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.