Proof of Gibbs Maximisers for the Regularised N-Particle Dyson Solutions Tested at a Local Extremum: Energy Bound, Concentration and the Score Identity

Each result cited is universally quantified over the data in its own statement, and is applied to the data named here. We prove clause 1 in full and then list the changes for clause 2.

Notation. The realising levels of Half-Relaxed Limits of the Sup- and Inf-Convolved N-Particle Dyson Solutions: Bounds, Continuity and Limits along Configurations of Bounded Energy §realising are strictly increasing with $N_{1}\ge N_{0}\ge2$, so $N_{k}\to\infty$. For a fixed $k$ we write $N=N_{k}$, $w=\overline{w}_{N,\tau}$, $\chi=\chi^{+}_{N}$, $p_{0}=p_{0,N}$, $W(x)=W_{2}(\mu^{N}_{x},\hat{\mu})$, $\lVert\cdot\rVert$ for the Euclidean norm of $\mathbb{R}^{N}$ and $\lVert\cdot\rVert_{\hat{\mu}}$, $\langle\cdot,\cdot\rangle_{\hat{\mu}}$ for those of $L^{2}(\hat{\mu};\mathbb{R})$; $B_{i}$, $T_{x}$, $\bar{q}_{i}$, $m_{i}$ are as in Quantile Blocks of an Atomless Measure on the Line: Block Maps, Distances to Ordered Empirical Measures, and the Block Test Function. Put

and, for $P\in\mathcal{D}_{N}$ such that $w-N\chi$ is $P$-integrable (integrals of functions on $W_{N}$ are over $W_{N}$, which has full $P$-measure), $\Phi_{k}(P)=\int(w-N\chi)\,dP-\delta\,\mathcal{E}_{N}(P)=\delta\bigl(\int\bar{f}_{k}\,dP-\mathcal{E}_{N}(P)\bigr)$, with $\bar{f}_{k}$ equal to $f_{k}$ on $W_{N}$ and to $0$ off it. Since $\mathcal{E}_{N}(P)=a_{N}\mathrm{Ent}(P)+\int P_{N}\,dP$ by The Relative Free Energy and the Relative Score of a Probability Measure for a Potential on an Open Set §energy,

By Half-Relaxed Limits of the Sup- and Inf-Convolved N-Particle Dyson Solutions: Bounds, Continuity and Limits along Configurations of Bounded Energy §bounds and $0<\delta<1$, $|\hat{m}|\le\lambda^{-1}b_{g}+|\mathcal{E}(\hat{\mu})|$. Let $e_{*},A_{0}$ be as in The Discrete Dyson Energy per Particle: Bounds, Compactness with a Lower Limit, and Recovery Configurations with Separated Particles §bounds; then $e_{*}N\le P_{N}$ on $W_{N}$, hence $e_{*}N\le p_{0}$.

Facts on the block test function. (F1) Expanding the formula of Quantile Blocks of an Atomless Measure on the Line: Block Maps, Distances to Ordered Empirical Measures, and the Block Test Function §test-function with $\hat{\mu}(B_{i})=\frac1N$ (Quantile Blocks of an Atomless Measure on the Line: Block Maps, Distances to Ordered Empirical Measures, and the Block Test Function §blocks) gives $N\chi(x)=K\lVert x\rVert^{2}+\sum_{i}(\bar{q}_{i}-2Km_{i})x_{i}+c_{\chi}$ for a real $c_{\chi}$. (F2) By Cauchy-Schwarz on each block, $\bar{q}_{i}^{2}\le N\int_{B_{i}}q^{2}\,d\hat{\mu}$ and $(x_{i}-m_{i})^{2}=\bigl(N\int_{B_{i}}(x_{i}-s)\,\hat{\mu}(ds)\bigr)^{2}\le N\int_{B_{i}}(x_{i}-s)^{2}\,\hat{\mu}(ds)$; summing, and using Quantile Blocks of an Atomless Measure on the Line: Block Maps, Distances to Ordered Empirical Measures, and the Block Test Function §optimal for ordered $x$, $\sum_{i}\bar{q}_{i}^{2}\le N\lVert q\rVert_{\hat{\mu}}^{2}$ and $\sum_{i}(x_{i}-m_{i})^{2}\le NW(x)^{2}$. With the gradient formula of Quantile Blocks of an Atomless Measure on the Line: Block Maps, Distances to Ordered Empirical Measures, and the Block Test Function §test-function, for ordered $x$

