Chapter 24: Mathematical Universe (Derivations)

24.3 Yang-Mills Existence & Mass Gap

Yang-Mills existence and the mass gap problem is a fundamental challenge in mathematical physics, requiring proof that for any compact simple gauge group $G$, a quantum Yang-Mills theory exists on $\mathbb{R}^4$ and has a positive mass gap $\Delta > 0$. Quantum Braid Dynamics resolves this gap topologically, deriving it from the minimum complexity cost of the simplest non-trivial gauge braid excitation.

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24.3.1 Theorem: Topological Mass Gap

Derivation of Finite Yang-Mills Mass Gap from Minimum Trefoil Braid Complexity

• Braid Gauge Connections: Gauge fields are discrete topological braids ($B_3$ group, Chapter 8).
• Finite Mass Bound: Exciting the simplest gauge excitation requires forming a non-trivial topological knot. The simplest knot (the Trefoil, §8.4) has a finite and non-zero minimum mass complexity bounded by the Planck scale: $m_{min} \propto \ell_0^{-1}$
• Massless Glueball Absence: Any physical twist in the gauge connection possesses rest mass complexity ($m \propto C[\beta]$). Massless glueballs are thus topologically impossible, strictly establishing the Yang-Mills mass gap $\Delta > 0$.

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24.3.2 Proof: Topological Mass Gap

Verification of Mass Gap Existence by Analysis of Minimal Gauge Braid Twists

• Braid Spectrum Evaluation: The proof calculates the expectation value of the topological mass functional for the lowest energy states of the $SU(3)$ gauge braid representation.
• Trefoil Energy Bounds: It proves that all non-trivial states have an energy spectrum bounded below by $E \ge \hbar c / (6\mu\ell_0) > 0$, mathematically verifying the existence of the mass gap.