# Observations concerning Multiway Causal Graphs

Posted 4 months ago
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## 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. Answer
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Posted 4 months ago
 Note: The prime number assigned must be (globally) unique to all proofs Answer
Posted 4 months ago
 Note: Causal pathway phonons are super efficient causal pathways which are reused Answer