Proof of Minus the Free Entropy Is Bounded Below on Bounded Laws and Has Weak-Star Closed Sublevel Sets

Each result cited is universally quantified over the data in its own statement. We work in Free Products, Semicircular Laws, the Free Heat Flow, Free Fisher Information and Free Entropy: Standing Notation; note $\mathcal{D}_{\chi}\subseteq\Sigma_{d}$ by Voiculescu's Non-Microstates Free Entropy of a Noncommutative Law §domain, and $-\chi^{*}$ is real-valued on $\mathcal{D}_{\chi}$ by Voiculescu's Non-Microstates Free Entropy of a Noncommutative Law §entropy, so $(\mathcal{D}_{\chi},-\chi^{*})$ is a pair of the kind considered in Free Entropy Penalties: Displacement Convexity with Minus the Conjugate Variables as Gradient. Since $d\in\mathbb{N}$, $d>0$.

Monotonicity of the logarithm. If $x,y$ are reals with $0<x\le y$, then $\log x\le\log y$. Indeed, otherwise $\log y<\log x$, and since $\exp$ is strictly increasing (claim 4 of Basic Properties of the Exponential Function), $y=\exp(\log y)<\exp(\log x)=x$ by The Natural Logarithm, contradicting $x\le y$.

Clause 1 (lower bound). Let $R>0$ be real and $\lambda\in\mathcal{D}_{\chi}\cap\Sigma_{d,R}$. For $j\in[d]$, the polynomial $x_{j}^{2}=x_{w}$ with $w=(j,j)$ a word of length $2$ is self-adjoint, so $\lambda(x_{j}^{2})$ is real by Basic Properties of Noncommutative Laws: Adjoints, the Self-Adjoint Pairing, Cauchy-Schwarz, Monotonicity in the Bound, and Laws of Constant Tuples §adjoint, and $\lambda(x_{j}^{2})=\operatorname{Re}\lambda(x_{j}^{2})\le|\lambda(x_{w})|\le R^{2}$ by claim 6 of Properties of Complex Conjugation and Modulus and Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §norm-bound. Summing over $j$, $M(\lambda)\le dR^{2}$. By The Free Entropy: the Semicircular Upper Bound, Semicircular Values, Smoothing, the de Bruijn Identity and Upper Semicontinuity §upper-bound, $M(\lambda)>0$ and

$$\chi^{*}(\lambda)\le\frac{d}{2}\log\Bigl(\frac{2\pi e\,M(\lambda)}{d}\Bigr).$$

Since $2\pi e>0$ (as recorded in Voiculescu's Non-Microstates Free Entropy of a Noncommutative Law) and $0<M(\lambda)/d\le R^{2}$, we get $0<2\pi e\,M(\lambda)/d\le2\pi e\,R^{2}$ by claim 5 of Elementary Arithmetic in an Ordered Field and claim 5 of Elementary Order Arithmetic in an Ordered Field. By the monotonicity of the logarithm and claim 5 of Elementary Arithmetic in an Ordered Field (multiplying by $d/2\ge0$), $\chi^{*}(\lambda)\le\frac{d}{2}\log(2\pi e\,R^{2})$, and by sign reversal (claim 4 of Elementary Order Arithmetic in an Ordered Field),

$$-\chi^{*}(\lambda)\ge-\frac{d}{2}\log(2\pi e\,R^{2}).$$

Thus, for every real $R>0$, the real number $-\frac{d}{2}\log(2\pi e\,R^{2})$ is at most $-\chi^{*}(\lambda)$ for every $\lambda\in\mathcal{D}_{\chi}\cap\Sigma_{d,R}$, which is exactly what the lower bound clause of a free entropy penalty requires of the pair $(\mathcal{D}_{\chi},-\chi^{*})$ (the real number called $e$ there being $-\frac{d}{2}\log(2\pi e\,R^{2})$, where $e$ inside the logarithm is $\exp(1)$).

Clause 2 (closed sublevel sets). Let $R>0$ and $c$ be reals, and let $(\lambda_{k})_{k\in\mathbb{N}}$ be a sequence in $\mathcal{D}_{\chi}\cap\Sigma_{d,R}$ with $-\chi^{*}(\lambda_{k})\le c$ for every $k$ that converges weak-star to some $\lambda\in\Sigma_{d}$. By sign reversal (claim 4 of Elementary Order Arithmetic in an Ordered Field), $\chi^{*}(\lambda_{k})\ge-c$ for every $k$. Apply The Free Entropy: the Semicircular Upper Bound, Semicircular Values, Smoothing, the de Bruijn Identity and Upper Semicontinuity §upper-semicontinuous to the law $\lambda$ (with any real $s>0$, say $s=1$, which that clause does not use), the reals $R$ and $-c$, and the sequence $(\lambda_{k})_{k\in\mathbb{N}}$: it gives $\lambda\in\mathcal{D}_{\chi}$ and $\chi^{*}(\lambda)\ge-c$, hence $-\chi^{*}(\lambda)\le c$ by sign reversal again. This is the closed sublevel set clause for the pair $(\mathcal{D}_{\chi},-\chi^{*})$, with $\mathcal{E}_{0}=-\chi^{*}$ and $\mathcal{D}_{0}=\mathcal{D}_{\chi}$.