi$. By Intrinsic Test Functions at a Maximiser of the Doubled Difference Linked through One-Particle Marginals §admitted fix $\mathbb{Y}\in\mathcal{S}(d)$ with $(\mathbb{X},\mathbb{Y})$ admitted at $\alpha$ and $a^{\oplus}\cdot(\mathbb{Y}_{N}a^{\oplus})=Na\cdot(\mathbb{Y}a)$ for every $a\in\mathbb{R}^{d}$. Put

The field $h$ lies in $T_{\nu^{*}}$: as $\nu^{*}\in\mathcal{D}$ and $\mathcal{D}$ has the map property, $\mathrm{id}-S'\in T_{\nu^{*}}$ (The Map Property of a Set of Probability Measures §map-property), and $T_{\nu^{*}}$ is a linear subspace of $L^{2}(\nu^{*};\mathbb{R}^{d})$ (Basic Properties of the Tangent Space: Closed Subspace, the Identity Map Belongs to It, Second-Moment Limits, and Representation of Bounded Functionals on Gradients §closed), so it contains $h=-\alpha(\mathrm{id}-S')$. Moreover $h^{\oplus}=\alpha(S'-\mathrm{id})^{\oplus}$ in $L^{2}(\sigma^{*};\mathbb{R}^{dN})$: if $\tilde{g}$ is a Borel representative of $S'-\mathrm{id}$, then $\alpha\tilde{g}$ is one of $h$, and by Particle Blocks of the Configuration Space: Block Maps, Configurations, Product Maps and Diagonal Points §product-map the coordinate with index $b(k,i)$ of $(\alpha\tilde{g})^{\oplus}(x)$ is the $i$th coordinate of $\alpha\tilde{g}(\mathfrak{p}_{k}(x))$, that is $\alpha$ times that of $\tilde{g}^{\oplus}(x)$; so $(\alpha\tilde{g})^{\oplus}=\alpha\tilde{g}^{\oplus}$, and passing to classes (Product Fields and the Projection onto One-Particle Tangent Fields §product-field) gives the claim.

(4b) The subsolution side. We apply Closure of Approximate Test Data for Viscosity Sub- and Supersolutions on the Wasserstein Space §subsolution at the particle level to the pair $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$, Wasserstein-coercive with closed score along couplings by (P2) and (P3), to $F$, which satisfies the shift-coercivity condition by (P5) and the shift-semicontinuity condition of The Shift-Semicontinuity Condition for an Equation Operator on the Wasserstein Space §semicontinuity by (P6), to $\delta\in I$, to $v$, bounded above and a viscosity subsolution of $F$ relative to the pair (Step 2), and to $\rho^{*}$, $V_{*}$ and $\mathbb{X}$. Its hypothesis on approximate test data is Intrinsic Test Functions at a Maximiser of the Doubled Difference Linked through One-Particle Marginals §subsolution, whose integrand $\alpha(y-S(y))$ is $V_{*}(y)$. Hence $\rho^{*}\in\mathcal{D}_{\Sigma}$ and

(4c) The supersolution side. We apply Closure of Approximate Test Data for Viscosity Sub- and Supersolutions on the Wasserstein Space §supersolution at the configuration level to the pair $(\mathcal{D}_{N},\mathcal{D}_{N,\Sigma},\mathcal{E}_{N},\Sigma_{N})$, Wasserstein-coercive with closed score along couplings by (P2) and (P3) for $n=dN$, to $F_{N}$, which satisfies the shift-coercivity condition and the shift-semicontinuity condition of The Shift-Semicontinuity Condition for an Equation Operator on the Wasserstein Space §semicontinuity by (P5) and (P6) for $n=dN$, to $\delta$, to $U$, bounded below and a viscosity supersolution of $F_{N}$ relative to that pair (Step 2), and to $\sigma^{*}$, $W_{*}=h^{\oplus}=\alpha(S'-\mathrm{id})^{\oplus}\in L^{2}(\sigma^{*};\mathbb{R}^{dN})$ and $\mathbb{Y}_{N}$. Its hypothesis on approximate test data is Intrinsic Test Functions at a Maximiser of the Doubled Difference Linked through One-Particle Marginals §supersolution. Writing $F^{+}_{N,\delta}$ for the $\delta$-shift $F^{+}_{\delta}$ of $F_{N}$ relative to the configuration-level pair, we get $\sigma^{*}\in\mathcal{D}_{N,\Sigma}$ and

(4d) Transfer to the marginal. By The Langevin Free-Energy Pair across the Particle and Configuration Levels: Tensor Powers, One-Particle Marginals and Diagonal Translation Hessians §marginal, $\sigma^{*}\in\mathcal{D}_{N,\Sigma}$ gives $\nu^{*}\in\mathcal{D}_{\Sigma}$. We apply Cross-Level Inequalities between the Shifted Lifted N-Particle and Mean-Field Langevin Operators at Tensor Powers and through One-Particle Marginals §marginal with the present $V$, $\lambda_{0}$, $\sigma$, $\theta$, $p$, $\Gamma$, the bounded Borel $c$ (Step 1) and $g$, so that its $F_{N}$ and $F$ are ours; with $\delta\in I$, the matrices $\mathbb{Y}_{N}$ and $\mathbb{Y}$ of (4a), which satisfy its diagonal identity; with $P=\sigma^{*}\in\mathcal{D}_{N,\Sigma}$, which satisfies $Ng(\nu^{*})\le\int_{\mathbb{R}^{dN}}c\,d\sigma^{*}$ by the cost domination hypothesis of the statement at $\sigma^{*}\in\mathcal{D}_{N}$; with $r=r_{*}$, so that $Nr=t_{*}$; and with $h\in T_{\nu^{*}}$. With (4c),

and multiplying by $N^{-1}>0$ (claim 5 of Elementary Arithmetic in an Ordered Field),

