The Rational Frequency Method: A Unified Proof for Bertrand's Postulate, Oppermann's, and Legendre's Conjectures

This paper introduces the Rational Frequency Method (RFM) as an algebraic framework to demonstrate the existence of prime numbers within defined intervals. By proving the theorem that any real number |N| ≥ 2 is a product of a positive prime p and a frequency scaling factor F ∈ R where |F| ≥ 1, we derive a Prime Squeeze Theorem. Through the strategic algebraic substitution of N as a function of perturbed frequency (f + t), we prove that these intervals are strictly nested within the boundaries of Bertrand’s Postulate, Oppermann’s Conjecture, and Legendre’s Conjecture for all n ≥ 2. The RFM successfully derives Bertrand’s Postulate as a verified base case, establishing a unified scaling property for prime distribution.