It looks like Theorem 1 can be improved slightly, by dropping the “only if” condition on . We can then code up something like Kolmogorov complexity by adding a probability transition from every site to our chosen UTM.
If you only want the weaker statement that there is no stationary distribution, it looks like there’s a cheaper argument: Since is aperiodic and irreducible the hypothetical stationary distribution is unique. is closed under the action of , and (2) implies that for any , the map is an automorphism of the Markov chain. If the (infinite) transition matrix is , then can be considered as a permutation matrix with (abusing notation) . Then and so by uniqueness. So is constant on orbits of , which are all countably infinite. Hence is everywhere , a contradiction.
The above still holds if (2) is restricted to only hold for a group such that every orbit under is infinite.
I think the above argument shows why (2) is too strong; we shouldn’t expect the world to look the same if you pick a “wrong” (ie. complicated) UTM to start off with. Weakening (2) might mean saying something like asserting only . To do this, we might define the measures and together (ie. finding a fixed point of a map from pairs to ). In such a model, constraints the transition probabilities, is stationary; it’s not clear how one might formalise a derivation of from but it seems plausible that there is a canonical way to do it.
Even if it’s the case that the statistics are as suggested, it would seem that a highly effective strategy is to ensure that there are multiple adults around all the time. I’ll accept your numbers ad arguendo (though I think they’re relevantly wrong).
If there’s a 4% chance that one adult is an abuser, there’s a 1⁄625 chance that two independent ones are, and one might reasonably assume that the other 96% of adults are unlikely to let abuse slide if they see any evidence of it. The failure modes are then things like abusers being able to greenbeard well enough that multiple abusers identify each other and then proceed to be all the adults in a given situation. Which is pretty conjunctive as failures go, and especially in a world where you insist that you know all the adults personally from before you started a baugruppe rather that letting Bob (and his 5 friends who are new to you) all join.
You also mention “selection for predators”, but that seems to run against the (admittedly folk) wisdom that children at risk of abuse are those that are isolated and vulnerable. Daycare centres are not the central tendency of abuse; quiet attics are.