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.