The Prime‑Dimensional Information Geometry of the Universe - Evolution in Discrete ln 2 Steps

This work presents the complete formulation of the Prime‑Dimensional ln 2 Information Geometry: a unified framework in which number theory, information theory, geometry, and physics all arise from the same discrete logarithmic structure. The theory identifies the twin‑prime gap as the smallest structural step in number space and shows that it corresponds exactly to the smallest informational step in physics, the rewrite unit ΔW=ln⁡2. This establishes a direct equivalence between prime gaps, twin‑prime spacing, and the ln 2 evolution of physical systems. At the physical scale, the framework shows that particles, black holes, and the universe all evolve through fixed powers of ln 2. Compression, expansion, curvature, rotation, entropy, information, and cosmic time each scale as integer powers of ln 2, forming a single discrete ladder that governs all physical change. Particle identity, weak‑force transitions, black‑hole accretion, jet emission, and cosmic expansion are revealed as different manifestations of the same ln 2 stepping behaviour. A key result is that information scales as ln(⁡n) while geometry scales as ln⁡2. The difference ln(⁡n)−ln(⁡2) acts as a dimensional factor, showing that the number of degrees of freedom in physical systems arises from the projection gap between informational and geometric scaling. This provides a natural explanation for dimensionality and unifies the behaviour of particles, black holes, and the universe under a single prime‑dimensional structure. The final outcome is a unified, testable, and falsifiable Theory of Everything in which nature is fundamentally discrete and evolves in logarithmic steps. Prime gaps define the structural ladder, ln 2 defines the informational ladder, and all physical quantities scale as fixed powers of ln 2. The result is a coherent architecture linking number theory, information geometry, and the evolution of the universe.