Biological hierarchies emerged from natural characteristics of number theory

TL;DR

This paper introduces the PzDom model, linking biological hierarchies and species formation to number theory and topological properties, with a novel connection to the zeros of the Riemann zeta function.

Contribution

It presents a new framework connecting biological organization with number-theoretic structures and topological analysis, including a novel link to the Riemann zeta zeros.

Findings

Hierarchical organization arises from a one-dimensional probability space.

Species are modeled as p-Sylow subgroups within a community.

Scaling parameters match the imaginary parts of Riemann zeta zeros.

Abstract

We show how biological grouping-particularly species formation-can emerge from interactions among populations governed by number-theoretic structure. In our framework, a species is identified with a $p$-Sylow subgroup of a community occupying a single niche; this identification is supported by a topological analysis. We call the resulting framework the patch with zeta dominance (PzDom) model. We then examine the system's topological properties in detail and demonstrate that both hierarchical organization and temporal ordering are induced by a one-dimensional probability space endowed with an appropriate topology. To clarify the appearance of induced fractal structure and its relation to renormalization, we develop a theoretical account based on a new observation: the scaling parameters that play the role of magnetization analogs coincide exactly with the imaginary parts of the nontrivial zeros of the Riemann zeta function. In the PzDom model, all required computations reduce to the time-dependent density of individuals.