Total Variation Mixing of Weighted Quadratic Gibbs Samplers on a Cube

We consider uniform random-scan coordinate Gibbs sampling on \([0,1]^d\) for a density proportional to \(\exp(-a\sum_{i<j}c_{ij}(x_i-x_j)^2)\), with symmetric nonnegative weights. For every fixed nonzero weight matrix, we prove a worst-start total-variation upper bound of order \(a\) as \(a\to\infty\), with an explicit finite-parameter bound and logarithmic dependence on the requested accuracy. Together with the Wasserstein lower bound of Gerencser and Ottolini, this yields the sharp order \(\Theta(a)\) at total-variation tolerance \(1/4\), including disconnected graphs and isolated vertices. The proof adapts the non-strongly-convex entropy interpolation of Ascolani, Lavenant and Zanella to the compact cube. A coordinate-coverage decomposition handles arbitrary deterministic starts without assigning finite entropy to the remaining singular branch. The result addresses the total-variation upper bound discussed for this weighted Gibbs model. It does not cover arbitrary Metropolis proposals or optimize dependence on dimension or graph parameters. This attributed application is AI-assisted, self-audited and unrefereed; no independent review, formal verification or absolute priority is claimed.