(F4) For ordered $x$ (in particular $x\in W_{N}$), Quantile Blocks of an Atomless Measure on the Line: Block Maps, Distances to Ordered Empirical Measures, and the Block Test Function §optimal with $\nu=\hat{\mu}$ and $T=\mathrm{id}$ gives $W(x)=\lVert T_{x}-\mathrm{id}\rVert_{\hat{\mu}}$, and $W(x)^{2}=\sum_{i}\int_{B_{i}}(x_{i}-s)^{2}\,\hat{\mu}(ds)$ is a polynomial in $x$; by Quantile Blocks of an Atomless Measure on the Line: Block Maps, Distances to Ordered Empirical Measures, and the Block Test Function §test-function, $\chi(x)=\langle q,T_{x}-\mathrm{id}\rangle_{\hat{\mu}}+KW(x)^{2}$, so $-\lVert q\rVert_{\hat{\mu}}W(x)+KW(x)^{2}\le\chi(x)\le\lVert q\rVert_{\hat{\mu}}W(x)+KW(x)^{2}$ and $\chi(x)\ge-\lVert q\rVert_{\hat{\mu}}^{2}/(4K)\ge-\lVert q\rVert_{\hat{\mu}}^{2}$ (as $K\ge1$). Also $NW(x)^{2}\le2\lVert x\rVert^{2}+2NM_{2}(\hat{\mu})$.

Step 1 (hypotheses; optimal maps; clause (a)). $W_{N}$ (The Weyl Chamber of Ordered Points in Euclidean Space) is open and convex, being a finite intersection of open half-spaces, and nonempty, containing $(N,N-1,\dots,1)$. $P_{N}$ is a penalty on $W_{N}$ with monotone gradient (The N-Particle Dyson Potential on the Weyl Chamber: Penalty, Monotone Gradient, Dissipation with Constants Uniform in N, and Gibbs Integrability §penalty), and the Gibbs integrability function of Gibbs Maximisers of the Relative Free Energy: the Variational Principle and the Relative Score for $U=P_{N}$, $a=a_{N}$ is the function $g_{N}$ of The N-Particle Dyson Potential on the Weyl Chamber: Penalty, Monotone Gradient, Dissipation with Constants Uniform in N, and Gibbs Integrability §gibbs, which is integrable. By Half-Relaxed Limits of the Sup- and Inf-Convolved N-Particle Dyson Solutions: Bounds, Continuity and Limits along Configurations of Bounded Energy §regularised (ii) and Semiconvex Function on a Convex Subset of $\mathbb{R}^n$, $w+\frac{1}{2\tau}\lVert x\rVert^{2}$ is convex on $W_{N}$; by (F1), $-N\chi+K\lVert x\rVert^{2}$ is affine; so $f_{k}+\frac{1}{2}\delta^{-1}(\tau^{-1}+2K)\lVert x\rVert^{2}=\delta^{-1}\bigl[(w+\frac{1}{2\tau}\lVert x\rVert^{2})+(-N\chi+K\lVert x\rVert^{2})\bigr]$ is convex, and $f_{k}$ is semiconvex on $W_{N}$ with constant $\delta^{-1}(\tau^{-1}+2K)$. By Half-Relaxed Limits of the Sup- and Inf-Convolved N-Particle Dyson Solutions: Bounds, Continuity and Limits along Configurations of Bounded Energy §regularised (i) and (F4), $f_{k}\le\delta^{-1}(\lambda^{-1}Nb_{g}+|e_{*}|+N\lVert q\rVert_{\hat{\mu}}^{2})$ on $W_{N}$. Since $\chi$ is $C^{2}$, $f_{k}$ is differentiable at $x\in W_{N}$ exactly when $w$ is, and then $Df_{k}(x)=\delta^{-1}(Dw(x)-N\nabla\chi(x))$. At such $x$, Half-Relaxed Limits of the Sup- and Inf-Convolved N-Particle Dyson Solutions: Bounds, Continuity and Limits along Configurations of Bounded Energy §regularised (iii) gives $\lVert Dw(x)\rVert^{2}\le\tau^{-2}r_{N}(x)^{2}=\frac{2}{\tau}\bigl(2\lambda^{-1}Nb_{g}+\frac{P_{N}(x)-p_{0}}{N}\bigr)$, and (F3), (F4) and The Discrete Dyson Energy per Particle: Bounds, Compactness with a Lower Limit, and Recovery Configurations with Separated Particles §bounds give $\lVert N\nabla\chi(x)\rVert^{2}\le2N\lVert q\rVert_{\hat{\mu}}^{2}+8K^{2}\bigl(2A_{0}(N+p_{0}-e_{*}N)+2A_{0}(P_{N}(x)-p_{0})+2NM_{2}(\hat{\mu})\bigr)$. Hence $\lVert Df_{k}(x)\rVert^{2}\le2\delta^{-2}(\lVert Dw(x)\rVert^{2}+\lVert N\nabla\chi(x)\rVert^{2})\le A+B(P_{N}(x)-p_{0})$ for nonnegative $A,B$ depending on $k$. All hypotheses of Gibbs Maximisers of the Relative Free Energy: the Variational Principle and the Relative Score hold, so $\pi_{k}$ is defined, $\pi_{k}\in\mathcal{D}_{N}$, $\pi_{k}(W_{N})=1$ and $\Phi_{k}(\pi_{k})=\delta a_{N}\log Z$ (Gibbs Maximisers of the Relative Free Energy: the Variational Principle and the Relative Score §gibbs), and $\Phi_{k}(P)\le\delta a_{N}\log Z=\Phi_{k}(\pi_{k})$ for every admissible $P$ (Gibbs Maximisers of the Relative Free Energy: the Variational Principle and the Relative Score §variational, multiplied by $\delta$). Moreover $G_{k}$ is $\pi_{k}$-integrable: $w$ is bounded (Half-Relaxed Limits of the Sup- and Inf-Convolved N-Particle Dyson Solutions: Bounds, Continuity and Limits along Configurations of Bounded Energy §regularised (i)), $\chi$ is a polynomial of degree at most $2$ and $\pi_{k}\in\mathcal{P}_{2}(\mathbb{R}^{N})$, and $P_{N}$ is $\pi_{k}$-integrable as $\pi_{k}\in\mathcal{D}_{N}$ (The Relative Free Energy and the Relative Score of a Probability Measure for a Potential on an Open Set §energy).

