Here is a revised version of my comments on this subject. I think it fixes several aspects of what I have had to say earlier.

## Advertising

Standalone deterministic evolving universes: Such a universe is describable as a concatenation of single output programs of the form: Rj(Aj) -> B; Rj(B) -> C; ... Rj(F) -> G; Rj(G) -> H; Rj(H) -> I; .... "Rj" is a subset of the rules of an N-bit FAS. "Aj" is an axiom of that FAS. There are no potential branches in or out. It is an elegant proof and defines the complexity of each of its objects B,C,...F,G,H,I ... Because the proof is everywhere elegant each successive object is more complex than its preimage. This is the type of theorem cascade I was trying to identify. It has a problem: It will - if it does not stop - eventually outrun the complexity of its N-bit FAS. Since the proof is known to be elegant this is [as I understand it] a contradiction of Chaitin's result. However, if it stopped at complexity N + c as required by Chaitin [again I think] the end state would be highly but finitely complex. Highly but finitely complex states are going to have some consequent under the rules of the cascade. It can not be both "provably highly but finitely complex" and "absent operative correlations" under the same set of finite rules. It might reach an absence of operative correlations when it becomes infinitely complex and the domains of any correlations fall below the resolution of the still finite Rj of the cascade. So the cascade can not stop prior to an infinity complex end state. This seems to be a contradiction. It can be cured if the FAS is able to spontaneously take in additional complexity from an outside source. I call such a source a Superverse. The new information would act to replace the latest object of the cascade with a new axiom acceptable to the Rj and the cascade proceeds. I use this to argue in favor of non deterministic standalone universes. Now this universe is actually an isomorphism linked to the successive strings [objects] B, C etc. The isomorphism has the same Rj as its laws of physics. This should continue to hold even if one shifts to a universal dovetailer running all universes simultaneously. Individual isomorphisms would still jump to the string currently containing as the latest sub string [string structure: ....;sub string;sub string;latest substring] that is the successor state of that isomorphism under its individual Rj. Thus each individual isomorphism remains a single output program or elegant proof under its Rj. A non deterministic result is sustained. Random histories are just isomorphisms with Rj = "do not care" and they jump as easily as any isomorphism. Well that is the current state of the argument. Hal