Dear Juergen and Bruno:
Clearly I have a problem when I try to use mathematical terminology in
which I am not formally trained to explain my approach.

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So here is an attempt to explain it in just a few normal words. My
"system" be it a FAS or not is modeled on the logistics equation process
not the equation itself.
Here is the cascade:
{Rules(String 0) + SD(0)} -> String 1;
{Rules(String 1) + SD(1)} -> String 2;
{Rules(String 2) + SD(2)} -> String 3;
etc.
"String 0" is like an axiom.
"Rules" define a cascade [universe] and is the entire rule set.
"String 0" contains the entire initial alphabet.
"SD(N) is the self-delimiter.
All the "Rules" apply to each "String N".
The cascade is considered to define a universe as opposed to imposing from
"outside" a restriction to "mathematical structure".
For this reason I prefer "Pattern N" instead of "String N".
The cascade is a self contained system. I call it a FAS because I believe
it meets the definition.
The cascade is initially assumed everywhere [each step and overall] single
valued and elegant = deterministic. As a result each "String N" is more
complex than String (N -1).
The assumption leads to a contradiction when "String N" exceeds the
complexity allowed by Chaitin. More information must be added to the
cascade for it to continue.
Add to this Godelian incompleteness and a touch of just plain "do not care"
as possible aspects of the Rules. The result is a succession of fresh
"String O" initiations to the cascade. Some cascades are sufficiently well
behaved to support SAS.
The cascade is modified by considering it to be an isomorphism and its
association with a particular pattern to be an isomorphic link. All
patterns as considered to exist simultaneously in infinite repetition in a
Superverse. The Rules act as a comparator mitigating isomorphic link
shifts between successor patterns.
The necessary gradients within the Superverse are provided and stirred by
the Superverse/Nothing alternation which is historyless and driven by an
incompleteness in both the Superverse and the Nothing.
I will expand my reading in logic to help my communication, but I believe
the above total Superverse be an infinite collection of "FAS" of all
complexities including those where the Rules are completely "do not care".
Hal