Proof of The Wall-Confined Free Energy: Norm Bound, Lower Bound, Weak-Star Compact Sublevel Sets and Displacement Monotonicity of the Score

Each result cited is applied with the data of its own statement. By The Wall-Confined Free Energy and Its Score §energy, $\mathcal{D}=\mathcal{D}_{0}\cap\mathcal{D}_{R}$ and $\mathcal{E}(\lambda)=\mathcal{E}_{0}(\lambda)+\mathcal{W}_{R}(\lambda)$ for $\lambda\in\mathcal{D}$.

Step 1 (Claim 1). By The Wall Energy: Finiteness Forces the Norm Bound, Closed Sublevel Sets, and Monotonicity of the Wall Force along Couplings §norm-bound, $\mathcal{D}_{R}\subseteq\Sigma_{d,R}$ and $\mathcal{W}_{R}(\lambda)\ge0$ for $\lambda\in\mathcal{D}_{R}$. Hence $\mathcal{D}\subseteq\mathcal{D}_{R}\subseteq\Sigma_{d,R}$ and $\mathcal{D}\subseteq\mathcal{D}_{0}\cap\Sigma_{d,R}$. By Free Entropy Penalties: Displacement Convexity with Minus the Conjugate Variables as Gradient §lower for this $R$ there is a real $e$ with $e\le\mathcal{E}_{0}(\lambda)$ for every $\lambda\in\mathcal{D}_{0}\cap\Sigma_{d,R}$. For $\lambda\in\mathcal{D}$ therefore $\mathcal{E}(\lambda)=\mathcal{E}_{0}(\lambda)+\mathcal{W}_{R}(\lambda)\ge e+0=e$. This proves claim 1; the $e$ fixed here is used below.

Step 2 (Facts on indices). Let $(n_{i})_{i\in\mathbb{N}}$ and $(r_{i})_{i\in\mathbb{N}}$ be strictly increasing sequences in $\mathbb{N}$ in the sense of Subsequence of a Sequence in a Set. (i) $n_{i}<n_{k}$ whenever $i<k$: by induction on $k$, the case $k=i+1$ is the definition, and $n_{i}<n_{k}<n_{k+1}$. Consequently $(n_{r_{i}})_{i\in\mathbb{N}}$ is strictly increasing, since $r_{i}<r_{i+1}$ gives $n_{r_{i}}<n_{r_{i+1}}$. (ii) $i\le r_{i}$ for every $i$: $1\le r_{1}$ since $r_{1}\in\mathbb{N}=\{1,2,3,\dots\}$ (Natural Numbers), and $i\le r_{i}<r_{i+1}$ gives $i+1\le r_{i+1}$. (iii) If a real sequence $(a_{i})$ converges to $a$ in the sense of Limit of a Sequence of Real Numbers, then so does $(a_{r_{i}})_{i}$: given $\varepsilon>0$ and $N$ with $|a_{i}-a|<\varepsilon$ for $i\ge N$, (ii) gives $r_{i}\ge i\ge N$, hence $|a_{r_{i}}-a|<\varepsilon$, for $i\ge N$. (iv) For $N\in\mathbb{N}$ the index sequence $t\mapsto N+t$ is strictly increasing, so a tail $(y_{N+t})_{t\in\mathbb{N}}$ of a sequence $(y_{i})$ is a subsequence of it.

Step 3 (Extraction). Let $c$ and $(\lambda_{m})$ be as in claim 2. For every $m$, Step 1 gives $\lambda_{m}\in\Sigma_{d,R}$, $e\le\mathcal{E}_{0}(\lambda_{m})$ and $0\le\mathcal{W}_{R}(\lambda_{m})$, and $\mathcal{E}_{0}(\lambda_{m})+\mathcal{W}_{R}(\lambda_{m})=\mathcal{E}(\lambda_{m})\le c$; hence

$$e\le\mathcal{E}_{0}(\lambda_{m})\le c,\qquad0\le\mathcal{W}_{R}(\lambda_{m})\le c-e.$$

