The Golden mean, scale free extension of Real number system, fuzzy sets and $1/f$ spectrum in Physics and Biology

TL;DR

This paper introduces a scale-free, fuzzy extension of calculus and physics models, explaining the origin of the 1/f spectrum and the significance of the golden mean in natural systems.

Contribution

It presents a novel scale-free solution framework to differential equations, linking 1/f spectra, fuzzy calculus, and the golden mean in physics and biology.

Findings

Existence and uniqueness of scale-free solutions to differential equations.

A fuzzy extension of calculus and physics frameworks.

Explanation of the golden mean's role in natural phenomena.

Abstract

We show that the generic $1/f$ spectrum problem acquires a natural explanation in a class of scale free solutions to the ordinary differential equations. We prove the existence and uniqueness of this class of solutions and show how this leads to a nonstandard, fuzzy extension of the ordinary framework of calculus, and hence, that of the classical dynamics and quantum mechanics. The exceptional role of the golden mean irrational number is also explained.