Learned Bow Control on a Measured Bowed-String Model: a Revised Minimum-Bow-Force Law, a Recurrent Controller, and the Domain of a Supervision Ceiling

A finite-difference bowed-string model with implicitly resolved Stribeck friction is presented, with a regime diagnostic. Without implicit resolution no stick phase forms at any bow force. With friction, impedance and quality factor from published measurement rather than fitted, all four strings return a stick fraction of 89.1% against an ideal 90%. Schelleng's maximum bow force is recovered on every string. His minimum, Fmin propto Z2 vb β-2, is replaced by a law, Fmin = C Z vb / β with a dimensionless C = 1.112 ± 0.017 that reproduces on sixty-four held-out operating points. Six controllers at matched capacity, on four strings at twenty seeds, place a gated recurrent network ahead of a feedforward one, its margin over the mean of the other five largest when the commanded bow speed is overridden at mid-stroke. The feedforward network completes more strokes, mostly from a cold start no player would use. A minimal gated variant fails because gates computed from the input alone cannot clear a latched state. Training loss selects neither the capacity nor the context length. No learned controller improves on the lookup rule that generated its labels where that rule is correct. Of the rule's 1089 cells, 57 are playable on a properly settled plant and unplayable by its labels, and the controller commands them where the lookup will not. The controller's score regresses on the rule's with a slope of 0.32, more than ten standard errors below unity, so it overtakes the rule where the rule fails and is bounded by it where it holds. Under a rigid finger stop the plant is provably invariant, so transfer loss between pitches is the controller's alone, traced to one feature. A regime classifier without a stick test mistakes small-amplitude periodic slipping for Helmholtz motion, and a harmonicity measure rates a string the bow never grips above it.