Thus, since they set 0 = 1, they should limit their numerical tests to where < 1. However, as their fifth error, they also used it for = 1, 10, 100, i.e. critically damped and overdamped cases. They also evaluated it in the limit . Errors #6: Also, in their derivation of log H they would have had to encounter log(ei), where tan = p/x = tan(t + ) in their case. This requires a careful treatment, since it is the inverse of a non-injective function; it is normally treated by restricting to (, ] and then using Riemann sheets . However, they made no mention of this issue, which may be related to the unusual swirling feature in their graphs (cf. Fig. 4 in ). This is their sixth error.

Finally, from their method they claimed to have derived a constant, labeled H1 (see Sec. III C of ). However, it is not actually constant versus time. This can be demonstrated using the correct x, p with 0 = 1, = 0.1, x(0) = 3, and p(0) = 0; the result is shown in Fig. 1. In the left plot of the figure is shown the trajectory for t [0, 14.9], with a red dot indicating the initial value; corresponding values of H1 are shown in the right plot. This clearly shows that their H1 is not a constant. In summary, they obtained an incorrect result using M. F. Zimmer, arXiv preprint arXiv:2110.06917v2 (2021), (See Appendix C). Z. Liu, V. Madhavan, and M. Tegmark, arXiv preprint arxiv:2203.12610 (2022). Z. Liu, V. Madhavan, and M. Tegmark, Phys. Rev. E 106, 045307 (2022). E. D. Rainville and P. E. Bedient, Elementary Differential Equations, 6th ed. (Macmillan Publishing Co., 1981). 2 an incorrect method; if their method is corrected, it can no longer be used.

Afterword

Intermediate steps: The reader should note that some of the work summarized in the Review portion has a counterpart in the earlier work of Zimmer . For example, Eqs. 3,4 in this Comment can be matched to similar equations in App. C of . (The author has since streamlined his approach in his latest preprint , and now uses different intermediate steps.) Their analytical approach: In their treatment of the 1D damped oscillator, they began from exact solutions for x, p, and then formed combinations of them to isolate the parameters of the solution, thereby determining a constant of motion. Such an approach can only be used, as they presented it, in the undamped case. To make it work in the damped case, they should have made a variable change (x w = x + p) that Zimmer recognized (see App. D.2 of ). However, Liu, et al. gave no indication they were aware of such a transformation. 2D oscillator: Liu, et al. also analyzed the 2D oscillator (undamped), and made similar omi