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Motivated by the limiting behavior of an explicit class of compact ancientcurve shortening flows, we prove codimension bounds for ancient mean curvatureflows by their tangent flow at $-\infty$, generalizing a theorem for cylindersin [CM19b]. In the case of the $m$-covered circle, we apply this bound to provea strong rigidity theorem. Furthermore, we extend this paradigm by showing thatunder the assumption of sufficiently rapid convergence, a compact ancient meancurvature flow is identical to its tangent flow at $-\infty$.

Codimension Bounds and Rigidity of Ancient Mean Curvature Flows by the Tangent Flow at $-\infty$