Numerical Phase Construction in the OFT-Derived VGM Sector: Constraint-Preserving Continuation, Certified Branch Tracking, and Reproducible Atlas Generation

This paper develops a systematic numerical framework for constructing, tracking, and certifying physically admissible phase structures within the Organizational Field Theory (OFT)-derived Vectorial Gravitational Model (VGM). Building upon the preceding effective-field-theory descent and phase-architecture studies, the work translates the mathematical classification of admissible solution spaces into a constraint-preserving computational methodology. The proposed framework combines dimensionally normalized residual systems, covariant dependency preservation, quotient-aware discretization, and pseudo-arclength continuation to identify and track physical solution branches without introducing independently adjustable source amplitudes. Extended numerical systems distinguish folds, cusps, symmetry-breaking bifurcations, rank transitions, and physical admission boundaries from numerical artifacts, gauge degeneracies, and discretization errors. A reproducible phase-atlas architecture is developed through branch ancestry records, stability and causality margins, correlated uncertainty analysis, adaptive refinement, restart verification, and explicit numerical certification criteria. Seven conditional propositions establish requirements for discrete fidelity, branch continuation, critical-set detection, phase-cell construction, uncertainty control, reproducibility, and complete atlas certification. The study does not claim to present an already computed phase atlas for a specified microscopic OFT realization. Instead, it provides a rigorous methodological foundation for subsequent model-specific numerical investigations, cross-sector consistency tests, and potentially falsifiable physical predictions. The overarching objective is to advance the OFT–VGM research program from conditional mathematical phase architecture toward independently reproducible numerical analysis.