Theorem - All mathematical proofs which can be modeled by a multiway causal graph can also be mapped to prime numbers and solved by a GrayCode counter in polynomial- or linear time:
Proposition 0000
All rewrite rules can be mapped to primes
Proof
Let N meet the requirement it must be an integer, and Let the rewrite rule: [ C->ABA,D->CBC,D ], and you wish to know if the pattern [ ABABABA ] is conserved by (ie. "in") this rule, it is enough to show that any mapping [ A,B,C,... ] => [ 2,3,2 3 2,... ] == N mod (P1 P2 ...Pn) holds.
Lemma 0001
All 2SAT problems can be solved in polynomial time
Proof
Let R be a problem in 2SAT. Iff R has atmost 2 free variables, and R can be modeled as a Boolean SAT problem, then R has a guaranteed upper bound polynomial runtime.
Proposition 0002
All prime number divisibility problems are 2SAT
Proof
The problem of Z == N mod P, where P is a prime and N, Z is an integer reduces to a YES/NO decision problem. All YES/NO decision problems are 2SAT.
Lemma 0003
GrayCode counters perform atmost 1 bit-flip during each rewrite
Proof
A GrayCode counter is defined as an ordered number system derived in such a way that any successive value differs in only a single binary digit
Lemma 0004
All GrayCode counting problems are decidable
Proof
Let R be a GrayCode algorithm. Any Z-Bit R is guaranteed to halt following 2^Z rewrites
Lemma 0005
All GrayCode counting problems perform in linear time, and are linear time optimal
Proof
2^Z bit rewrites are polynomial, with the GrayCode counter generating a result each clock cycle, however, with the selection of appropriate prime numbers -- one's exact position after any computed step is known
Proposition 0006
Any rewrite rule mapped to appropriate primes can be mapped onto a GrayCode
Proof
Let R be the rewrite rule [ C->ABA,D->CBC,D ] be mapped appropriately to the prime numbers A=2,B=3, where C and D are composites: C=12 (C = ABA = 2 3 2 = 12),D=432 ( D = CBC = 12 3 12 = 432 ), its GrayCode value ( and foliation step, later proven ) corresponds to
...,
[ 2 3 2 3 2 3 2 ],
...
Proposition 0007
Any GrayCode counter can solve any foliation step in polynomial or linear time
Proof
By Proof (3) and Proof (5) and Proof (6), any rewrite step appropriately mapped onto primes, mapped onto a GrayCode counter-- can produce one proofstep each clock cycle
Lemma 0008
for each foliation step in the Multiway Causal Graph, the rewrite step is reversible
Proof
All causal graphs are causal
Lemma 0009
for each foliation step in the Multiway Causal Graph, iff the rewrite step is reversible, energy is conserved, and each foliation step -- when represented as time -- is emergent and not fundamental; and is also reversible
Proof
By Proof (8), assuming each rewrite step requires energy, the state of entropy is completely knowable and reversible, because of Proof (8)
Proposition 0010
iff all rewrite rules can be mapped to appropriate primes numbers,
and all prime number divisibility problems are 2SAT
and each foliation step performs atleast 1 rewrite,
and GrayCode counters can solve foliation steps in polynomial or linear time, then
All proofs which can be modeled by multiway causal graphs can be mapped to- and solved by reversible GrayCode Counters in polynomial or linear time
Proof
Each foliation step is a subsequent hash into the GrayCode counter
Q.E.D.