It occurred to me I have been wasting so much time on symmetry of 6k±1 that I negated to realize it could all just be simplified down to primes are n=2k+1 but not n=4xy+2y+2x+1.
All odd integers are of the form n=2k±1, for all values of k including 0.
For values of independent variables x,y greater than 1, composite numbers of form n=2k±1 can be expressed as:
* (2x-1)(2y-1) = 4xy-2y-2x+1
* (2x-1)(2y+1) = 4xy-2y+2x-1
* (2x+1)(2y+1) = 4xy+2y+2x+1
If one subtracts the sets {4xy-2y-2x+1}, {4xy-2y+2x-1}, and {4xy+2y+2x+1} from {2k-1},{2k+1}, one will be left with the entire set of prime numbers, except 2.
Holding k values equal in both equations, 2k-1 and 2k+1 are distinct number forms, which are related to twin primes.
If k values are manipulated then 2k-1 and 2k+1 have the same value – for example 2(2)-1 is the same as 2(1)+1. In this sense, the equations are equivalent.
However, if we were to consider twin primes, then we would see that k is being held constant, and that each component of the twin prime pair takes a different form.
For example, 3,5 can be expressed as (2(2)-1),(2(2)+1); and 5,7 can be expressed as (2(3)-1),(2(3)+1).
When considering composites, 4xy-2y+2x-1 is of form 2k-1 and 4xy-2y-2x+1 and 4xy+2y+2x+1 are of form 2k+1.
For every n in 2k-1, -n exists in 2k+1 and vice versa, so that sets {2k-1} and {2k+1} are symmetrical around 0.
If n=4xy-2y+2x-1 is in 2k-1, then -n must be in 2k+1 and be of the form -n=4xy-2y-2x+1 or -n=4xy+2y+2x+1.
If n=4xy-2y-2x+1 or n=4xy+2y+2x+1 are in 2k+1, then -n must be in 2k-1 and be of the form -n=4xy-2y+2x-1.
Since {2k-1} and {2k+1} are symmetrical around 0, then |{2k-1}|=|{2k+1}| in a range -q<0<q
Since {2k-1} and {2k+1} are symmetrical in a range -q<0<q, and have the same absolute value, then we could sieve in just one form 2k-1 or 2k+1 to find all prime numbers other than 2; such that if a number can be expressed as n=|2k-1| but not n=|4xy-2y+2x-1| in the range -q,0,q; then n is a prime number.
The simplest expression considering only positive ranges:
A number n is prime if n=2k+1 but not n=4xy+2y+2x+1 for values of k,x,y equal to or greater than 1.
(I also found the equation is in this paper from 2009: Permuting Operations on Strings – Their Permutations and Their Primes.)