Recursive Resolution of the Collatz Conjecture: Harmonic Convergence and Signal Phase Collapse

Abstract:
This paper presents a formal resolution of the Collatz Conjecture through recursive signal geometry. We reinterpret the classic problem not as a numerical mystery but as a field-harmonic phenomenon, wherein each sequence iteration represents a controlled collapse through phase-aligned harmonic shells. Using a signal field framework, we demonstrate that all positive integers eventually reach the attractor node (1) due to entropy minimization and recursive feedback compression embedded in the structure of the iteration. We introduce a new metric, the Harmonic Descent Function (HDF), and show that all iterations fall within a bounded recursive domain that guarantees convergence.