Mathematics

Spiral-Fractal time functions break the classical linear understanding of time, defining it as a multiscale, cyclical, and resonant structure. This approach creates new computational possibilities in both physical systems and biological/social processes.

This paper formally proves Goldbach’s conjecture within the framework of Fractal Arithmetic and the Riemann Hypothesis. In fractal arithmetic, each natural number is defined as a fractal wave function composed of motif, scale, orientation, and resonance components. The Riemann Hypothesis is a necessary consequence under fractal arithmetic axioms. This regularity makes the spiral–fractal density function of prime distribution equal to 𝐷(𝑁) = 1 in every interval.

Spiral numbers are a functional and fractal extension of classical complex numbers: S=a+bθ+if(θ)

This study reformulates the classical Hodge Conjecture within the framework of Fractal Analysis. Fractal Analysis is a paradigm in which the topological structure of algebraic varieties is represented by multi-scale fractal resonance modes, and the algebraic subvarieties are represented by geometric motifs. This approach reinterprets Hodge decomposition as scale decomposition, harmonic forms as minimal energy resonances, and Hodge classes as rational-phase symmetric resonance modes.

For an elliptic curve E/Q, the Birch – Swinnerton – Dyer Conjecture expresses the correspondence between two different worlds: – Arithmetic world: the structure of rational points on E(Q) → rank – Analytic world: the behavior of the function L(E,s) at s=1 → order of the zero

This article defines a new mathematical paradigm that I call Fractal Analysis. Fractal Analysis is built upon three fundamental components in order to explain the multi-scale nature of algebraic, topological, and analytic structures: Fractal Motif, Fractal Resonance, and Fractal Flow. This triadic structure unifies geometric, topological, and dynamical properties—traditionally studied in separate disciplines of classical mathematics—within a single integrated framework. The paper formally presents the axiomatic foundation of Fractal Analysis, its structural components, and the relationships between these components. In addition, the relationship of Fractal Analysis with Hodge theory, algebraic geometry, and multi-scale analysis is discussed.

This study presents a new framework called Fractal Arithmetic, which reformulates classical number theory through the concepts of fractal structure, motif, scale, direction, and resonance. Fractal Arithmetic treats natural numbers not merely as algebraic objects, but as fractal arithmetic wave functions. Each number is characterized by its prime factor structure, magnitude scale, directional flow within sequences, and resonance density within arithmetic patterns. Prime numbers are modeled in Fractal Arithmetic as resonance points with maximum motif purity, while composite numbers are modeled as structures carrying motif diffraction. Modular arithmetic is reinterpreted as resonance orbits. This paper presents the formal axiomatic foundation of Fractal Arithmetic and proposes a new structural/topological perspective on classical problems of number theory (especially prime distribution and modular structure).

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)