A Resolution of the abc Conjecture via the Prime-Structured Functional Framework (PSFF)

This manuscript provides a rigorous resolution of the longstanding abc conjecture through a novel mathematical approach, the Prime-Structured Functional Framework (PSFF). Central to number theory, the abc conjecture addresses a profound relationship between additive properties of integers and their multiplicative prime-factorization structures, particularly through the radical function. In response to unresolved logical ambiguities inherent in existing approaches—most notably Shinichi Mochizuki’s Inter-Universal Teichmüller Theory (IUT)—we introduce the PSFF, anchored by two innovative structures: a deterministic Prime Discovery System (PDS) for prime generation, and Structured Entropy Layering (SEL) to control arithmetic complexity. We establish two key lemmas, the Radical Control Lemma and the Finiteness Lemma, to demonstrate controlled radical growth and the finiteness of exceptional integer triples explicitly, validating these claims with comprehensive computational evidence. Our results conclusively affirm the abc conjecture, offering transparent logic, computational verifiability, and broader implications for arithmetic geometry and number theory.