A Deterministic Proof of the Riemann Hypothesis viathe Prime-Function Framework and the Prime Discovery System

This manuscript presents a deterministic proof of the Riemann Hypothesis (RH)—one of the most significant and longstanding open problems in mathematics. The proof introduces two innovative mathematical structures: the Prime Discovery System (PDS), which systematically generates prime numbers through wheel factorization, segmented sieving, and deterministic primality testing; and the Prime-Function Framework (PFF), which directly connects prime distribution to computational complexity via prime-indexed functional inversions. By assuming RH to be false, the manuscript demonstrates that inversion entropy—representing computational complexity of prime-indexed transformations—exhibits super-polynomial growth, inevitably leading to deterministic intervals devoid of prime numbers. This result conflicts explicitly with the independently established completeness properties of the PDS and polynomial boundedness conditions of the PFF. Thus, the contradiction conclusively confirms that the Riemann Hypothesis must be true, verifying that all nontrivial zeros of the Riemann zeta function ζ(s) lie precisely on the critical line Re(s) = 1/2. The manuscript rigorously details each step, lemma, and proof structure, supported by computational validations to reinforce theoretical conclusions. This result has profound implications across analytic number theory, computational complexity theory, cryptography, and mathematical physics, opening numerous avenues for further research.