Proof of The Half-Relaxed Limits of the Regularised N-Particle Dyson Solutions are Viscosity Sub- and Supersolutions of the Limit Dyson Equation up to a Cost Defect

Each result cited is universally quantified over the data in its own statement, and is applied to the data named here. Throughout, $N_{0}$, $\varepsilon_{0}$, $C_{0}$, $e_{*}$, $p_{0,N}$, $r_{N}$, $\theta_{1,N}$, $\theta_{2,N}$, $g^{\pm}_{N}$, $\overline{w}_{N,\tau}$ and $\underline{w}_{N,\tau}$ are those of Half-Relaxed Limits of the Sup- and Inf-Convolved N-Particle Dyson Solutions: Bounds, Continuity and Limits along Configurations of Bounded Energy, and $e_{0}$, the block test functions $\chi^{\pm}_{N}$, the sets $\mathcal{D}_{N}$, $\mathcal{D}^{\Sigma}_{N}$ and the relative scores $\Sigma_{N}$ are those of Gibbs Maximisers for the Regularised N-Particle Dyson Solutions Tested at a Local Extremum: Energy Bound, Concentration and the Score Identity. For a function $f$ on $W_{N}$, $\bar{f}$ denotes its extension by $0$ to $\mathbb{R}^{N}$. Every law $Q\in\mathcal{D}_{N}$ satisfies $Q(W_{N})=1$ (The Relative Free Energy and the Relative Score of a Probability Measure for a Potential on an Open Set §energy), so integrals against it may be computed on $W_{N}$.

Step 0 (clause 1). Each set whose least upper bound defines $\omega_{g}(s)$ consists of nonnegative numbers and is bounded above by $2b_{g}$, so $0\le\omega_{g}(s)\le2b_{g}$ for $s\ge0$; in particular $0\le h_{\tau}\le2b_{g}$. Let $\varepsilon>0$. If $b_{g}=0$ then $\omega_{g}=0$, so $h_{\tau}=0$ for every $\tau$, and $\tau_{0}=1$ serves. Otherwise, $g$ being uniformly continuous (The N-Particle Dyson Game and its Mean-Field Limit: Standing Data and Equations §data, Uniformly Continuous Map Between Metric Spaces), there is $s_{\varepsilon}>0$ with $|g(\mu)-g(\nu)|<\varepsilon$ whenever $W_{2}(\mu,\nu)<s_{\varepsilon}$; hence every element of the set defining $\omega_{g}(s)$ is less than $\varepsilon$ when $0\le s\le s_{\varepsilon}/2$, and $\omega_{g}(s)\le\varepsilon$ for such $s$. Put $\tau_{0}=\lambda s_{\varepsilon}^{2}/(32b_{g})$. For $0<\tau\le\tau_{0}$, $\sqrt{2}R_{\tau}=(8\tau\lambda^{-1}b_{g})^{1/2}\le(8\tau_{0}\lambda^{-1}b_{g})^{1/2}=s_{\varepsilon}/2$, and $\omega_{g}$ is nondecreasing, so $h_{\tau}\le\varepsilon$.

Clause 2: data and order of choice. Fix $\tau>0$ and write $\bar{u}=\bar{u}_{\tau}$ and $\overline{w}_{N}=\overline{w}_{N,\tau}$. Following Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space §subsolution, fix $\delta$ with $0<\delta<1$, an intrinsic test function $\varphi$ on $\mathcal{D}$, a point $\hat{\mu}\in\mathcal{D}$ at which $\bar{u}^{-}_{\delta}-\varphi$ has a local maximum relative to $\mathcal{D}$, with a radius $\rho>0$ as in Local Maximum of a Function Relative to a Subset of a Metric Space, and $\varepsilon>0$. Put $q=\nabla\varphi(\hat{\mu})$. The choices are then made in this order: the radii $\theta(\eta)$ (Step 2); the realising levels $(N_{k})_{k}$, then for each $j$ the numbers $\eta_{j},r_{j},K_{j}$, the index $k_{j}$ and the law $Q_{j}$ (Step 3); the recovery points $x^{k}$ (Step 6); the witnesses (Step 8).