(4e) Bounds. Since $\Psi(\rho^{*},\sigma^{*})=M\ge0$, Existence, Penalty Bounds and Monotonicity of the Maximum of a Two-Space Doubled Difference with a Continuous Link §penalty gives $N\delta|\mathcal{E}(\rho^{*})|\le B_{\delta}$ and $\delta|\mathcal{E}_{N}(\sigma^{*})|\le B_{\delta}$, with $B_{\delta}=N|b_{v}|+|b_{U}|+(N+1)\delta|e_{0}|$; as $\delta<1$, $\delta|e_{0}|\le|e_{0}|$ (claim 5 of Elementary Arithmetic in an Ordered Field), so $B_{\delta}\le B$. By (0), $\delta|\mathcal{E}(\rho^{*})|\le N\delta|\mathcal{E}(\rho^{*})|\le B$. By (2a) with $P=\sigma^{*}$, $\delta|\mathcal{E}(\nu^{*})|\le\delta|\mathcal{E}_{N}(\sigma^{*})|+2\delta|e_{0}|\le B+2|e_{0}|\le2B$, the last because $2|e_{0}|\le(N+1)|e_{0}|\le B$. Hence

By the definition of $\Psi$, $Ns_{*}-t_{*}=M+\tfrac{N\alpha}{2}W_{2}(\rho^{*},\nu^{*})^{2}\ge M\ge0$, so

the second after multiplying $t_{*}\le Ns_{*}$ by $N^{-1}$. By The Delta-Envelopes of Bounded Functions and Their Monotonicity in the Weight, for a Wasserstein-Coercive Penalty Pair §bounded for each pair, $s_{*}\le b_{v}-\delta\mathcal{E}(\rho^{*})\le b_{v}-\delta e_{0}\le|b_{v}|+|e_{0}|\le B$ and $t_{*}\ge b_{U}+\delta\mathcal{E}_{N}(\sigma^{*})\ge b_{U}+\delta e_{0}\ge-(|b_{U}|+|e_{0}|)$, using $0<\delta<1$ and claims 3 and 4 of Properties of the Absolute Value in an Ordered Field; multiplying the last by $N^{-1}\le1$ gives $r_{*}\ge-N^{-1}(|b_{U}|+|e_{0}|)\ge-(|b_{U}|+|e_{0}|)\ge-B$. So $-B\le r_{*}\le s_{*}\le B$; in particular $-R\le r_{*}\le R$. With $a_{*}=\delta\mathcal{E}(\rho^{*})$, $|a_{*}|\le B$, hence

by claim 6 of Properties of the Absolute Value in an Ordered Field.

(4f) Properness and the structure pair. By The Bundle of Vector Fields over a Set of Measures, Second-Order Equation Operators on the Wasserstein Space, and Their Delta-Shifts §shifted, $F^{-}_{\delta}(\rho^{*},x,V_{*},\mathbb{X})=F(\rho^{*},x+a_{*},V_{*}+\delta\Sigma(\rho^{*}),\mathbb{X}+\delta H_{\mathcal{E}}(\rho^{*}))$ for every $x\in\mathbb{R}$, the last two arguments not depending on $x$, and $(\rho^{*},V_{*}+\delta\Sigma(\rho^{*}))\in\mathcal{V}(\mathcal{D}_{\Sigma})$ as $\rho^{*}\in\mathcal{D}_{\Sigma}$. By (4e3) and Locally Strictly Proper Second-Order Equation Operator on the Wasserstein Space §constant (with $Q=\mathcal{D}_{\Sigma}$ and the values $s_{*}+a_{*}$ and $r_{*}+a_{*}$),

Now $\rho^{*},\nu^{*}\in\mathcal{D}_{\Sigma}$, both ordered pairs $(\rho^{*},\nu^{*})$, $(\nu^{*},\rho^{*})$ are uniquely mapped with the optimal maps $S$ and $S'$, (4e1) holds, $-R\le r_{*}\le R$, $(\mathbb{X},\mathbb{Y})$ is admitted at $\alpha$, $1<\alpha$ and $\delta\in I$. So The Second-Order Structure Condition at Uniquely Mapped Pairs on the Wasserstein Space §pair, for $(\omega_{1},\omega_{2})$ at $R$ with the value slot $r_{*}$, gives

Adding the two displays, and using (4b) and (4d),

Finally (4e2) and $0<\lambda$ give $\lambda M\le\lambda N(s_{*}-r_{*})=N\lambda(s_{*}-r_{*})$ (claim 5 of Elementary Arithmetic in an Ordered Field), and multiplying the last display by $N>0$ gives (4).