Classification of ancient finite-entropy curve shortening flows

We prove that any ancient smooth embedded finite-entropy curve shortening flow is one of the following: a static line, a shrinking circle, a paper clip, a translating grim reaper, or a graphical ancient trombone. An ancient trombone is an immersed ancient flow, either compact or non-compact, obtained by gluing together $m$ translating grim reaper curves. For each $m$, there exists a $(2m-1)$-parameter family of graphical ancient trombones, up to rigid motions and time shifts as constructed by Angenent-You. In particular, our result implies that any compact ancient smooth embedded finite-entropy flow is convex. Moreover, any non-compact ancient smooth embedded finite-entropy flow is either a static line or a complete graph over a fixed open interval.