Zhu-Liang Yang–Mills Existence and Mass Gap Necessity Inference ——Based on the Qiankun Quantum Recurron Framework: Entropy Minimization in Causal Networks

Within the meta-logical framework established in the Qiankun Quantum Recurron, this paper presents a complete necessity inference for the Yang–Mills existence and mass gap problem. This problem asks to prove that for any compact simple Lie group \ (G\), the quantum Yang–Mills theory on four-dimensional spacetime exists and has a strictly positive mass gap \ (> 0\). We first identify four core reccurons directly related to the Yang–Mills theory: the gauge field recurron \ (\), the spacetime recurron \ (\), the quantum state recurron \ (\), and the renormalization group recurron \ (\). We rigorously define the state sets and morphism sets for each recurron, and prove that they are coupled into a strongly connected causal network through deterministic relations such as the Yang–Mills equations, gauge invariance, path integral quantization, and renormalization group flow. We then construct a network entropy functional \ (ₘ₌\) that vanishes if and only if the theory exists and has a positive mass gap. By the Truth Metric Theorem, there exists a unique global entropy-minimizing state, which necessarily forces the Yang–Mills theory to exist and the mass gap to be positive. This paper also reveals the fundamental limitations of reductionist methods in addressing such global problems.