arXiv:2610.31493v1 Announce Type: new Abstract: Let $\mu$ be a log-concave isotropic probability measure on $R^d$ , and let $\mu^n$ denote the law of the normalized sum of $n$ independent random vectors with law $\mu$. We prove that \[ H(\mu^n|\gamma)\leq \frac{C}{n} \min\left\{1+\log(ed)^2H(\mu|\gamma)\ ,\ d\right\}. \] Compared to the previous best known bound, our estimate replaces the polynomial dependence on the dimension by a polylogarithmic one. The proof relies on a new combination of the De Bruijn identity with Eldan's stochastic localization process.