Regarding Prime-Preserving Bijections and Their Rigidity

Abstract

Bijections f: N → N that preserve primes are considered multiplicatively. We demonstrate that if this function f
is uniformly gapped on primes p such that f(p) /= p and if the set of these primes that are shifted has a
divergent reciprocal sum, then there is a limited number of primes that f may permute. These bijections may
nontrivially permute an infinite number of composites linked to a limited number of moving primes, contrary
to previous assertions. Our outcome strengthens earlier claims, thereby resolving a complex issue regarding
mathematical systems' rigidity.