The Unity Theorem: Prime Emergence via 4D Hub Activation and the Completeness Identity
Abstract
We establish a geometric characterization of prime numbers through the Hub Activation Ratio Λ_hub(n) := n/(n+4). Defining Q(n) := Λ_hub(d(n)) = d(n)/(d(n)+4), where d(n) is the divisor count, we prove: n is prime if and only if Q(n) = 1/3 (exact fixed point), while composites satisfy Q(n) ≥ 3/7 with a perfect gap. The completeness identity Λ_hub(n) + β(n) = 1 connects prime emergence to the spectral constants τ* ≈ 0.145 and κ ≈ 0.8509 via the Bosonic string limit n=22.prime numbers, divisor function, dimensional activation, spectral partition, proximity graphs, Phase-OS
Fields of Study & Subjects
Analytic Number Theory Research
Advanced Mathematical Theories and Applications
Smarandache Functions and Recreational Number Theory
Publication details
- Journal / source
- Zenodo (CERN European Organization for Nuclear Research)
- Publication date
- 1 March 2026
- Language
- EN
- Document type
- Preprint