Explicit error bounds and guaranteed convergence of the Koopman-Hill projection stability method for linear time-periodic dynamics

The paper proves the bound ||Φ(t) − C e^{Ht} W|| ≤ (2 e^{−b})^N e^{|4 a t|} and the subharmonic bound (2 e^{−b})^{2N} e^{|4 a t|}, together with the N* sizing rule, by a series/comb combinatorial argument (Theorem 6, eqs. (81)–(82), and Theorem 9) that is internally consistent and fully derived in the text . The candidate solution reaches the same numerical bounds but relies on a key false step: a projected semiconjugacy claim that implies C e^{H∞ t} E = C e^{H t} and hence that the only error source is the dropped initial tail. This would force zero error whenever the initial condition is supported on the kept modes, contradicting the very phenomenon the paper carefully counts (multi-index paths that leave and re-enter the truncated set) . Without that identity, the model’s proof lacks a mechanism to control the cumulative effect of off-band excursions that feed back into the center, so its argument is invalid even though its final bound matches the paper’s result.