The Closed-Score Hypothesis Appears to Fail for the Free Entropy of One Variable: a Sketched Example, Conditional on the Tangent Inequality

In the setting of Free Products, Semicircular Laws, the Free Heat Flow, Free Fisher Information and Free Entropy: Standing Notation, let $d=1$. The pair $(\mathcal{D}_{\chi},-\chi^{*})$ has the lower bound and closed sublevel clauses of a free entropy penalty by Minus the Free Entropy Is Bounded Below on Bounded Laws and Has Weak-Star Closed Sublevel Sets §lower and Minus the Free Entropy Is Bounded Below on Bounded Laws and Has Weak-Star Closed Sublevel Sets §closed, and its tangent clause is Displacement Convexity of Minus the Free Entropy: the Tangent Inequality along Optimal Couplings §tangent. Suppose that clause holds, and let $R>1$ be real. The construction below indicates that the wall-confined free energy of $(\mathcal{D}_{\chi},-\chi^{*})$ with radius $R$ does not have closed score, so that this hypothesis of Perron's Method for Envelope Viscosity Solutions of the Discounted HJB Equation with Free Langevin Noise in a Wall: Existence and Well-Posedness would not be met. The construction is a sketch: its analytic steps are classical one-variable harmonic analysis, and one checks along the way that the laws involved lie in the score domain.

One variable. A tracial state on $\mathcal{P}_{1}$ with norm bound $R'$ corresponds to a probability measure on $[-R',R']$ through its moments. For such a law with a density $p$ in $L^{3}$, the conjugate variable is $2\pi(Hp)(x)$ (Voiculescu), where $Hf(x)=\frac{1}{\pi}\,\mathrm{p.v.}\int f(y)(x-y)^{-1}\,dy$ is the Hilbert transform. Its squared norm $4\pi^{2}\int(Hp)^{2}p$ is therefore at most a constant times $\int p^{3}$, because $H$ is bounded on $L^{3}$.

Oscillating densities. Let $p$ be a smooth density supported in $[-1,1]$, let $b$ be a smooth $1$-periodic function with mean zero, $|b|<1$ and nonzero periodic Hilbert transform $H_{\mathrm{per}}b$, and let $\varepsilon_{n}\to0$. Let $p_{n}$ be $p(x)(1+b(x/\varepsilon_{n}))$, renormalised on each cell $[k\varepsilon_{n},(k+1)\varepsilon_{n})$ so that it carries the mass $p$ gives that cell. The $p_{n}$ are bounded in $L^{3}$, so their free Fisher information stays bounded. Their supports lie in a fixed compact subset of $(-R,R)$, so their wall forces are uniformly bounded continuous functions of the position. But $2\pi Hp_{n}$ differs from $2\pi Hp$ by approximately $2\pi p(x)(H_{\mathrm{per}}b)(x/\varepsilon_{n})$, an oscillation of unit size.

The coupling. Realise everything in the commutative tracial W*-probability space $L^{\infty}([0,1]^{2})$ acting on $L^{2}([0,1]^{2})$, whose real-valued elements are the self-adjoint vectors. Let $X$ have density $p$, and let $V$ be uniform and independent of $X$. Let $X_{n}$ be the point of the cell of $X$ whose quantile under the normalised restriction of $p_{n}$ to that cell is $V$. Then $|X_{n}-X|\le\varepsilon_{n}$. The realised scores converge in $L^{2}$ to the score of $p$ at $X$ plus a multiple of $p(X)(H_{\mathrm{per}}b)(U)$, where $U$ is the quantile transform of $V$ under the density $1+b$. This extra term is nonzero and orthogonal to every function of $X$. A tuple $Q$ with $\mathrm{law}(X,Q)$ equal to the score plan of the limit is unique by Realising a Square-Integrable Field of the GNS Space of a Bounded Law next to Its Positions: Uniqueness and the Joint Law with a Momentum §unique, and it is the score of $p$ at $X$. So the weak limit of the realised scores is not that $Q$, and the clause of Wall-Confined Free Energies with Closed Score §closed fails.

What survives. The realised conjugate variables do converge against every field of the limit realised next to $X$; this is Projected Closedness of Conjugate Variables along Square-Integrable Realisations with Bounded Free Fisher Information. The Wasserstein counterpart Penalty Pairs with Closed Score Along Couplings likewise tests only against fields of the limit, through Strong and Weak Convergence of Vector Fields Along Couplings of Vanishing Cost §weak. The noncommutative closed score tests against every tuple, and this is used in the Perron proofs, where the limit score is paired with a test momentum that is not a function of the limit position.