ALT We give a polynomial-time algorithm to sample from the Gibbs measure of the Sherrington–Kirkpatrick model with negligible total-variation distance (TVD) error up to inverse temperature β< 1/2. Prior work obtained TVD error guarantees only up to β≈ 0.295, while results covering the entire replica-symmetric regime β< 1 gave guarantees only in Wasserstein distance. Our approach demonstrates that the same potential Hessian ascent previously developed for optimization also functions as a sampling algorithm by implementing algorithmic stochastic localization at high temperature. By estimating the covariance of the tilted Gibbs distribution via Gaussian integration by parts, overlap concentration, and precise cavity estimates, we show that a Hessian-ascent process achieves an O(1) Wasserstein error guarantee for finite-time localization, improving on the previous o(n). A careful comparison of stochastic localization with the Hessian ascent process and a free probability argument controlling t