Step 1 (the envelope is exact). By Half-Relaxed Limits of the Sup- and Inf-Convolved N-Particle Dyson Solutions: Bounds, Continuity and Limits along Configurations of Bounded Energy §modulus, applied in both orders, $|\bar{u}(\mu)-\bar{u}(\mu')|\le\frac{1}{2\tau}W(3W+2R_{\tau})$ with $W=W_{2}(\mu,\mu')$, for $\mu,\mu'\in\mathcal{D}$; so $\bar{u}$ is continuous on $\mathcal{D}$. $\mathcal{E}$ is lower semicontinuous on $\mathcal{D}$ by The Confined Logarithmic-Energy Pair is a Displacement Convex, Wasserstein-Coercive Penalty Pair with Closed Score and Regular Penalised Maxima §growth, and $\bar{u}$ has penalty-subordinate growth from above, as recorded in the statement. By Basic Properties of the Delta-Envelopes on the Wasserstein Space §exact, $\bar{u}^{-}_{\delta}(\nu)=\bar{u}(\nu)-\delta\mathcal{E}(\nu)$ for every $\nu\in\mathcal{D}$, so for every $\nu\in\mathcal{D}$ with $W_{2}(\nu,\hat{\mu})<\rho$

Step 2 (the local linear bound). By property (b) of Intrinsic Test Functions on the Wasserstein Space and Their Translation Hessians §test, $\varphi$ is differentiable along couplings at $\hat{\mu}\in\mathcal{D}$ in the sense of The Intrinsic Calculus on the Wasserstein Space: Standing Notation §gradients, and $q\in T_{\hat{\mu}}\subseteq L^{2}(\hat{\mu};\mathbb{R})$. For $\eta>0$ let $\theta(\eta)>0$ be as in Differentiability of a Function on the Wasserstein Space Along Couplings, and Its Gradient §differentiable at $\hat{\mu}$ with gradient $q$ and with $\eta$ in place of its $\varepsilon$. Let $\nu\in\mathcal{D}$ with $W_{2}(\nu,\hat{\mu})<\min(\rho,\theta(\eta))$ and let $T$ be an optimal map from $\hat{\mu}$ to $\nu$. As recorded in the statement of Gibbs Maximisers for the Regularised N-Particle Dyson Solutions Tested at a Local Extremum: Energy Bound, Concentration and the Score Identity, the class of $T$ lies in $L^{2}(\hat{\mu};\mathbb{R})$ and $\lVert T-\mathrm{id}\rVert_{\hat{\mu}}=W_{2}(\nu,\hat{\mu})$; so The Displacement Pairing of a Square-Integrable Vector Field Along a Coupling §displacement with $S=T$ gives $\pi_{T}=(\mathrm{id},T)_{\#}\hat{\mu}\in\Pi(\hat{\mu},\nu)$ with $I(\pi_{T})=W_{2}(\nu,\hat{\mu})^{2}<\theta(\eta)^{2}$ and $\mathcal{J}(q,\pi_{T})=\langle q,T-\mathrm{id}\rangle_{\hat{\mu}}$. Hence $\varphi(\nu)-\varphi(\hat{\mu})\le\langle q,T-\mathrm{id}\rangle_{\hat{\mu}}+\eta W_{2}(\nu,\hat{\mu})$, and with (1)

This is the hypothesis of Gibbs Maximisers for the Regularised N-Particle Dyson Solutions Tested at a Local Extremum: Energy Bound, Concentration and the Score Identity §sub with $\eta'=\eta$ and $r_{0}=\min(\rho,\theta(\eta))$.

