You asserted that there was something wrong in principle with many-to-one dynamics. I pointed out that we have consistent irreversible theories that are considered in-principle acceptable theories, and in particular are strictly preferred over one-to-one theories in their respective domains.

> You can conjecture that our current reality could have been arrived at from two different paths. What you cannot do is assert the truth of a theory which selects one path in favor of the other, because the existence of any information which would make that determination would contradict that these both histories led to the same present.

Just because two different pasts would have led to identical presents does not mean we don't have criteria (such as simplicity) to favor one past over another. Indeed, all of science is based on this. Ultimately all we can use to check our theories are our observations, and the actual microscopic state of the universe is vastly under-determined from these observations alone. Assumptions about simplicity and regularity must be deployed.

As an extreme example: it's logically possible a cheesecake materialized in the center of the sun 1 second ago. (You can either think of this as a thermodynamic fluke, or as a new proposed fundamental theory where the standard model of physics is temporarily suspended 13.8 billion years after the big bang and exactly 1 cheesecake appears in the center of each star.) The cheesecake would be instantly destroyed, and it would not influence my observations, and thus there are two possible pasts (cheesecake vs. no cheesecake) which give the same present state and observations and we cannot categorically rule one out. Nevertheless, we assign astronomically low probability to the cheesecake past.