Convergence of potential-preserving annealed Sinkhorn under non-summable temperature schedules

Annealed Sinkhorn solves entropic optimal transport by progressively decreasing the regularisation parameter over its iterations. When the regularisation changes, one may preserve either the multiplicative scalings or the physical dual potentials. Recent work has shown that scaling-preserving annealed Sinkhorn converges to balanced optimal transport under sufficiently slow annealing, yet already under a linear inverse-temperature schedule it converges instead to a semi-unbalanced solution. We prove that this failure is specific to scaling preservation: for potential-preserving annealed Sinkhorn, if the regularisation schedule tends to zero and is non-summable, the iterates admit a subsequence converging to an optimal transport solution. This includes linear inverse-temperature schedules. The condition is sharp: for summable regularisation schedules, we give an explicit example for which no subsequence converges to an optimal transport solution.