Step 3 (diagonal choice of Gibbs laws). Fix realising levels $(N_{k})_{k}$ for $\bar{u}$ at $\hat{\mu}$ (Half-Relaxed Limits of the Sup- and Inf-Convolved N-Particle Dyson Solutions: Bounds, Continuity and Limits along Configurations of Bounded Energy §realising). For $j\in\mathbb{N}$ put $\eta_{j}=1/j$, $r_{j}=\min(\rho,\theta(1/j))$ and

so that $(\tau,\delta,\hat{\mu},q,\eta_{j},r_{j},K_{j})$ are admissible data of Gibbs Maximisers for the Regularised N-Particle Dyson Solutions Tested at a Local Extremum: Energy Bound, Concentration and the Score Identity and, by Step 2, clause Gibbs Maximisers for the Regularised N-Particle Dyson Solutions Tested at a Local Extremum: Energy Bound, Concentration and the Score Identity §sub applies; let $(\pi^{(j)}_{k})_{k}$ be its Gibbs laws. By Gibbs Maximisers for the Regularised N-Particle Dyson Solutions Tested at a Local Extremum: Energy Bound, Concentration and the Score Identity §sub-energy and Gibbs Maximisers for the Regularised N-Particle Dyson Solutions Tested at a Local Extremum: Energy Bound, Concentration and the Score Identity §sub-concentration, the latter with $1/(jK_{j})$ in place of its $\varepsilon$, there is $k_{0}(j)$ such that both conclusions hold for $k\ge k_{0}(j)$. Choose recursively $k_{j}\ge k_{0}(j)$ with $k_{j}>k_{j-1}$, and put $M_{j}=N_{k_{j}}$, strictly increasing with $M_{1}\ge N_{0}\ge2$, and $Q_{j}=\pi^{(j)}_{k_{j}}$, which lies in $\mathcal{D}^{\Sigma}_{M_{j}}$ by Gibbs Maximisers for the Regularised N-Particle Dyson Solutions Tested at a Local Extremum: Energy Bound, Concentration and the Score Identity §sub-score. Write $W(x)=W_{2}(\mu^{M_{j}}_{x},\hat{\mu})$. Since $K_{j}\ge j\ge1$,

both tending to $0$; with $E_{1}$ as in Gibbs Maximisers for the Regularised N-Particle Dyson Solutions Tested at a Local Extremum: Energy Bound, Concentration and the Score Identity §sub-energy, which does not depend on $j$,

By The Discrete Dyson Energy per Particle: Bounds, Compactness with a Lower Limit, and Recovery Configurations with Separated Particles §bounds, $p_{0,N}\ge e_{*}N$, so (4) gives

