Generalized Singularity Degradation and Phase-Transition Mechanics (GSDPM): Classifying Coordinate, Topological, and Capacity-Saturated Divergences within the ZFC Axiomatic Bridge

In general relativity, nonlinear continuum mechanics, and quantum field theory, classical descriptions break down at singularities and ultraviolet divergences. However, the mere mathematical smoothing or ad-hoc cutoff of an infinity does not automatically resolve the underlying physical singularity; rather, a rigorous theory must classify which algebraic, geometric, and dynamical structures degenerate at the boundary of validity and how they transition to more fundamental degrees of freedom. In this paper, we establish Generalized Singularity Degradation and Phase-Transition Mechanics (GSDPM), a deductive physical framework anchored in our 71-page ZFC Master Reference Monograph (Lee, 2026, DOI: 10.5281/zenodo.23486681) [1] and synthesized with the Causal Operator Learning System (COLS, DOI: 10.5281/zenodo.23556758) [2] and the Dynamic Topological Critical Transition System (DTCTS, DOI: 10.5281/zenodo.23557507) [5]. Governed by the universal quadruple 𝔘 = (𝒯_0x8A2C, 𝐂_UPTC, η_friction, ϵ_SHID) with strict order-of-limits separation from the Colombeau mollifier scale ϵ(α) = exp(−1/(1 − α)), we formulate six core axioms (GSDPM-1 through GSDPM-6) and prove six fundamental theorems: (1) Exhaustive Trichotomy of Physical Divergences into Type I (Coordinate/Gauge Null-Ideal Artifacts in 𝒩^α(Ω)), Type II (Topological Free-Boundary Pinch-Offs with ΔPHₖ ≠ 0), and Type III (True Capacity-Saturated Phase Transitions at 𝐂_UPTC); (2) Exact Quotient Elimination of Type I Divergences with Order-of-Limits Commutation; (3) Sobolev H³ Regularity and Positive Topological Precursor Lead Time (Δt_lead > 0) for Type II Pinch-Offs; (4) Universal Curvature Saturation Plateau (|𝒦|_ϵ ≤ 𝐂_UPTC²) and 16D Sedenion Spin–Torsion Bounce for Type III Gravitational and Fluid Singularities; (5) Ultraviolet Fixed-Point Stability without Landau Poles on the Condensed Takesaki Type II₁ Factor ℳ_ground; and (6) Quantitative Discriminating Predictions (Saturation Plateau, Echo Delay, and Hysteresis Scar Irr > 0) with Exact Classical Limit Recovery. Complete Python numerical verification is provided in Appendix A.