Optimal maps. Let $\nu\in\mathcal{P}_{2}(\mathbb{R})$ and let $T$ be an optimal map from $\hat{\mu}$ to $\nu$. By the change-of-variables formula of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward, $\int T^{2}\,d\hat{\mu}=\int t^{2}\,\nu(dt)<\infty$, so the class of $T$ lies in $L^{2}(\hat{\mu};\mathbb{R})$, and the quadratic cost of $(\mathrm{id},T)_{\#}\hat{\mu}$ is $\int|s-T(s)|^{2}\,\hat{\mu}(ds)=\lVert T-\mathrm{id}\rVert_{\hat{\mu}}^{2}$; this coupling being optimal, $\lVert T-\mathrm{id}\rVert_{\hat{\mu}}=W_{2}(\hat{\mu},\nu)=W_{2}(\nu,\hat{\mu})$.

Clause (a). By Gibbs Maximisers of the Relative Free Energy: the Variational Principle and the Relative Score §score, $\pi_{k}\in\mathcal{D}^{\Sigma}_{N}$ and $\Sigma_{N}(\pi_{k})=\nabla f_{k}$ in $L^{2}(\pi_{k};\mathbb{R}^{N})$ for every gradient map $\nabla f_{k}$ of $f_{k}$. Let $\nabla w$ be a gradient map of $w$ (Gradient Maps of a Function on an Open Subset of Euclidean Space §gradient-map), with null set $N'$. The map $\delta^{-1}(\nabla w-N\nabla\chi)$ is Borel ($\nabla\chi$ being continuous), and at every $x\in W_{N}\setminus N'$ the function $w$, hence $f_{k}$, is differentiable with $Df_{k}(x)=\delta^{-1}(\nabla w(x)-N\nabla\chi(x))$; so it is a gradient map of $f_{k}$, and $\delta^{-1}(\nabla w-N\nabla\chi)=\Sigma_{N}(\pi_{k})$ in $L^{2}(\pi_{k};\mathbb{R}^{N})$. As $N\nabla\chi$ is affine and $\pi_{k}\in\mathcal{P}_{2}(\mathbb{R}^{N})$, its class lies in $L^{2}(\pi_{k};\mathbb{R}^{N})$, hence so does that of $\nabla w=N\nabla\chi+\delta\nabla f_{k}$, and $\nabla w=N\nabla\chi+\delta\Sigma_{N}(\pi_{k})$ there.

Step 2 (value lower bound). Claim: for every positive $\varepsilon$ there is $k_{0}$ with $\Phi_{k}(\pi_{k})/N\ge\hat{m}-\varepsilon$ for every $k\ge k_{0}$.

