Emergence of Gauge Structure, Flavor Hierarchy, and Ground State Selection on the Lambert-W Branch (Information-Geometric Physics System IV)

This paper constitutes Part IV of the Information-Geometric Physics System (IGPS) series. It develops the IGPS into a rigorous effective field-theoretic framework where the Standard Model's gauge symmetries, flavor mass hierarchies, and mixing matrices emerge strictly from the topology and geometry of "informational seams" (topological defects), drastically reducing parametric freedom. Key Contributions and Discoveries: Emergent Gauge Structure: Demonstrates that gauge symmetry is not postulated a priori but derived dynamically from the representations of the fundamental group of the seam complement. The minimal nontrivial configuration (a 3-seam link) uniquely selects SU(3) as the emergent gauge group. Flavor Hierarchy & Mixing: Reveals that fermionic degrees of freedom arise as localized WKB modes along the seams. Their mass spectrum exhibits an exponential hierarchy governed by geometric action functionals, while flavor mixing is derived from spatial overlap integrals. CKM and PMNS Unification: The framework naturally reproduces the hierarchical structure of the quark sector (CKM) and the democratic large mixing of the lepton sector (PMNS) as two geometric regimes (localized vs. delocalized) of the exact same underlying mechanism. Parameter-Free Cabibbo Prediction: By deriving the effective transverse potential from extrinsic curvature and incorporating topological measure shifts, the geometric stiffness parameter is strictly determined. The framework predicts the Cabibbo mixing parameter lambda approx 0.2245 conditional on the seam geometry locking at the transcendental Lambert-W branch point (K^2 R_c^2 = 1/e). * Hidden Supersymmetry (Open Problem): The paper openly identifies that the dynamical RG flow required to lock the system at this exact 1/e fixed point corresponds structurally to a hidden N=2 supersymmetry on the seam worldsheet. The verification of the Dirac index for these modes is established as a precise, calculable open problem for future research. Cosmological Implications: The same holonomy structures responsible for CP violation intrinsically induce baryon asymmetry (eta_B ~ 10^{-10}) and resolve classical singularities via geometric saturation (Born-Infeld dynamics). By replacing ad hoc parameter fitting with strict topological and geometric conditions, this work provides a clear, falsifiable pathway toward structural precision predictions in fundamental physics.