From "theory of everything" forum at megafoundation; reply to Andrew:
(this may clarify re: gravitational fields):
Chess model interpretation:
You are about to make a Chess move. You weigh up several possible options for that NEXT move.
For each option there is a "virtual world" of possible Chess games that COULD be played following such a move.
Moves in some of these virtual Chess games may interfere with moves in alternative virtual Chess games.
Consider the SUM of the virtual Chess games generated by the COMPARING OF COMPARISONS among various competing Chess strategies (game-plans).
A "comparison" is 2-D. A "comparing of comparisons" gives 4-D with a "uncertainty zone" namely the region of Chess moves that are logically possible in any of the virtual game-possibility-worlds that radiate from each "NEXT MOVE" option considered.
"Quadratic invariant" would be the 4-ness of this; the "hyperspace" aspect is the region of moves available to each 4-ness viewpoint (as these are movers all game-plans COULD contain.
Suppose you look at a SEQUENCE of several Chess moves: so: next; next; next; next, next, next,next.
Consider the "menu" of moves you might have made at each step.
Consider the "virtual Chess games" radiating from each menu. Consider just those moves that do not cancel among a bundle of virtual Chess games emanating from a particular "next move" decision.
Look at this for each step. (This looks so much like doing Quantum electro-dynamics; it is!)
Now: consider at each step we have a bundle of moves that COULD occur in any of the virtual Chess games associated with that step's menu of "immediate next move" options.
Over the several steps of "next" we have a space for each of logically possible moves in all local game-plans; and we can construct a space across all the steps of logically possible moves common to each step and still there (not cancelled) from first step to last step.
This gives a "quadratic invariant", that is a "4-ness" invariant integrated over the "tension" of weighing up alternatives across several steps.
This gives an "uncertainty region" that applies to the whole sequence of steps: you could make any of those surviving moves.
Why is it "gravitational entropy"?
"Gravitational" because it involves the freedom surface (force) associated with "old time" (just as centrifugal force is the "freedom surface" (force) associated with (where an object twirled in a circle around you on a string was being pulled a quarter turn ago) ; the survival of previous compared and matched patterns in new comparisons bundles them together in the binding of the new comparisons from a statistical perspective.
It is "gravitational entropy" because it involves "the common ground surviving over all "next" steps" specified only in terms of an uncertainty region where "gravity" itself is loosely defined within the freedom of definition this surviving uncertainty zone provides.
Re: cosmological dust modelling:
it appears that one can specify a dust field and calculate a "mutual definition space" between how the "dust field" defines "space-time" and how "space-time" defines the dust field. Any results may be self-fulfilling prophesies as they will cover "all bases" in a circular reasoning manner?
(I can show how appears that QED involves alleged circular reasoning) .
"The nonequilibrium entropy discussed here is a measure of loss of information about the system."
Looks like it!
calculation worth its salt is linked to
gravitational and net density fluctuations
in a cosmic background."
Exactly what the Chess model does! The "net density fluctuation" is the surviving "uncertainty" about what moves you COULD make in the future of your Chess game; even after doing "quantum electro-dynamics" on your changing "virtual game possibilities" at each step of your last say 5 moves and taking the net surviving "future moves options still available" to simultaneously DEFINE what you COULD do in the game now in terms of what you DID do during the recent series of moves.
This mixing of "initial conditions: final conditions" to DEFINE "every way space-time could happen" in-between is classic circular reasoning exposed in the structure of QED (I can show how).
The "gravitational fluctuation" (ways gravity COULD be defined between initial and final moves in a sequence of moves) is mixed with the "density fluctuation" (ways density (move-survival through layers) can be defined as the moves concerned are available right through from initial to final conditions so who knows which moves will be taken early and build up density through conservation over subsequent moves) as they define each other here it appears (against the background of the wider game of Chess.
" We discuss the origin of the entropy in our cosmological models and compare the definition of entropy in terms of correlation functions with the microcanonical definition in quantum statistical mechanics"
"Note though the first reference.It makes
a linkage with Entropy behavior and the
net 'arrow of time'."
Exactly! The "arrow" of "time" is the "next, next, next, next, next, next, next" I needed to generate this pattern. The "entropy" is associated with the concept of the linear sequence.
But "linear sequence" is associated with "number line", or "counting".
And "counting" involves "defining numbers".
And "defining numbers" involves "Zeno's Arrow" like self-referential constructions as I mentioned elsewhere, it seems.
But any attempt to "count" the "Zeno-structure" itself involves "counting.
Which opens a possibility to look at this in terms of Chris Langan's "conspansive duality" which is very pre-math.
In Christianity we know "God is Three and God is One".
In the Chess model analysis "t" as in "initial to final conditions" covering a sequence of "next moves" is approximately "S" in that the two concepts are mutually defining within mutual error margins which is looks like a classic scenario depicted by the idea of "conspansive duality"?
Because of this "mutual definition space" I placed two terms in a previous post to give:
S approximately equal to cct where "c" is "comparison."
Hopefully you can see the bridges I have built between different ways of looking at this:
your requirement of "fine tuning" I agreed with as in my own different way of approaching this subject I found to make it work I had to "fine tune" as by definition "comparison" gives more space.