Diagonal choice. For each natural $j\ge1$, The Discrete Dyson Energy per Particle: Bounds, Compactness with a Lower Limit, and Recovery Configurations with Separated Particles §recovery applied to $\hat{\mu}$ and $1/j$ gives positive $c_{j},R_{j}$ and $N_{1}(j)\ge2$; replacing $c_{j}$ by $\min(c_{j},1)$ keeps its gap condition, so we assume $c_{j}\le1$. Let $L_{j}$ be the maximum of $|V'|$ on $[-R_{j}-1,R_{j}+1]$ ($V'$ is continuous, $V$ being $C^{2}$ by The N-Particle Dyson Potential on the Weyl Chamber: Penalty, Monotone Gradient, Dissipation with Constants Uniform in N, and Gibbs Integrability §hypotheses) and $\Lambda_{j}=\log(1/c_{j})+L_{j}\ge0$. For each $k$ let $S_{k}$ be the set of $j\in[k]$ with $N_{k}\ge N_{1}(i)$ and $N_{k}\ge i(1+\Lambda_{i})$ for every $i\in[j]$. Since $N_{k}\to\infty$, each $j$ lies in $S_{k}$ for all large $k$; let $k_{1}$ be such that $1\in S_{k}$ for $k\ge k_{1}$, and put $j(k)=\max S_{k}$ for $k\ge k_{1}$. Then $j(k)\to\infty$, $N_{k}\ge N_{1}(j(k))$ and $\Lambda_{j(k)}\le N_{k}/j(k)$. For $k\ge k_{1}$ let $y^{k}\in W_{N_{k}}$ be the point of The Discrete Dyson Energy per Particle: Bounds, Compactness with a Lower Limit, and Recovery Configurations with Separated Particles §recovery for $j=j(k)$ and $N=N_{k}$, and for $k<k_{1}$ any point of $W_{N_{k}}$. Then $W(y^{k})<1/j(k)\to0$ and $P_{N_{k}}(y^{k})\le c'N_{k}$ for all $k$, with $c'$ the largest of $\mathcal{E}(\hat{\mu})+1$ and the finitely many $P_{N_{k}}(y^{k})/N_{k}$, $k<k_{1}$; so $w(y^{k})/N_{k}\to\bar{u}_{\tau}(\hat{\mu})$ by Half-Relaxed Limits of the Sup- and Inf-Convolved N-Particle Dyson Solutions: Bounds, Continuity and Limits along Configurations of Bounded Energy §realising.

Fix $k\ge k_{1}$ and write $j=j(k)$, $c=c_{j}$, $L=L_{j}$, $\Lambda=\Lambda_{j}$, $y=y^{k}$, $h=c/(4N)\le1/(4N)$ and $Q=\{z\in\mathbb{R}^{N}:|z_{i}-y_{i}|\le h\text{ for every }i\}$. For $i<l$, summing gaps, $y_{i}-y_{l}\ge(l-i)c/N\ge4h$, so for $z\in Q$, $z_{i}-z_{l}\ge y_{i}-y_{l}-2h\ge\frac12(y_{i}-y_{l})>0$; thus $Q\subseteq W_{N}$. Also $\lVert z-y\rVert\le h\sqrt{N}$ for $z\in Q$.

(2a) Energy on $Q$. Let $z\in Q$ and $i<l$; with $t=(z_{i}-z_{l})/(y_{i}-y_{l})$ and $u=2h/(y_{i}-y_{l})\le\frac12$ we have $t\ge1-u$, and by The Function $s\log s$: Continuity, Young's Inequality and Lower Bounds, with the Elementary Bounds for the Exponential and the Logarithm §log, $-\log t\le t^{-1}-1\le u/(1-u)\le2u=4h/(y_{i}-y_{l})\le1/(l-i)$. By The Logarithmic Energy of N Ordered Particles on the Weyl Chamber §energy with strength $b_{N}=\beta/(2(N-1))$,

using $\frac1m\le\log\frac{m}{m-1}$ for $m\ge2$ (The Function $s\log s$: Continuity, Young's Inequality and Lower Bounds, with the Elementary Bounds for the Exponential and the Logarithm §log). As $|z_{i}|,|y_{i}|\le R_{j}+1$, the mean value theorem gives $|V(z_{i})-V(y_{i})|\le Lh\le L/(4N)\le\Lambda/N\le1/j$. Hence, with the bound on $P_{N}(y)$ from The Discrete Dyson Energy per Particle: Bounds, Compactness with a Lower Limit, and Recovery Configurations with Separated Particles §recovery,

using $1+\log N\le N\le2(N-1)$ (The Function $s\log s$: Continuity, Young's Inequality and Lower Bounds, with the Elementary Bounds for the Exponential and the Logarithm §log).

