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The standard recipe -- truncate to a finite cap N and exponentiate -- pays cubic cost in N and a time-growing boundary-feedback bias. We identify a structural condition on the rates, L_{n+r,n} = alpha_r n + beta_r (\"linear-rate\"), under which the generating function G(z,t) = sum_n x_n(t) z^n satisfies a first-order linear PDE in z, and the method of characteristics yields a composition-multiplier representation G(z,t) = K_t(z) G(Phi_t(z), 0). 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