Entropy and determinants for unitary representations

The paper proves the determinantal formula h_π(ϕ) = log Δ_τ(ϕ_ac) under tracial convergence and asymptotic association (Theorem A), by reducing to k=1 (Proposition 6.4) and establishing matching lower/upper exponential volume bounds (Theorems 8.1 and 8.2) via Kaplansky-type approximation (Proposition 7.4) and a precise volume normalization c(k,d) from Proposition 3.19 and its asymptotics (3.3.7) . The candidate solution reproduces the high-level idea (singular part contributes no exponential volume; absolutely continuous part contributes a log-determinant via a linear change of variables), but it makes two critical errors: (i) it replaces the paper’s exact normalization c(k,d) by the volume of a Euclidean ball v(d)^k, which shifts the (1/d) log-scale by k log k and thus changes the limit; the paper’s choice of c(k,d) is essential to obtain the correct constant-free rate ; and (ii) it postulates positive L_n in the commutant with moment-matching and determinant convergence but gives no justification from the paper’s hypotheses. The paper, instead, uses positive elements a ∈ A (via Proposition 7.4) and the determinant-Jacobian lemma (Lemma 8.3) to control volumes, avoiding any nontrivial commutant modeling . There are also smaller issues (using L_n instead of L_n^2 in the trace-moment identity; using a heuristic ‘thick base set’ without the paper’s concentration and polar-integration framework). Because the paper’s argument is complete and correct, while the model’s proof depends on an incorrect normalization and an unproved commutant approximation, the appropriate verdict is Paper correct, model wrong.