Step 4 (limits of the score terms). Let $G_{j}=M_{j}\nabla\chi^{+}_{M_{j}}$, $\chi^{+}_{M_{j}}$ having data $(q,K_{j})$; by Quantile Blocks of an Atomless Measure on the Line: Block Maps, Distances to Ordered Empirical Measures, and the Block Test Function §test-function its $i$-th component is $\bar{q}_{i}+2K_{j}(x_{i}-m_{i})$, an affine function of $x$, so $G_{j}\in L^{2}(Q_{j};\mathbb{R}^{M_{j}})$. By (3) and (4), The Mean-Field Limit of the Relative Score of Laws of Dyson Particles Concentrating at a Measure applies to $(M_{j})_{j}$, $(Q_{j})_{j}$, $\hat{\mu}$ and $A_{*}$; by The Mean-Field Limit of the Relative Score of Laws of Dyson Particles Concentrating at a Measure §domain, $\hat{\mu}\in\mathcal{D}_{\Sigma}$; write $\hat{\Sigma}=\Sigma(\hat{\mu})$. We check the hypothesis of The Mean-Field Limit of the Relative Score of Laws of Dyson Particles Concentrating at a Measure §pairing. Let $\varepsilon'>0$. As $q\in T_{\hat{\mu}}$, the closure of the gradients of test functions (The Tangent Space of the Wasserstein Space at a Probability Measure §tangent), there is $\psi\in C_{c}^{\infty}(\mathbb{R})$ with $\lVert q-\psi'\rVert_{\hat{\mu}}<\varepsilon'/3$; $\psi'$ is bounded and Lipschitz with constant $L_{\psi}=\max|\psi''|$. For $x\in W_{M_{j}}$ (ordered) and $N=M_{j}$: since $\hat{\mu}(B_{i})=1/N$ (Quantile Blocks of an Atomless Measure on the Line: Block Maps, Distances to Ordered Empirical Measures, and the Block Test Function §blocks), $x_{i}-m_{i}=N\int_{B_{i}}(x_{i}-s)\,\hat{\mu}(ds)$ and the Cauchy-Schwarz inequality gives $(x_{i}-m_{i})^{2}\le N\int_{B_{i}}(x_{i}-s)^{2}\hat{\mu}(ds)$, whence $\frac{1}{N}\sum_{i}(x_{i}-m_{i})^{2}\le W(x)^{2}$ by Quantile Blocks of an Atomless Measure on the Line: Block Maps, Distances to Ordered Empirical Measures, and the Block Test Function §optimal; with Quantile Blocks of an Atomless Measure on the Line: Block Maps, Distances to Ordered Empirical Measures, and the Block Test Function §averages for $h=\psi'$ and $(a+b)^{2}\le2a^{2}+2b^{2}$,

which is below $\varepsilon'^{2}$ for all large $j$ by (3). Hence

and by The Mean-Field Limit of the Relative Score of Laws of Dyson Particles Concentrating at a Measure §liminf, for every $\varepsilon'>0$, $\lVert\hat{\Sigma}\rVert_{\hat{\mu}}^{2}\le\alpha_{j}+\varepsilon'$ for all large $j$.

Step 5 (the lifted inequality). Fix $j$, put $N=M_{j}$ and $w=\overline{w}_{N}$. We apply Integrating a Semiconvex Viscosity Subsolution of the Penalty-Drift Equation against a Measure of Finite Relative Fisher Information with $d=N$, $D=W_{N}$, $U=P_{N}$, discount $\lambda$, $\kappa=\kappa_{N}$, $\theta'=\theta_{1,N}$, $\tilde{g}=g^{+}_{N}$, this $w$, and $p_{0}=p_{0,N}$. Its hypotheses hold: $W_{N}$ is open and convex (it is cut out by finitely many strict linear inequalities, The Weyl Chamber of Ordered Points in Euclidean Space); $P_{N}$ is a penalty on $W_{N}$ (The N-Particle Dyson Potential on the Weyl Chamber: Penalty, Monotone Gradient, Dissipation with Constants Uniform in N, and Gibbs Integrability §penalty), hence of class $C^{2}$ (Penalty on an Open Subset of Euclidean Space §regularity); $\lambda>0$ (The N-Particle Dyson Game and its Mean-Field Limit: Standing Data and Equations §data) and $\kappa_{N}>0$ (The N-Particle Dyson Potential on the Weyl Chamber: Penalty, Monotone Gradient, Dissipation with Constants Uniform in N, and Gibbs Integrability §penalty), and $a=\kappa_{N}/2=a_{N}$, so $\mathcal{D}^{\Sigma}_{U,a}=\mathcal{D}^{\Sigma}_{N}$ and $\Sigma_{U,a}=\Sigma_{N}$; $\theta_{1,N}\ge\frac{1}{2}$ as $N\varepsilon_{0}\ge N_{0}\varepsilon_{0}\ge1$; $g^{+}_{N}$ is bounded and Borel on the open set $W_{N}$, so its zero extension is Borel, and $w$ is bounded, semiconvex with constant $\tau^{-1}$ and a viscosity subsolution of the operator in question (Half-Relaxed Limits of the Sup- and Inf-Convolved N-Particle Dyson Solutions: Bounds, Continuity and Limits along Configurations of Bounded Energy §regularised (i), (ii), (v)); and by (iii) there, $\lVert Dw(x)\rVert^{2}\le\tau^{-2}r_{N}(x)^{2}=\frac{4Nb_{g}}{\lambda\tau}+\frac{2}{\tau N}\bigl(P_{N}(x)-p_{0,N}\bigr)$ at every point of differentiability. By Integrating a Semiconvex Viscosity Subsolution of the Penalty-Drift Equation against a Measure of Finite Relative Fisher Information §subsolution at $\mu=Q_{j}$,

