Let $K=\{x\in\R^d:Ax\ge b\}$ be a bounded full-dimensional polytope with $m$ inequalities. For targets on $K$ with density proportional to $\exp(-\vartheta^\transpose x)$, we prove a $\widetilde O(d^2)$ bound, measured in Markov chain iterations, for the chi-square mixing of the $1/2$-lazy Metropolis Dikin walk based on a polylogarithmically rescaled Lee--Sidford metric. The estimate holds for every initial law with finite chi-square divergence. The proof controls the reverse Gaussian proposal through a quantitative average self-concordance estimate at the required $O(1/d)$ scale. We establish this estimate from uniform analytic derivative bounds for the Lewis-weight subspace and an all-order projection-weighted contraction inequality for Gaussian chaoses. For order $j$, the latter bounds each base-point polynomial at scale $d^{j/2}$, up to $j$-dependent and logarithmic factors. Truncating at logarithmic order then absorbs the remaining pathwise loss. This resolves, in the affirmative, the open problem of whether the Gaussian Lee--Sidford Dikin walk mixes in $\widetilde O(d^2)$ iterations: the previously stated quadratic bounds \cite[Theorem~1.4]{LLV20}, \cite[Theorem~1.2]{GKM24} rest on reverse-proposal steps identified as unresolved in \cite[Section~1]{Kook26}, whose $\widetilde O(d^{9/4})$ bound was the best rigorously established rate prior to this work. The proof is found by ChatGPT 5.6 Pro.

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