shape_the_sound

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Kac's question in reverse: design a polygonal drumhead whose lowest tones play requested frequency ratios (a fifth, a triad, a pentatonic scale), each accepted within ±tolerance_cents. proven_impossible is returned only when proven theorems exclude every ratio combination inside those tolerance windows for a connected uniform membrane with a fixed edge: simple fundamental, f2/f1 ≤ j11/j01 ≈ 1.5933 with equality for the disk (Ashbaugh–Benguria 1992), and Yang's universal inequality minimized exactly over the window box; an exact-target exclusion is reported separately. Otherwise multi-start L-BFGS with exact finite-element shape derivatives designs a 5–16-vertex polygon, then reanalyses it on three refined meshes with the shared hear_the_shape solver: achieved ratios, errors in cents with estimated uncertainty (not rigorous bounds), resolution warnings, optional numerical tuning of the fundamental in Hz. not_achieved is a bounded-search result, never an impossibility proof. Ideal membrane only.