With $M=|e|+|c|+1>0$ both real sequences $(\mathcal{E}_{0}(\lambda_{m}))_{m}$ and $(\mathcal{W}_{R}(\lambda_{m}))_{m}$ satisfy $|\cdot|\le M$, so they and all their subsequences are bounded in the sense of Bounded Sequence of Real Numbers. By Sequential Weak-Star Compactness of the Noncommutative Laws with a Given Norm Bound §compact there are a strictly increasing $(k_{l})_{l\in\mathbb{N}}$ and $\lambda\in\Sigma_{d,R}$ with $\lambda_{k_{l}}\to\lambda$ weak-star. By Bolzano-Weierstrass Theorem for Real Sequences, applied to $(\mathcal{E}_{0}(\lambda_{k_{l}}))_{l}$, there are a strictly increasing $(l_{i})_{i\in\mathbb{N}}$ and a real $a$ with $\mathcal{E}_{0}(\lambda_{k_{l_{i}}})\to a$ as $i\to\infty$. By the same theorem, applied to $(\mathcal{W}_{R}(\lambda_{k_{l_{i}}}))_{i}$, there are a strictly increasing $(r_{t})_{t\in\mathbb{N}}$ and a real $b$ with $\mathcal{W}_{R}(\lambda_{k_{l_{r_{t}}}})\to b$ as $t\to\infty$. In both applications the conclusion is a subsequence in the sense of Subsequence of a Sequence of Real Numbers, given by a strictly increasing sequence of positive integers indexed by the positive integers; as the positive integers are the natural numbers $\mathbb{N}=\{1,2,3,\dots\}$ of Natural Numbers, $(l_{i})$ and $(r_{t})$ are strictly increasing sequences in $\mathbb{N}$ and the extracted sequences are subsequences in the sense of Subsequence of a Sequence in a Set. Put $m_{t}=k_{l_{r_{t}}}$, $a_{t}=\mathcal{E}_{0}(\lambda_{m_{t}})$ and $b_{t}=\mathcal{W}_{R}(\lambda_{m_{t}})$. By Step 2(i), applied twice, $(m_{t})$ and $(l_{r_{t}})$ are strictly increasing. Thus $(\lambda_{m_{t}})_{t}$ is a subsequence of $(\lambda_{k_{l}})_{l}$ and converges weak-star to $\lambda$ by Sequential Weak-Star Compactness of the Noncommutative Laws with a Given Norm Bound §unique; $a_{t}\to a$ by Step 2(iii); and $b_{t}\to b$ by construction. Also $\lambda\in\Sigma_{d}$ by Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §law.

Step 4 ($\lambda\in\mathcal{D}_{0}$ and $\mathcal{E}_{0}(\lambda)\le a$). Let $\varepsilon>0$ be real, and choose $N$ with $|a_{t}-a|<\varepsilon$, hence $a_{t}\le a+\varepsilon$, for $t\ge N$. By Step 2(iv) and Sequential Weak-Star Compactness of the Noncommutative Laws with a Given Norm Bound §unique, the tail $(\lambda_{m_{N+t}})_{t\in\mathbb{N}}$ converges weak-star to $\lambda\in\Sigma_{d}$; it lies in $\mathcal{D}\subseteq\mathcal{D}_{0}\cap\Sigma_{d,R}$ by Step 1, and $\mathcal{E}_{0}(\lambda_{m_{N+t}})\le a+\varepsilon$ for every $t$. By Free Entropy Penalties: Displacement Convexity with Minus the Conjugate Variables as Gradient §closed, applied with $R$ and the real $a+\varepsilon$ in place of $c$, $\lambda\in\mathcal{D}_{0}$ and $\mathcal{E}_{0}(\lambda)\le a+\varepsilon$. As $\varepsilon>0$ was arbitrary, $\mathcal{E}_{0}(\lambda)\le a$ by Comparison of Real Numbers with Arbitrary Positive Slack §slack-above.

Step 5 ($\lambda\in\mathcal{D}_{R}$ and $\mathcal{W}_{R}(\lambda)\le b$). Let $\varepsilon>0$ be real, and choose $N$ with $b_{t}\le b+\varepsilon$ for $t\ge N$. As in Step 4 the tail $(\lambda_{m_{N+t}})_{t\in\mathbb{N}}$ lies in $\mathcal{D}_{R}$, satisfies $\mathcal{W}_{R}(\lambda_{m_{N+t}})\le b+\varepsilon$ and converges weak-star to $\lambda\in\Sigma_{d}$. By The Wall Energy: Finiteness Forces the Norm Bound, Closed Sublevel Sets, and Monotonicity of the Wall Force along Couplings §closed, applied with the real $b+\varepsilon$ in place of $c$, $\lambda\in\mathcal{D}_{R}$ and $\mathcal{W}_{R}(\lambda)\le b+\varepsilon$. Hence $\mathcal{W}_{R}(\lambda)\le b$ by Comparison of Real Numbers with Arbitrary Positive Slack §slack-above.

