Let me correct one little issue which I think helps to clarify what I am saying. I add a comment on the universal dove-tailer.
1) Yesterday I said that the cascade 1+1=2, 2+1=3, 3+1=4, etc. was not everywhere elegant. But I went outside my identification process for "determinism" = "everywhere elegant proof" to do so. The error was to slip into full number theory and think: "84" one of the strings this cascade will eventually reach for example has more than one proof. That is true because number theory is richer than just "data + 1 =". But here the working definition of "deterministic" I use is that all of the selected set of rules of a cascade act at each step on all the data. You should not have within the idea of "deterministic" some of the rules active today and some others active tomorrow, some of the data regions exempt from some rules today and other data regions partially exempt tomorrow unless that was itself in Rj. Deterministic as I understand it = all the rules of physics always apply to the entire state of the universe. So for the above cascade we have selected "data + 1 =" as the exclusive expression - the entire rule set of the operative FAS - for which we are seeking the cascade of values given some start data [effectively the axiom of that cascade] in this case "1". The fact that this rule may also belong to a different and richer FAS is not germane to the cascade viewed as an attempted deterministic sequence. The operative FAS contains just this one rule as its Rj and applies it to all the data at each step. That makes this cascade everywhere elegant because there is no other proof of any of the output strings available in the operative FAS. Thus hits the complexity wall established by the complexity of the operative FAS. 2) A universal dove-tailer generating all strings using a fixed algorithm every part of which applies to all the current data in the same way at each step seems an odd thing. A dove-tailer is not directly generating the "whole" ensemble. What it is doing is selecting by fixed rules a particular string out of the ensemble and adding some quantity of bits, putting that back in and selecting another and adding some quantity of bits to it etc., etc. That is a very selective and complex process on a step by step basis. You wind up constructing this incredibly complex everywhere elegant proof of what must then be an incredibly complex object that is nevertheless considered to be a very low complexity object. If I have the ideas of a UD, elegant proof, and AIT complexity correct the UD appears to be a contradiction. The contradiction again is that we have a FAS that constructs a proof it knows is elegant that is nevertheless far more complex than any proof it can know is elegant. Hal