The map $\nabla w$ there is Borel (Integrating a Semiconvex Viscosity Subsolution of the Penalty-Drift Equation against a Measure of Finite Relative Fisher Information §borel) and equals $Dw$ wherever $w$ is differentiable; $w$ is twice differentiable, hence differentiable, off a null subset of $W_{N}$ (Alexandrov's Theorem for Semiconvex Functions on an Open Convex Set §ae). So $\nabla w$ is a gradient map of $w$, and Gibbs Maximisers for the Regularised N-Particle Dyson Solutions Tested at a Local Extremum: Energy Bound, Concentration and the Score Identity §sub-score gives $\nabla w=G_{j}+\delta\Sigma_{N}(Q_{j})$ in $L^{2}(Q_{j};\mathbb{R}^{N})$. Expanding the square and the pairing and dividing by $N$:

Step 6 (the zeroth-order terms). (i) $\liminf_{j}\int\bar{w}/M_{j}\,dQ_{j}\ge\bar{u}(\hat{\mu})$. For $m\in\mathbb{N}$, The Discrete Dyson Energy per Particle: Bounds, Compactness with a Lower Limit, and Recovery Configurations with Separated Particles §recovery with $\mu=\hat{\mu}$ and $1/m$ in place of its $\varepsilon$ gives $L_{m}$ such that for every $N\ge L_{m}$ some $y\in W_{N}$ has $W_{2}(\mu^{N}_{y},\hat{\mu})<1/m$ and $P_{N}(y)\le N(\mathcal{E}(\hat{\mu})+1)$; put $L'_{m}=\max(L_{1},\dots,L_{m})$. For $k$ with $N_{k}\ge L'_{1}$ let $m(k)$ be the largest $m\le k$ with $N_{k}\ge L'_{m}$ and let $x^{k}\in W_{N_{k}}$ be such a point for $1/m(k)$; for the finitely many other $k$ ($N_{k}\ge k$) let $x^{k}\in W_{N_{k}}$ be arbitrary. Given $m$, $m(k)\ge m$ whenever $k\ge m$ and $N_{k}\ge L'_{m}$, so $W_{2}(\mu^{N_{k}}_{x^{k}},\hat{\mu})\to0$, and $P_{N_{k}}(x^{k})\le c'N_{k}$ for every $k$, with $c'$ the maximum of $\mathcal{E}(\hat{\mu})+1$ and the finitely many exceptional ratios. By Half-Relaxed Limits of the Sup- and Inf-Convolved N-Particle Dyson Solutions: Bounds, Continuity and Limits along Configurations of Bounded Energy §realising, $\overline{w}_{N_{k}}(x^{k})/N_{k}\to\bar{u}(\hat{\mu})$; put $y^{j}=x^{k_{j}}\in W_{M_{j}}$. For $x\in W_{N}$, $N=M_{j}$, Half-Relaxed Limits of the Sup- and Inf-Convolved N-Particle Dyson Solutions: Bounds, Continuity and Limits along Configurations of Bounded Energy §regularised (iv), first inequality read with $y^{j}$ in place of its $x$ and $x$ in place of its $y$, divided by $N$, gives

