Exponentials with a Moving Sound Speed: The Ansatz Identity Re-Run Against c_s(t)

Executes item D-495-501-02: packet #496's ansatz identity re-run with a time-dependent sound speed, gated on #496's exhibits reproduced at the printed digits (Gram determinant 5.35e-02; RMS residuals 4.48/5.08/6.66; Bessel deviation 8.38e-11). With c_s(t) free the identity becomes a determination: e^{gamma t} is a solution iff c_s(t)^2 k^2 = -gamma^2 t^{4/3} - (4 gamma/3) t^{1/3} + (2/3) t^{-2/3} at every t. Existence survives in exactly two shapes: gamma = 0 on the whole half-line under the single decaying profile (2/3) t^{-2/3}, with general solution c1 + c2 t^{-1/3}; and gamma != 0 real on a finite interval ending at a computed crossing t*(gamma) (0.7749 at gamma = 0.5) beyond which the dictated profile demands c_s^2 < 0. Non-real gamma is never sustainable by a real sound speed. Among power-law sound speeds exactly one member admits an exponential, at rate zero. Quantum gravity NOT SOLVED; vacuum catastrophe NOT SOLVED.