Articles

This study reformulates the analytic structure of the Riemann Zeta Function within the framework of Fractal Arithmetic. Fractal Arithmetic is a new axiomatic system that treats natural numbers not merely as algebraic objects, but as fractal arithmetic wave functions composed of motif (M: motif), scale (S: scale), direction (Y: direction), and resonance (R: resonance) components. Under this structure, the zeta function is redefined as a resonance-weighted energy operator. Prime numbers are modeled as atomic resonance points in Fractal Arithmetic, and their resonance spectra are defined in the form ​. This model derives the critical line  of the zeta function as a scale–resonance equilibrium manifold. Thus, the Riemann Hypothesis becomes a necessary consequence under the axioms of Fractal Arithmetic.

This study reformulates the fundamental open problem of computer science, P vs NP, within the framework of Fractal Mechanics, independently of classical computational models. Fractal Mechanics is a novel mathematical paradigm that models each problem as a fractal wave function, composed of motif–scale–direction–resonance components. This approach demonstrates that the distinction between P-class and NP-class problems is not solely computational time, but also the topological resonance structure. Under the axioms of Fractal Mechanics, NP problems carrying multi-directional spiral resonance cannot be reduced to a unidirectional spiral structure. Therefore, within the FM framework, P ≠ NP is a necessary outcome.

This law appears with the same motif in: physical fields (spin, polarities, flow directions), atomic structure (shells, orbital orientations), planetary systems (stable resonance zones), galactic dynamics (spiral arm directions), information theory (bit strings, number of states), mathematics (number of functions, number of subsets), FM (spiral–fractal energy distribution, minimum-energy directions)

Fractal geometry abandons the “flat, fixed, scale-independent” structure of classical Euclidean geometry and instead describes a geometry that is: scale-dependent, self-repeating, composed of spiral or multi-layered motifs, preserving the same structure as scale increases. This suggests that the universe is not built from “straight lines and circles,” but from spiral-scaled motifs.