(2b) $w$ on $Q$. By 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 pair $(y,z)$ in place of $(x,y)$, $w(y)\le w(z)+\frac{1}{2\tau}\lVert y-z\rVert(3\lVert y-z\rVert+2r_{N}(z))$, so $w(z)/N\ge w(y)/N-\frac{1}{2\tau}(3h^{2}+2hr_{N}(z)/\sqrt{N})$. By (2a) and $p_{0}\ge e_{*}N$, $r_{N}(z)^{2}/N=2\tau\bigl(2\lambda^{-1}b_{g}+(P_{N}(z)-p_{0})/N^{2}\bigr)\le2\tau(2\lambda^{-1}b_{g}+\bar{e}-e_{*})=:\rho^{2}$, so with $h\le1$, $w(z)/N\ge w(y)/N-(3+2\rho)h/(2\tau)$.

(2c) $\chi$ on $Q$. By Quantile Blocks of an Atomless Measure on the Line: Block Maps, Distances to Ordered Empirical Measures, and the Block Test Function §block-maps, $\lVert T_{z}-T_{y}\rVert_{\hat{\mu}}=\lVert z-y\rVert/\sqrt{N}\le h$; with (F4), $\chi(z)\le\chi(y)+\lVert q\rVert_{\hat{\mu}}h+Kh(2W(y)+h)$ and $\chi(y)\le\lVert q\rVert_{\hat{\mu}}W(y)+KW(y)^{2}$.

(2d) The competitor. Let $P$ be the measure with density $(2h)^{-N}\mathbf{1}_{Q}$ with respect to Lebesgue measure ($Q$ has Lebesgue measure $(2h)^{N}$). It is a probability measure with bounded support, so $P\in\mathcal{P}_{2}(\mathbb{R}^{N})$, with $P(W_{N})=1$; by The Entropy of a Probability Measure on Euclidean Space §entropy (with $\phi(0)=0$) it has finite entropy $\mathrm{Ent}(P)=-N\log(2h)=N\log(2N/c)$; $P_{N}$ is bounded on $Q$ by (2a), so $P\in\mathcal{D}_{N}$; and $w-N\chi$ is bounded on $Q$ by Half-Relaxed Limits of the Sup- and Inf-Convolved N-Particle Dyson Solutions: Bounds, Continuity and Limits along Configurations of Bounded Energy §regularised (i) and (2c). By Step 1, (0) and (2a)--(2c),

As $k\to\infty$: $w(y^{k})/N_{k}\to\bar{u}_{\tau}(\hat{\mu})$, $W(y^{k})\to0$, $h\le1/(4N_{k})\to0$, $e_{k}\to\mathcal{E}(\hat{\mu})$, $a_{N}\log(2N)\to0$, and $a_{N}\Lambda\le\sigma^{2}\Lambda/N\le\sigma^{2}/j(k)\to0$. The right side tends to $\hat{m}$, which proves the claim.

Step 3 (entropy bound; clause (b)). Let $c_{1}$ be the Gaussian constant of Basic Properties of the Entropy on the Wasserstein Space: Comparison with the Gaussian Relative Entropy, Lower Bound, Translation Invariance, Absolute Continuity, Closed Sublevel Sets and Lower Semicontinuity in dimension $N$. By The Gaussian Kernels on Euclidean Space: Scaling, Derivatives up to Order Three, the Convolution Identity, Moments and Tails §moments with $q=N$, $s=1$ and $R=\sqrt{2N}$, the Gaussian weight $g_{1}\le c_{1}$ has mass at most $\frac12$ outside the ball of radius $R$, which lies in the cube $[-R,R]^{N}$ of Lebesgue measure $(2R)^{N}$; so $\frac12\le c_{1}(2R)^{N}$ and $\log c_{1}\ge-\log2-\frac N2\log(8N)$. For $P\in\mathcal{D}_{N}$ put $X(P)=\frac1N\int P_{N}\,dP$; by The Discrete Dyson Energy per Particle: Bounds, Compactness with a Lower Limit, and Recovery Configurations with Separated Particles §bounds, $M_{2}(P)\le A_{0}N(1+X(P)-e_{*})$, and Basic Properties of the Entropy on the Wasserstein Space: Comparison with the Gaussian Relative Entropy, Lower Bound, Translation Invariance, Absolute Continuity, Closed Sublevel Sets and Lower Semicontinuity §lower gives

where $\varepsilon_{N},\theta_{N}\to0$ as $k\to\infty$. Write $X_{k}=X(\pi_{k})$. By (0), (3a) and Step 2, for every positive $\varepsilon$ and $k\ge k_{0}(\varepsilon)$,

By Half-Relaxed Limits of the Sup- and Inf-Convolved N-Particle Dyson Solutions: Bounds, Continuity and Limits along Configurations of Bounded Energy §regularised (i) and (F4), $G_{k}\le\lambda^{-1}b_{g}+|e_{*}|+\lVert q\rVert_{\hat{\mu}}^{2}-\delta P_{N}/N$ on $W_{N}$, so

