Emergent Scale-Dependent Geometry and π from a Fractal Prime Network: Pre-geometric Unification of General Relativity and Quantum Mechanics

In this work, a novel model of emergent geometry and the constant π is presented, arising from a discrete, fractal network of prime numbers organized according to the modulo-15 structure. It is proposed that the pre-geometric prime network provides a sufficient foundation for reconstructing both classical Euclidean geometry and the curved spacetime of Einstein’s general relativity, without the need to introduce additional dimensions, fields, or free parameters. At the fundamental level, at scales r → 0, the ratio of circumference to diameter approaches unity (π_eff → 1), corresponding to a pre-geometric state in which no well-defined metric yet exists. As the scale increases, through recursive spiral interactions and the modulo-15 wheel, the value of π_eff(r) continuously transitions to the classical constant π, while the fractal dimension D(r) evolves from 1 to 2. The model generates emergent curvature profiles consistent with the Schwarzschild and Kerr solutions, as well as vortex structures interpreted as analogues of dark matter and conformal effects corresponding to dark energy — without requiring these components to be introduced as separate physical entities. The presented numerical results and the derivation of the main field equation indicate that the unification of general relativity and quantum mechanics may occur at the arithmetic level rather than the geometric one. The model is fully falsifiable and opens new directions of research into the role of prime numbers in the structure of spacetime.