{
  "abstract": "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].\n\nFormulating 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "name": "Seonggil Lee"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/14zmd8pkvn57a.bib",
  "cited_by": 0,
  "doi": "10.5281/zenodo.23557507",
  "fulltext_url": null,
  "id": "14zmd8pkvn57a",
  "issns": [],
  "landing_url": "https://zenodo.org/records/23557507",
  "language": "eng",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [],
  "openalex": null,
  "pdf_url": "https://zenodo.org/api/records/23557507/files/The%20Dynamic%20Topological%20Critical%20Transition%20System%20(DTCTS)_A%20Deductive%20Framework%20Unifying%20Fast%E2%80%93Slow%20Stochastic%20Tipping,%20Dynamic%20Persistent%20Homology,%20and%20Hysteresis%20Irreversibility%20Bounds%20within%20the%20ZFC%20Axiomatic%20Bridge.pdf/content",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/14zmd8pkvn57a.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T12:20:43Z",
      "native_id": "23557507",
      "source": "zenodo",
      "url": "https://zenodo.org/records/23557507"
    }
  ],
  "subjects": [],
  "title": "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",
  "title_text": "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",
  "type": "preprint",
  "updated_at": "2026-11-05T12:20:43Z",
  "url": "https://research.draconitic.com/paper/14zmd8pkvn57a",
  "venue": "Zenodo",
  "versions": []
}