Energy bound. Let $C_{1}=2\lambda^{-1}b_{g}+|e_{*}|+|\mathcal{E}(\hat{\mu})|+\lVert q\rVert_{\hat{\mu}}^{2}$, so $E_{1}=\delta^{-1}(C_{1}+2)$. Take $\varepsilon=\frac12$ in (3b) and $k_{0}\ge k_{0}(\frac12)$ with $\varepsilon_{N}\le\frac12$ and $\theta_{N}(C_{1}+2)\le1$ for $k\ge k_{0}$. For such $k$, (3b), (3c), $-\hat{m}\le\lambda^{-1}b_{g}+|\mathcal{E}(\hat{\mu})|$ and $\delta<1$ give $\delta X_{k}\le C_{1}+1+\delta\theta_{N}X_{k}$. If $X_{k}\le0$ then $X_{k}\le E_{1}$; otherwise $\delta X_{k}\le(C_{1}+1)/(1-\theta_{N})\le C_{1}+2$, as $\theta_{N}(C_{1}+2)\le1$. So $X_{k}\le E_{1}$.

Fisher bound. Let $\nabla f_{k}$ be a gradient map of $f_{k}$ (Gibbs Maximisers of the Relative Free Energy: the Variational Principle and the Relative Score §score) with null set $N'$. Since $\pi_{k}$ has a density with respect to Lebesgue measure, $\pi_{k}(N')=0$; and $\pi_{k}(W_{N})=1$. For $x\in W_{N}\setminus N'$, by Step 1, (iii) of Half-Relaxed Limits of the Sup- and Inf-Convolved N-Particle Dyson Solutions: Bounds, Continuity and Limits along Configurations of Bounded Energy §regularised and (F3),

Integrating against $\pi_{k}$, with $\lVert\Sigma_{N}(\pi_{k})\rVert_{\pi_{k}}^{2}=\int\lVert\nabla f_{k}\rVert^{2}\,d\pi_{k}$ and $0\le\frac1{N^{2}}\int(P_{N}-p_{0})\,d\pi_{k}\le X_{k}-p_{0}/N\le E_{1}+|e_{*}|$ (as $N\ge1$, $p_{0}\ge e_{*}N$), gives the Fisher bound, since $\frac2\tau(2\lambda^{-1}b_{g}+E_{1}+|e_{*}|)\le\frac4\tau(\lambda^{-1}b_{g}+E_{1}+|e_{*}|)$.

Step 4 (uniform upper estimate). Claim: for every positive $\varepsilon$ there is $k_{0}$ such that for every $k\ge k_{0}$ and every $x\in W_{N}$

Suppose not. Then there are a positive $\varepsilon$, a strictly increasing sequence $(k_{l})_{l}$ and $x^{l}\in W_{N_{k_{l}}}$ violating the inequality at $k=k_{l}$; write $N=N_{k_{l}}$ and $W_{l}=W(x^{l})$. By Half-Relaxed Limits of the Sup- and Inf-Convolved N-Particle Dyson Solutions: Bounds, Continuity and Limits along Configurations of Bounded Energy §regularised (i) and (F4), $G_{k}(x)\le\lambda^{-1}b_{g}+|e_{*}|+\lVert q\rVert_{\hat{\mu}}W(x)-KW(x)^{2}-\delta P_{N}(x)/N$, so the violation gives

So $P_{N_{k_{l}}}(x^{l})\le c''N_{k_{l}}$ for a constant $c''$, and The Discrete Dyson Energy per Particle: Bounds, Compactness with a Lower Limit, and Recovery Configurations with Separated Particles §liminf gives a further strictly increasing subsequence, along which we continue to index by $l$, and $\nu\in\mathcal{D}$ with $W_{2}(\mu^{N}_{x^{l}},\nu)\to0$ and, for every positive $\varepsilon''$, $\mathcal{E}(\nu)\le P_{N}(x^{l})/N+\varepsilon''$ for large $l$. By Half-Relaxed Limits of the Sup- and Inf-Convolved N-Particle Dyson Solutions: Bounds, Continuity and Limits along Configurations of Bounded Energy §sequences (the levels $N_{k_{l}}$ being strictly increasing and at least $N_{0}$), $w(x^{l})/N\le\bar{u}_{\tau}(\nu)+\varepsilon''$ for large $l$. By On the Real Line an Atomless Source is Uniquely Mapped, by a Nondecreasing Optimal Map §monotone ($\hat{\mu}$ atomless) there is an optimal map $T$ from $\hat{\mu}$ to $\nu$, Borel and nondecreasing on a Borel set of full $\hat{\mu}$-measure; Quantile Blocks of an Atomless Measure on the Line: Block Maps, Distances to Ordered Empirical Measures, and the Block Test Function §optimal gives $\lVert T_{x^{l}}-T\rVert_{\hat{\mu}}=W_{2}(\mu^{N}_{x^{l}},\nu)\to0$. With (F4) and Step 1, $W_{l}=\lVert T_{x^{l}}-\mathrm{id}\rVert_{\hat{\mu}}\to\lVert T-\mathrm{id}\rVert_{\hat{\mu}}=W_{\nu}:=W_{2}(\nu,\hat{\mu})$ and $\chi(x^{l})\to\langle q,T-\mathrm{id}\rangle_{\hat{\mu}}+KW_{\nu}^{2}$. Letting $l\to\infty$ in the violated inequality and then $\varepsilon''\to0$,

