Phase Architecture of the OFT-Derived VGM Sector: Control Manifolds, Stability Strata, and Critical Branch Transitions

Building upon the normalized effective-field-theory descent from Organizational Field Theory (OFT) to the Vectorial Gravitational Model (VGM), this paper develops a systematic mathematical framework for identifying, classifying, and testing physically admissible phase structures within the VGM solution space. The study introduces control manifolds, stability strata, invariant phase criteria, and critical branch transitions while distinguishing genuine physical phases from coordinate artifacts, parameter redefinitions, and violations of effective-field-theory validity. Universal Wilson coefficients are separated from state-dependent controls and derived source amplitudes. A unified admission framework incorporates constraint preservation, hyperbolicity, energetic stability, causal consistency, effective-field-theory bounds, and correspondence with General Relativity. Lyapunov–Schmidt reduction is employed to characterize candidate fold, cusp, and symmetry-breaking bifurcations, subject to additional dynamical and physical admissibility conditions. The work further establishes a reproducible numerical strategy involving pseudo-arclength continuation, spectral stability tracking, branch fingerprints, correlated uncertainties, and cross-sector validation. Rather than claiming the existence of experimentally established phase transitions, the paper defines the mathematical conditions under which distinct VGM phases may emerge, remain stable, undergo critical transitions, or be rejected. This research provides a theoretical foundation for the subsequent construction of a Numerical Phase Atlas and advances the OFT–VGM program toward mathematically controlled, falsifiable predictions.