Harristotelian Logic

Prime residue axiom:

Given differing primes p_1 and p_2, there is no preference for a particular residue of p_2 for p_1 mod p_2.

What's intriguing about this axiom is that with it you can trivially prove the Twin Primes Conjecture and trivially disprove Goldbach's Conjecture.

The argument for disagreement then is on whether or not it is an axiom.

Note such disagreement would require a residue preference, for instance, say, mod 3, primes would need to tend to say, have a residue of 1. For instance 7 mod 3 = 1. So a preference for a particular residue would mean that primes in general might prefer 1, as in other prime numbers out to infinity would want to have 1 as a residue modulo 3, if there were a preference.

Maybe you could call it a postulate and split mathematics around variations, but then you'd need primes to agree with you.

So far all evidence is that they do not care.

However, if you don't want Goldbach's gone and Twin primes solved, then you need a prime preference.

Otherwise those problems are gone. Solved with probabilistic proofs relying on the axiom above.