If $W_{\nu}<r_{0}$, the hypothesis of clause 1 and $\eta'W_{\nu}\le\frac{\eta'^{2}}{2K}+\frac K2W_{\nu}^{2}$ bound the left side of (4a) by $\hat{m}+\frac{\eta'^{2}}{2K}-\frac K2W_{\nu}^{2}$, whence $\varepsilon\le-\frac K4W_{\nu}^{2}\le0$, a contradiction. If $W_{\nu}\ge r_{0}$, then $\bar{u}_{\tau}(\nu)\le\lambda^{-1}b_{g}$ (Half-Relaxed Limits of the Sup- and Inf-Convolved N-Particle Dyson Solutions: Bounds, Continuity and Limits along Configurations of Bounded Energy §bounds), $-\delta\mathcal{E}(\nu)\le|e_{0}|$, and $-\langle q,T-\mathrm{id}\rangle_{\hat{\mu}}\le\lVert q\rVert_{\hat{\mu}}W_{\nu}\le\frac K4W_{\nu}^{2}+\frac{\lVert q\rVert_{\hat{\mu}}^{2}}{K}$; with $-\hat{m}\le\lambda^{-1}b_{g}+|\mathcal{E}(\hat{\mu})|$ and $K\ge1$, (4a) gives $\varepsilon\le S-\frac K2W_{\nu}^{2}\le S-\frac K2r_{0}^{2}\le0$, where $S=2\lambda^{-1}b_{g}+|e_{0}|+|\mathcal{E}(\hat{\mu})|+\lVert q\rVert_{\hat{\mu}}^{2}$ and $Kr_{0}^{2}\ge2S$ by hypothesis; a contradiction.

Step 5 (clause (c)). Let $\varepsilon>0$ and $\varepsilon'=\varepsilon/12$. Choose, in this order, $k_{0}$ from Step 4 for $\varepsilon'$, then $k_{0}'\ge k_{0}$ at least $k_{0}(\varepsilon')$ of (3b) and the $k_{0}$ of the energy bound, and such that $\varepsilon_{N}+\theta_{N}E_{1}\le\varepsilon'$ for $k\ge k_{0}'$. For $k\ge k_{0}'$, (3b) with $X_{k}\le E_{1}$ gives $\int G_{k}\,d\pi_{k}\ge\hat{m}-2\varepsilon'$, while integrating Step 4 against $\pi_{k}$ (all terms are integrable, $W^{2}$ being a polynomial on $W_{N}$ by (F4)) gives $\int G_{k}\,d\pi_{k}\le\hat{m}+\frac{\eta'^{2}}{2K}+\varepsilon'-\frac K4\int W^{2}\,d\pi_{k}$. Hence $K\int W^{2}\,d\pi_{k}\le\frac{2\eta'^{2}}{K}+12\varepsilon'=\frac{2\eta'^{2}}{K}+\varepsilon$.