the bound on $d_{j}$ by Quantile Blocks of an Atomless Measure on the Line: Block Maps, Distances to Ordered Empirical Measures, and the Block Test Function §optimal. Integrating and using Cauchy-Schwarz, $\int\bar{w}/N\,dQ_{j}\ge w(y^{j})/N-\frac{1}{2\tau}\bigl(3D_{j}+2D_{j}^{1/2}R_{j}^{1/2}\bigr)$ with $D_{j}=\int d_{j}^{2}\,dQ_{j}\le2\int W^{2}dQ_{j}+2W_{2}(\mu^{N}_{y^{j}},\hat{\mu})^{2}\to0$ by (3), and $R_{j}=\int r_{N}^{2}/N\,dQ_{j}=2\tau\bigl(2\lambda^{-1}b_{g}+\int(P_{N}-p_{0,N})/N^{2}\,dQ_{j}\bigr)\le2\tau\bigl(2\lambda^{-1}b_{g}+E_{1}+|e_{*}|\bigr)$ by (5). This proves (i).

(ii) $\limsup_{j}\int\bar{g}^{+}_{M_{j}}/M_{j}\,dQ_{j}\le g(\hat{\mu})+h_{\tau}$. On $W_{N}$, $g^{+}_{N}(x)/N=g(\mu^{N}_{x})+C_{0}/N+\omega_{g}(r_{N}(x)/\sqrt{N})$; $x\mapsto g(\mu^{N}_{x})$ is continuous on $W_{N}$ (The N-Particle Dyson Game and its Mean-Field Limit: Standing Data and Equations §particle-equation) and $g^{+}_{N}$ is Borel, so each term is Borel and bounded. Let $\theta''>0$ and $s>0$ with $|g(\mu)-g(\nu)|<\theta''$ when $W_{2}(\mu,\nu)<s$; then $|g(\mu^{N}_{x})-g(\hat{\mu})|\le\theta''+2b_{g}\mathbf{1}_{\{W\ge s\}}(x)$, so by Markov's inequality $\int g(\mu^{N}_{x})\,Q_{j}(dx)\le g(\hat{\mu})+\theta''+2b_{g}s^{-2}\int W^{2}dQ_{j}$. On $S_{j}=\{x\in W_{N}:(P_{N}(x)-p_{0,N})/N^{2}\le2\lambda^{-1}b_{g}\}$ one has $r_{N}^{2}/N\le8\tau\lambda^{-1}b_{g}=2R_{\tau}^{2}$, so $\omega_{g}(r_{N}/\sqrt{N})\le h_{\tau}$; off $S_{j}$, $\omega_{g}\le2b_{g}$, and as $P_{N}-p_{0,N}\ge0$, Markov's inequality and (5) give $2b_{g}Q_{j}(W_{N}\setminus S_{j})\le\lambda(E_{1}+|e_{*}|)/N$. Altogether

and (ii) follows from (3), $\theta''$ being arbitrary.

Step 7 (passage to the limit). $\theta_{1,M_{j}}=1-\frac{1}{2M_{j}\varepsilon_{0}}\to1$. Put $c=\frac{\delta^{2}}{2}+\delta>0$ and $c_{j}=\frac{\theta_{1,M_{j}}\delta^{2}}{2}+\delta$. As $0\le\alpha_{j}\le A_{*}$, $(c_{j}-c)\alpha_{j}\to0$, and by Step 4 $c\alpha_{j}\ge c(\lVert\hat{\Sigma}\rVert^{2}_{\hat{\mu}}-\varepsilon')$ for large $j$, for each $\varepsilon'>0$; so $\liminf c_{j}\alpha_{j}\ge c\lVert\hat{\Sigma}\rVert_{\hat{\mu}}^{2}$. By Step 4 the third and fourth terms of (6) tend to $(\delta+1)\langle\hat{\Sigma},q\rangle_{\hat{\mu}}$ and $\frac{1}{2}\lVert q\rVert_{\hat{\mu}}^{2}$. Taking the lower limit of (6) with Step 6:

