The Dynamic Topological Critical Transition System (DTCTS): A Deductive Framework Unifying Fast–Slow Stochastic Tipping, Dynamic Persistent Homology, and Hysteresis Irreversibility Bounds within the ZFC Axiomatic Bridge

Abrupt, irreversible critical transitions (tipping points) across climate, ecological, material, and neurobiological systems pose a profound mathematical challenge: classical early warning indicators based on local linear stability (critical slowing down, variance divergence, and lag-1 autocorrelation) react only in the immediate vicinity of a local bifurcation and fail to capture global basin geometry or noise-induced topological restructuring. In this paper, we establish the Dynamic Topological Critical Transition System (DTCTS), a unified deductive framework synthesizing fast–slow stochastic bifurcation theory, dynamic persistent homology (PH_k(X_t)), and free-boundary phase transitions, anchored directly in our 71-page ZFC Master Reference Monograph (Lee, 2026, DOI: 10.5281/zenodo.23486681) [1]. All foundational spaces and universal constants 𝔘 = (𝒯_0x8A2C, C_UPTC, η_friction, ε_SHID) follow Proposition 1.5, Theorem 2.9, Theorem 3.4, Lemma 4.5, Lemma 6.3, Lemma 6.5, Theorem 6.6, Lemma 7.3, and Lemma 8.6 of [1]. Formulating the coupled state Σ_t = (x_t, λ_t, Dgm(t), ℰ_t, K) over the η_friction-relaxed Lawvere quasi-metric space and the Fractional Sobolev–Colombeau Quotient Algebra 𝓖^α(Ω), we establish six core axioms (DTCTS-1 through DTCTS-6) and prove six fundamental theorems: (1) Quantitative Lead-Time Superiority (Δt_lead > 0) of Dynamic Persistent Homology over classical variance and autocorrelation indicators; (2) Information-Theoretic Necessary and Sufficient Signal-to-Noise Ratio Conditions (s/σ > C√(ln(1/α))) for separating topological noise from structural tipping; (3) Finite-Window Identifiability Limits (T · ε · d ≳ ln(1/δ)) on the Takesaki Type II_1 ground factor 𝓜_ground; (4) Topological Lower Bounds on Hysteresis Irreversibility (Irr ≥ c ∑ (d_i - b_i)) governed by vanished homology generators and the URRHC monad; (5) Exact Correspondence between Fold/Hopf Bifurcations and PH_0 / PH_1 Betti Jumps; and (6) Free-Boundary Topological Phase Tracking via Colombeau–UROA Fractal Brakes 𝓑_ε. Complete Python numerical verification is provided in Appendix A.