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I introduce the Collatz-Octave Framework (COM) as a universal structuring principle for mathematical periodicity, number theory, and quantum scaling.
This model describes recursive harmonic scaling in numerical sequences, revealing deep connections between prime distributions, the Riemann Hypothesis (RH), and modular renormalization techniques.
Through COM, I construct a spectral operator whose eigenvalues correspond to RH zeros, satisfying the Hilbert-Polya conjecture.
Additionally, I demonstrate that prime residues exhibit quantum wave structuring, enforcing their alignment along the critical line.
Using renormalization scaling, I establish that modular periodicity in prime distributions follows self-organized criticality, leading to a stable attractor structure that supports RH.
My findings suggest that harmonic oscillations within the Collatz-Octave sequences encode fundamental energy states analogous to quantum field interactions.
This interdisciplinary approach unites number theory, quantum mechanics, and renormalization physics, providing a new perspective on the foundational principles governing mathematical structures.