Yeah, we're crazy. I'm going to go through the thread, but I would first like to offer that Time Bastard doesn't specifically interfere with the Crono's cat point. Rather, it would fall under the principle named 'Flow Theory,' which dictates that time flows in a linear fashion and is a rudimentary form of Leebot's "Time Error" idea. Time changes only after time traveler's do the deed and invoke change; the future is not written, and the world is tanget to the observation of the traveler's. This would make it impossible for the cat to travel back as a result of the earlier quest, unless we invoked predetermination, fate, etc.
I'll get the small stuff first. "Marle: But the future that was supposed to have disappeared is about to be restored" is largely a metaphoric statement, though it is true that the future of Home World, destroyed by Lavos, will probably be almost completely like the Lavos Timeline's future. While this may not be explicitly proved in the game, it's just more likely than the resurrection of a timeline from the Darkness. The difference is thin, though.
Honestly, this statement is not in a tone of annoyance, but the words postulate, theorem, theory, hypothesis, etc. are used loosely around here precisely because we aren't dealing with the real world and cannot evoke the various meanings of those terms beyond "theory," as hypotheses cannot be formed, as they lack the possibility experiments to evaluate them. This protection against detailed, real world physics can be invoked elsewhere, but I think we will be capable of explaining "time" in the Chrono sense. The worst problem with this state of affairs is that the Axioms & Corollaries is intended to align the phenomena of the Chrono series into a single, definitive statement (referencing the best theories designed to fit with the games and themselves perfectly) that can build and support itself. If other theories are represented, they are done so as an aside - ideally. However, the Compendium is still young, and the revision of the A&C article (along with Secret of Nu and Many Mysteries of Magus) simply hasn't been attempted yet due to the long term planning schedule.
Since I am not truly a student of physics or other sciences, besides my own wonderings, I find it fitting that I draw up a rudimentary definition of "time" as it applies to the Chrono series. I'm going to try and avoid entropic definitions and anything else that detracts from the free will spirit. Thanks to Radical_Dreamer, Aitrus, Hadriel and CTCronoboy for assistance with this.
Time is the chronotic, naturally linear, endless flow and recording of the state of the universe; it is eternal, and in its infinite span, is only stablized by the observer as per the Chronopolis person's statement. At any given time, there exists one 'timeline' per a universe, there being multiple universes as proposed in Chrono Cross (called dimensions). As a defense mechanism against paradoxes, nature, when confronted with one who goes against the natural flow of time, and to the person's own view, realigns history according to that person. History forward, in relation to that grand, singular timeline in the linear progression of time is entirely rewritten according to their actions. Traveling forward in time is less complicated, as one is merely moving at an accelerated rate - history and time still occur, but without the traveler's influence. This does not preclude TB, as it works forward or backward; at the original, personal time of the original person's departure, the new person would fall victim to Time Traveler's Immunity/Bastard. As for measuring it, that could be up to anyone. Regardless of how many paradoxes I can invoke in my head about units of time becoming smaller and smaller per the second, it is nonetheless true that as I type this, time is moving forward. Thus, I do not have an answer for units of time, aside from the human second. But it is tangent to the observer even in that regard, no?
Thus, when time travel occurs, an intrinsic memory of the universe is invoked according to the observation of the time traveler, and linear time (as per Flow Theory) becomes dependent on his actions.
The timeline will propagate outward and generate an entire history instantly after the point of alteration. All the other histories will cease to exist, not the matter occupying them. (Hadriel)
As for the mechanism knowing that Serge X is the one to get shunted, I'll allow better explanations on this side of the argument to provide a scientific basis, but one that I will tout is this (and keep in mind that spirits and ghosts do exist in the series, so this is not baseless): it operates on the principle of a unique, identifiable 'soul' within a person. The soul is rooted in the existence of the universe. When a timeline is sent to the Darkness Beyond Time, matter and life forces are merely rearranged as a new state of the universe is invoked. There cannot be two congruent "unique" souls in rooted existence. When one time travels, Time Traveler's Immunity (under this theory) is extended as a universal (physics context, not "expansive" or "all encompassing") checkpoint for his the arrangement of his soul. Whatever new versions of himself that arise because of his meddling with time still have the same life essence; at TTI's invocation, however, it is rearranged, and so on for every instance of his time travel. And when the traveler is ready to return to his life as it was, there is no problem. While his life essence goes through many changes to adhere to his original progression as a person (which is saved forever in TTI), his history is always there, even if it exists "simultaneously." Like Robo, it is the same soul viewed in two contexts of time.
I'll call this the spiritual basis of time (above paragraph only). I'm sure this will follow with other, more scientific explanations for the mechanism that decides who goes to the DBT per Time Bastard.