Step 8 (the witnesses). Take $\nu=\hat{\mu}\in\mathcal{D}_{\Sigma}$, $\pi=(\mathrm{id},\mathrm{id})_{\#}\hat{\mu}\in\Pi(\hat{\mu},\hat{\mu})$, $s=\bar{u}^{-}_{\delta}(\hat{\mu})=\bar{u}(\hat{\mu})-\delta\mathcal{E}(\hat{\mu})$ (Step 1), the field $q=\nabla\varphi(\hat{\mu})$ and $Y=H_{\varphi}(\hat{\mu})$. Then $I(\pi)=\lVert\mathrm{id}-\mathrm{id}\rVert_{\hat{\mu}}^{2}=0$ by The Displacement Pairing of a Square-Integrable Vector Field Along a Coupling §displacement; the discrepancy of $q$ and $\nabla\varphi(\hat{\mu})$ along $\pi$ equals $\int|q(t)-q(t)|^{2}\,\hat{\mu}(dt)=0$ by the integration formula for push-forwards; and the remaining three closeness quantities of Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space §subsolution vanish. By The Bundle of Vector Fields over a Set of Measures, Second-Order Equation Operators on the Wasserstein Space, and Their Delta-Shifts §shifted, the formula of the statement for $F^{h_{\tau}}$ (an instance of The Dyson Hamilton-Jacobi Equation with Common Noise and a Confining Potential on the Wasserstein Space of the Real Line §operator with $\kappa=0$, so that the matrix argument plays no role) and $\hat{\Sigma}=V'-\frac{\beta}{4}\Xi_{\hat{\mu}}$ (The Confined Logarithmic-Energy Pair on the Wasserstein Space of the Real Line §pair), the $\delta$-shift $(F^{h_{\tau}})^{-}_{\delta}$ satisfies

which, expanded, is the left side of (7); hence it is at most $0<\varepsilon$. So $\bar{u}_{\tau}$ is a viscosity subsolution of $F^{h_{\tau}}$ relative to the pair.

Clause 3. Fix $\tau$, $\delta\in(0,1)$, $\varphi$, a point $\hat{\mu}$ at which $\underline{u}^{+}_{\delta}-\varphi$ has a local minimum relative to $\mathcal{D}$ with radius $\rho$ (Local Minimum of a Function Relative to a Subset of a Metric Space), and $\varepsilon>0$, as in Viscosity Subsolution, Supersolution and Solution of a Second-Order Equation on the Wasserstein Space §supersolution; write $\underline{u}=\underline{u}_{\tau}$, $w=\underline{w}_{N,\tau}$. The proof of clause 2 applies with the following changes, and no others.

(a) Steps 1-2. $\underline{u}$ is continuous by the second inequality of Half-Relaxed Limits of the Sup- and Inf-Convolved N-Particle Dyson Solutions: Bounds, Continuity and Limits along Configurations of Bounded Energy §modulus, so the second half of Basic Properties of the Delta-Envelopes on the Wasserstein Space §exact gives $\underline{u}^{+}_{\delta}=\underline{u}+\delta\mathcal{E}$ on $\mathcal{D}$; (1) is reversed with $+\delta\mathcal{E}$, differentiability gives $\varphi(\nu)-\varphi(\hat{\mu})\ge\langle q,T-\mathrm{id}\rangle_{\hat{\mu}}-\eta W_{2}(\nu,\hat{\mu})$, and (2) becomes the hypothesis of Gibbs Maximisers for the Regularised N-Particle Dyson Solutions Tested at a Local Extremum: Energy Bound, Concentration and the Score Identity §super.

