Spiral–Fractal Unified Theory 1: Triadic Scaling Manifolds and Physical Consequences of k = 3

Building upon the geometric foundations of fractal-spiral manifolds, we establish the physicalmanifestation of triadic scaling symmetry through the Triadic Scaling Theorem (Theorem 9).We demonstrate that discrete self-similarity in 120◦ tri-branch spiral manifolds requires a uniquescaling parameter k = 3, which emerges naturally from SU(3) center symmetry.The key contributions bridging geometric theory to observable physics are:(i) Geometric Necessity: We prove that SU(3) center symmetry requires k = 3 scaling in discrete spiralstructures, establishing the geometric foundation for triadic physics;(ii) Physical Scaling Laws: The triadic geometry yields parameter-free testable predictions for galacticrotation curves, quantum dimensional factors, and fundamental constant relationships;(iii) Transcendental Emergence: The discrete spiral structure naturally gives rise to classical constants π ande through geometric integration limits, providing a geometric origin for these universal constants.This geometric framework provides the theoretical foundation for unified spacetime-energy-information theory, offering a natural bridge between quantum mechanics and gravitational phenom-ena. The theory’s predictions are consistent with mathematical results on discrete scaling constraintsand align with observational data across multiple scales.