Step 6 (Claim 2). By Steps 4 and 5, $\lambda\in\mathcal{D}_{0}\cap\mathcal{D}_{R}=\mathcal{D}$. The real sequence $\mathcal{E}(\lambda_{m_{t}})=a_{t}+b_{t}$ converges to $a+b$ by claim 1 (Sums) of Arithmetic of Limits of Real Sequences, and

$$\mathcal{E}(\lambda)=\mathcal{E}_{0}(\lambda)+\mathcal{W}_{R}(\lambda)\le a+b=\lim_{t\to\infty}\mathcal{E}(\lambda_{m_{t}}).$$

Together with Step 3 this proves claim 2, with the strictly increasing sequence $(m_{t})$ and the law $\lambda$.

Step 7 (Real linearity of the coupling pairings). Let $\mu,\nu\in\Sigma_{d}$, $\gamma\in\Pi(\mu,\nu)$, and $\zeta,\zeta'\in\mathcal{H}_{\mu}^{d}$; write $\zeta-\zeta'$ for the componentwise difference. For $j\in[d]$, $V^{1}_{\gamma}(\zeta_{j}-\zeta'_{j})=V^{1}_{\gamma}\zeta_{j}-V^{1}_{\gamma}\zeta'_{j}$ since $V^{1}_{\gamma}$ is linear (Marginal Isometries, Bounded Plans and Displacement Pairings for Noncommutative Laws §isometries); the inner product is conjugate-linear in its first argument (Complex Hilbert Spaces and Bounded Linear Maps: Standing Notation §spaces) and $\overline{-1}=-1$, so $\langle u-u',v\rangle=\langle u,v\rangle-\langle u',v\rangle$; and $\operatorname{Re}$ is additive with $\operatorname{Re}(-z)=-\operatorname{Re}z$. Summing over $j$ in Marginal Isometries, Bounded Plans and Displacement Pairings for Noncommutative Laws §coupling-pairing,

$$\mathcal{J}^{1}_{\gamma}(\zeta-\zeta')=\mathcal{J}^{1}_{\gamma}(\zeta)-\mathcal{J}^{1}_{\gamma}(\zeta'),$$

and in the same way $\mathcal{J}^{2}_{\gamma}(\eta-\eta')=\mathcal{J}^{2}_{\gamma}(\eta)-\mathcal{J}^{2}_{\gamma}(\eta')$ for $\eta,\eta'\in\mathcal{H}_{\nu}^{d}$, using $V^{2}_{\gamma}$.

Step 8 (Claim 3). Let $\mu,\nu\in\mathcal{D}_{\Xi}$ and let $\gamma\in\Pi(\mu,\nu)$ be optimal. By The Wall-Confined Free Energy and Its Score §score, $\mu,\nu\in\mathcal{D}\subseteq\mathcal{D}_{0}\subseteq\Sigma_{d}$, both have conjugate variables $\xi_{\mu},\xi_{\nu}$ and square-integrable wall forces $F^{R}(\mu),F^{R}(\nu)$ of radius $R$, and $\Xi(\mu)=F^{R}(\mu)-\xi_{\mu}$, $\Xi(\nu)=F^{R}(\nu)-\xi_{\nu}$ componentwise. By Step 7,

$$\mathcal{J}^{1}_{\gamma}\bigl(\Xi(\mu)\bigr)-\mathcal{J}^{2}_{\gamma}\bigl(\Xi(\nu)\bigr)=\Bigl[\mathcal{J}^{1}_{\gamma}\bigl(F^{R}(\mu)\bigr)-\mathcal{J}^{2}_{\gamma}\bigl(F^{R}(\nu)\bigr)\Bigr]-\Bigl[\mathcal{J}^{1}_{\gamma}(\xi_{\mu})-\mathcal{J}^{2}_{\gamma}(\xi_{\nu})\Bigr].$$

The first bracket is $\ge0$ by The Wall Energy: Finiteness Forces the Norm Bound, Closed Sublevel Sets, and Monotonicity of the Wall Force along Couplings §monotone, which applies to the coupling $\gamma$ of the laws $\mu,\nu$ with square-integrable wall forces of radius $R$. The second bracket is $\le0$ by The Conjugate Variables of a Free Entropy Penalty are Displacement Monotone along Optimal Couplings §monotone, which applies since $\mu,\nu\in\mathcal{D}_{0}$ have conjugate variables and $\gamma$ is optimal. Hence the left side is $\ge0$, which is claim 3. $\square$