(b) Steps 3-4. The realising levels are those of the second half of Half-Relaxed Limits of the Sup- and Inf-Convolved N-Particle Dyson Solutions: Bounds, Continuity and Limits along Configurations of Bounded Energy §realising for $\underline{u}$ at $\hat{\mu}$, the Gibbs laws those of Gibbs Maximisers for the Regularised N-Particle Dyson Solutions Tested at a Local Extremum: Energy Bound, Concentration and the Score Identity §super, whose (a), (b), (c) give (3), (4), (5) unchanged and $\nabla w=G_{j}-\delta\Sigma_{N}(Q_{j})$ with $G_{j}=M_{j}\nabla\chi^{-}_{M_{j}}$, of components $\bar{q}_{i}-2K_{j}(x_{i}-m_{i})$ (Quantile Blocks of an Atomless Measure on the Line: Block Maps, Distances to Ordered Empirical Measures, and the Block Test Function §test-function with $-K_{j}$); the estimate of Step 4 is unchanged.

(c) Step 5. Integrating a Semiconvex Viscosity Subsolution of the Penalty-Drift Equation against a Measure of Finite Relative Fisher Information §supersolution is used, with $-w$ semiconvex, $\theta'=\theta_{2,N}$ and $\tilde{g}=g^{-}_{N}$ (Half-Relaxed Limits of the Sup- and Inf-Convolved N-Particle Dyson Solutions: Bounds, Continuity and Limits along Configurations of Bounded Energy §regularised (i), (ii), (iii), (v)); $\nabla w$ is a gradient map by Alexandrov's Theorem for Semiconvex Functions on an Open Convex Set §ae applied to $-w$. Inequality (6) becomes

(d) Step 6. (i) becomes $\limsup_{j}\int\bar{w}/M_{j}\,dQ_{j}\le\underline{u}(\hat{\mu})$: the recovery points are built identically, and the second inequality of Half-Relaxed Limits of the Sup- and Inf-Convolved N-Particle Dyson Solutions: Bounds, Continuity and Limits along Configurations of Bounded Energy §regularised (iv), with the same substitution, gives $w(x)/N\le w(y^{j})/N+\frac{1}{2\tau}(3d_{j}^{2}+2d_{j}r_{N}/\sqrt{N})$. (ii) becomes $\liminf_{j}\int\bar{g}^{-}_{M_{j}}/M_{j}\,dQ_{j}\ge g(\hat{\mu})-h_{\tau}$, since $g^{-}_{N}/N=g(\mu^{N}_{x})-C_{0}/N-\omega_{g}(r_{N}/\sqrt{N})$ and the same bounds apply.

(e) Step 7. $\theta_{2,M_{j}}=1+\frac{1}{M_{j}\varepsilon_{0}}\to1$, and the limit coefficient of $\alpha_{j}$ is $c=\frac{\delta^{2}}{2}-\delta=\delta(\frac{\delta}{2}-1)<0$, so $\limsup c_{j}\alpha_{j}=c\liminf\alpha_{j}\le c\lVert\hat{\Sigma}\rVert_{\hat{\mu}}^{2}$. Taking the upper limit, (7) becomes

(f) Step 8. The witnesses are the same with $s=\underline{u}^{+}_{\delta}(\hat{\mu})=\underline{u}(\hat{\mu})+\delta\mathcal{E}(\hat{\mu})$, and $(F^{-h_{\tau}})^{+}_{\delta}(\hat{\mu},s,q,Y)=\lambda\underline{u}(\hat{\mu})+\frac{1}{2}\lVert q-\delta\hat{\Sigma}\rVert_{\hat{\mu}}^{2}+\langle\hat{\Sigma},q-\delta\hat{\Sigma}\rangle_{\hat{\mu}}-g(\hat{\mu})+h_{\tau}$, which, expanded, is the right side above; hence $-\varepsilon<0\le(F^{-h_{\tau}})^{+}_{\delta}(\hat{\mu},s,q,Y)$, and $\underline{u}_{\tau}$ is a viscosity supersolution of $F^{-h_{\tau}}$ relative to the pair.