Yes, Paul, I think your phase plot match is good. Those corners must be your anti-entropy filtering. Amusing how they add a bit of low frequency entropy in removing high frequency noise.
"Nonlocal-in-time" is a very awkward expression. Newtonian time began nonlocal by presuming a single uniform time across the universe. Relativity properly undermined this assumption. We are now redefining time to be inherent oscillation as such. That waves are integer-countable make them QM at any scale or frequency.
Wilczek has proposed that time-symmetric oscillations may be defined as "time crystals". While this began as exotic nanoscopic QM, it serves as well for Galileo's Pendulum, or any orderly periodic wave-form, including atmospheric oscillations.
Lets posit "conservation of time" (and space), in that one may locally dilate or compress time, but the spacetime field does not gain or lose time overall; there is opposite and equal time warpage when things fly apart.
We need not insist any physics ideas are fundamental truths, merely effective interpretations to ponder.