A General Framework for Metropolis-Adjusted Dikin Walks: Dimension-Square Mixing on Polytopes and Log-Det Walks on Spectrahedra

We analyze exact-metric, Metropolis-adjusted Dikin walks by keeping the proposal determinant and reverse quadratic form together. Their leading uncentered terms cancel in the complete logarithmic acceptance ratio, leaving centered fluctuations that can be controlled with second-order tools. For a polytope given by n inequalities and a convex L-Lipschitz potential, this yields warm-start mixing in widetilde O((d2+dL2R2)log(w/δ)) steps for the regularized Lee--Sidford walk. For a spectrahedron with n× n blocks, the log-det walk mixes in widetilde O((psi^star nd+dL2R2)log(w/δ)) steps, where psi^star measures matrix leverage. The two analyses share an acceptance-to-mixing reduction. A proposal-comparison argument transfers the polytope bound to an appropriately padded O(1/d)-accurate metric computed from high-precision Lewis weights. For spectrahedra, given widehatpsi≥psi^star, a direct-or-two-seed TensorSRHT construction gives an exact-arithmetic implementation with psi^star replaced by widehatpsi in the mixing bound.