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

Building upon the rigorous mathematical foundations established in Paper 0 through breakthrough 2024-2025 advances in automorphism classification (Ginzburg, 2025) and spectral gap theory (Viana, 2025), we demonstrate the physical manifestation of triadic scaling symmetry through the Triadic Scaling Theorem (Theorem 9). We establish that discrete self-similarity in 120-degree tri-branch spiral manifolds requires a unique scaling parameter k = 3, which emerges naturally from modern SU(3) center symmetry analysis. 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 spiral structures, establishing the geometric foundation for triadic physics;(ii) Physical Scaling Laws: The triadic geometry yields parameter-free testable predictions for galactic rotation curves, quantum dimensional factors, and fundamental constant relationships;(iii) Transcendental Emergence: The discrete spiral structure naturally gives rise to classical constants pi and e 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 phenomena. The theory’s predictions are consistent with mathematical results on discrete scaling constraints and align with observational data across multiple scales.