Abstract

Gödel’s Incompleteness Theorems define the limits of formal systems by revealing the existence of true statements that cannot be proven within their own axiomatic structure. These theorems rely on self-referential paradoxes, which assume formal logic operates within a closed, static framework. However, this assumption overlooks the deeper nature of structured emergence. Using the Chirality of Dynamic Emergent Systems (CODES), we propose a fundamental shift: mathematical truth is not a fixed binary but a coherence-weighted phenomenon, governed by structured resonance rather than rigid self-reference. In this reframing, incompleteness is not an intrinsic flaw but an emergent spectral gap within prime-structured systems. What Gödel treated as paradoxical undecidability is instead a nonlinear coherence alignment problem, resolved dynamically through phase-locking mechanisms. This perspective dissolves incompleteness not by proving all truths within a system, but by embedding systems in a structured resonance hierarchy where coherence replaces paradox.