The Infinite Field of Rational-Conforming Integer Sets

This paper explores special prime classes, S, such that if q is a prime in S, then there exists rational functions, α_q=(f(q))/(g(q)), such that S= {q∈℘: ∃f,g∈Z,g(q)≠0,α_q∈Z,} have asymptotic distribution with a prime density coefficient, C_℘ (α_q,α_p )=〖α_q〗^2/〖α_p〗^2 [ A(α_p )]^2/[ A(α_q)]^2 , where, the primes p, Prime classes include Sophie primes. The rational functions, (f(q))/(g(q)) , form a prime index class for ℘. I demonstrate that the set of primes conforming to a rational mapping (f(q))/(g(q))∈Z and the associated cotangent–tangent phase condition tan⁡((f(q))/(g(q))) cot⁡〖β_p^m 〗=1 constitutes an infinite field of phase-locked primes, p. Each prime corresponds to a unique angular phase, β_q^m on a cotangent lattice, and the iteration among successive phases defines a self-referential, recursively stable prime ladder. The distribution of phased lock sets of primes is not unique to Sophie primes. Indeed, any sets of integers that have the rational function property tan⁡((f(q))/(g(q))) cot⁡〖β_p^m 〗=1, for which the phase is a rational set of values (f(q))/(g(q))=constant ∈Z including Perfec numbers also follow the successive phases defines a self-referential, recursively stable infinite ladder.