Spiral–Fractal Unified Theory 0: A Tripartite Proof of k = 3 in Quantum Gauge Theory

We present a comprehensive proof that the scaling parameter k = 3 is uniquelydetermined in quantum gauge theory through three independent mathematicalpathways. Path I employs group-theoretic constraints from SU(3) center analysis, es-tablishing that k must be a multiple of 3. Path II utilizes Chern-Simons quantizationand effective action minimization to demonstrate that k = 3 minimizes the physicalaction among valid candidates. Path III verifies renormalization group stabilitythrough Ruelle spectral analysis. The convergence of these disparate mathematicalframeworks provides robust evidence for k = 3 as a fundamental constant. Allresults are computationally verified and fully reproducible. Complete computationalframework and data available at: https://doi.org/10.5281/zenodo.15837578.本研究通过三条独立的数学路径，严谨证明了量子规范场论（quantum gauge theory）中的标度参数k=3具有唯一性。路径I基于SU(3)中心分析得到的群论约束条件，证明k必为3的整数倍；路径II借助陈-西蒙斯（Chern-Simons）量子化与有效作用量极小化方法，证实于所有合法候选取值中，k=3可使物理作用量取得极小值；路径III通过吕埃尔（Ruelle）谱分析验证了重整化群（renormalization group）的稳定性。上述三类迥异的数学框架彼此印证，为k=3作为基本常数提供了坚实的证据支撑。所有研究结果均经过计算验证，且可完全复现。完整的计算框架与数据集可通过以下链接获取：https://doi.org/10.5281/zenodo.15837578。