Clause 2 (supersolution side). The proof is the same with the following changes. $w=\underline{w}_{N,\tau}$, $\chi=\chi^{-}_{N}$ (data $(q,-K)$), $f_{k}=\delta^{-1}(N\chi-w)$, $\hat{m}=-(\underline{u}_{\tau}(\hat{\mu})+\delta\mathcal{E}(\hat{\mu}))$, $G_{k}=-w/N+\chi-\delta P_{N}/N$ and $\Phi_{k}(P)=\int(N\chi-w)\,dP-\delta\mathcal{E}_{N}(P)$, so that (0) holds and maximising $\Phi_{k}$ is minimising $\int(w-N\chi)\,dP+\delta\mathcal{E}_{N}(P)$; $|\hat{m}|\le\lambda^{-1}b_{g}+|\mathcal{E}(\hat{\mu})|$ by Half-Relaxed Limits of the Sup- and Inf-Convolved N-Particle Dyson Solutions: Bounds, Continuity and Limits along Configurations of Bounded Energy §bounds. In (F1), $N\chi(x)=-K\lVert x\rVert^{2}+\sum_{i}(\bar{q}_{i}+2Km_{i})x_{i}+c_{\chi}$; in (F3), $\bar{q}_{i}-2K(x_{i}-m_{i})$ replaces $\bar{q}_{i}+2K(x_{i}-m_{i})$, with the same bound; in (F4), $\chi(x)=\langle q,T_{x}-\mathrm{id}\rangle_{\hat{\mu}}-KW(x)^{2}$, so $-\lVert q\rVert_{\hat{\mu}}W-KW^{2}\le\chi\le\lVert q\rVert_{\hat{\mu}}W-KW^{2}\le\lVert q\rVert_{\hat{\mu}}^{2}$. Step 1: $-w$ is semiconvex with constant $\tau^{-1}$ and $-w\le\lambda^{-1}Nb_{g}+|e_{*}|$ (Half-Relaxed Limits of the Sup- and Inf-Convolved N-Particle Dyson Solutions: Bounds, Continuity and Limits along Configurations of Bounded Energy §regularised (ii), (i)), $N\chi+K\lVert x\rVert^{2}$ is affine, $N\chi\le N\lVert q\rVert_{\hat{\mu}}^{2}$ on $W_{N}$, and (iii) applies to $\underline{w}_{N,\tau}$; clause (a) becomes $\delta^{-1}(N\nabla\chi-\nabla w)=\Sigma_{N}(\pi_{k})$, that is, $\nabla w=N\nabla\chi-\delta\Sigma_{N}(\pi_{k})$. Step 2: the levels are realising for $\underline{u}_{\tau}$, so $w(y^{k})/N_{k}\to\underline{u}_{\tau}(\hat{\mu})$ by the second part of Half-Relaxed Limits of the Sup- and Inf-Convolved N-Particle Dyson Solutions: Bounds, Continuity and Limits along Configurations of Bounded Energy §realising; (2b) uses the second inequality of (iv), $w(y)\ge w(z)-\frac{1}{2\tau}\lVert y-z\rVert(3\lVert y-z\rVert+2r_{N}(z))$, giving $-w(z)/N\ge-w(y)/N-(3+2\rho)h/(2\tau)$; (2c) becomes $\chi(z)\ge\chi(y)-\lVert q\rVert_{\hat{\mu}}h-Kh(2W(y)+h)$ and $\chi(y)\ge-\lVert q\rVert_{\hat{\mu}}W(y)-KW(y)^{2}$; the lower bound for $\Phi_{k}(\pi_{k})/N$ tends to $-\underline{u}_{\tau}(\hat{\mu})-\delta\mathcal{E}(\hat{\mu})=\hat{m}$. Step 3 is unchanged, (3c) following from $-w/N\le\lambda^{-1}b_{g}+|e_{*}|$ and $\chi\le\lVert q\rVert_{\hat{\mu}}^{2}$. Step 4: $G_{k}(x)\le\lambda^{-1}b_{g}+|e_{*}|+\lVert q\rVert_{\hat{\mu}}W(x)-KW(x)^{2}-\delta P_{N}(x)/N$ as before; the first inequality of Half-Relaxed Limits of the Sup- and Inf-Convolved N-Particle Dyson Solutions: Bounds, Continuity and Limits along Configurations of Bounded Energy §sequences gives $-w(x^{l})/N\le-\underline{u}_{\tau}(\nu)+\varepsilon''$ for large $l$; $\chi(x^{l})\to\langle q,T-\mathrm{id}\rangle_{\hat{\mu}}-KW_{\nu}^{2}$; and (4a) reads $-\underline{u}_{\tau}(\nu)-\delta\mathcal{E}(\nu)+\langle q,T-\mathrm{id}\rangle_{\hat{\mu}}-KW_{\nu}^{2}\ge\hat{m}+\frac{\eta'^{2}}{2K}+\varepsilon-\frac K4W_{\nu}^{2}$. If $W_{\nu}<r_{0}$, the hypothesis of clause 2 gives $-\underline{u}_{\tau}(\nu)-\delta\mathcal{E}(\nu)+\langle q,T-\mathrm{id}\rangle_{\hat{\mu}}\le\hat{m}+\eta'W_{\nu}$, and if $W_{\nu}\ge r_{0}$ one uses $-\underline{u}_{\tau}(\nu)\le\lambda^{-1}b_{g}$, $-\delta\mathcal{E}(\nu)\le|e_{0}|$ and $\langle q,T-\mathrm{id}\rangle_{\hat{\mu}}\le\lVert q\rVert_{\hat{\mu}}W_{\nu}$; the contradictions follow verbatim. Step 5 is unchanged.