Mathematics

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.

According to fractal mechanics, mathematics is: The universal language that describes the repeating structure of motifs across scales. In other words, mathematics is not the science of numbers, but the science of how scales behave.

Euler’s identity establishes the equivalence of complex exponents with trigonometric functions by combining fundamental constants such as e, i, and π:

Phase–duality algebra is a unique structure that combines the geometric, algebraic and physical properties of trigonometric functions (sin, cos, sec, csc, tan, cot) and covers both circular and hyperbolic rotations. This algebra is reinterpreted within the framework of Clifford algebra and Lie groups, providing a strong basis for both mathematical consistency and physical modelling.

Traditionally, π\pi is defined as the ratio of a circle’s circumference to its diameter: 𝜋=circumference/diameter
This is a fundamental constant in geometric and trigonometric operations. However, based on our analysis of mathematical focal points and optical-electronic systems, π\pi is not just a geometric constant; it may be a critical point where the energy density is focused!

Traditionally, \frac{\pi}{2} is the critical point of trigonometric functions and is associated with maximum signal amplitude: it plays a special role in wave mechanics, optical systems, and quantum field theory. However, according to our analyses with mathematical focal points and optical-electronic systems, \frac{\pi}{2} is not just a trigonometric transition point, but a critical mathematical focal point where the energy density is maximum!

Traditionally, the definition of 𝑒 represents exponential growth and continuous systems. However, our mathematical and optical analyses show that ee is not just an abstract constant; it acts as a focal point for energy!