Spontaneous Dimensional Symmetry Breaking and the Yang-Mills Mass Gap

A foundational challenge in modern quantum gravity is explaining the macroscopic dimensionality of the observable universe without relying on a priori geometric assumptions, while simultaneously accounting for the quantized mass spectrum of elementary particles.

Building upon the Information-Topological Register Model (Phase A and B), this manuscript proves that a discrete, one-dimensional network of entangled information bits must thermodynamically collapse into exactly three spatial dimensions.

By defining the network's spectral dimension (D_s) through information diffusion and introducing a macroscopic Topological Free Energy Functional, we demonstrate that the emergence of 3D space is driven by Spontaneous Dimensional Symmetry Breaking. The universe systematically cools until it reaches D_s = 3, plunging into a global thermodynamic minimum.

Crucially, this manuscript bridges cosmological thermodynamics with microscopic knot theory. Relying on Zeeman's Unknotting Theorem, we show that 3D is the unique integer geometry allowing 1D strands to form stable knots (matter). By linking the bound topological stress to the minimal crossing numbers of these knots (c ≥ 3 for stable fermions), we provide a rigorous mathematical origin for the discrete quantization of mass, naturally deriving the Yang-Mills Mass Gap from first topological principles.

This work includes two computational visualizations:

1. The Topological Mass Gap (Discrete Mass Spectrum)

2. The Thermodynamic Potential of Spontaneous Symmetry Breaking