The Unified Master Action of Quantum Gravity and Emergent Spacetime: Sedenion Zero-Divisor Control, Nonassociative Einstein–Cartan Cosmological Bounce, and Takesaki–Lambert W_0 Black Hole Condensation within the ZFC Axiomatic Bridge

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A complete Theory of Everything (TOE) must simultaneously resolve the ultraviolet (UV) divergences of quantum field theory, the spacetime singularities of general relativity (both the primordial Big Bang and black hole interiors), the origin of Standard Model gauge groups and fermion mass hierarchies, and the macroscopic nature of dark matter and dark energy within a rigorous ZFC set-theoretic foundation. In this final flagship paper, we present the definitive unified synthesis of the Seonggil Field Equations (SFE), Seonggil Theory of Composite Torsion (STCT), and Universal Rough Operator Algebra (UROA), anchored directly in the 71-page ZFC Master Reference Monograph (Lee, 2026, DOI: 10.5281/zenodo.23486681) [1]. All foundational spaces, universal structures 𝔘 = (𝒯_0x8A2C, C_UPTC, η_friction, ε_SHID), and operator domains follow Proposition 1.5, Theorem 2.9, Lemma 4.5, Lemma 4.6, Lemma 5.3, Theorem 5.4, Lemma 8.4, Lemma 8.6, Theorem 8.7, Lemmas 9.1–9.3, Theorem 9.4, and Theorem 9.5 of [1]. We prove five core theorems: (1) Sedenion Zero-Divisor Control (𝓛_𝒯(M̂) ≥ c_𝒯 > 0) governing the 𝕊 = 𝕆_1 ⊕ 𝕆_2 e_8 → 𝕆_1 symmetry breaking into G_2 ⊃ SU(3)_c, exactly yielding the charged lepton mass hierarchy (m_e : m_μ : m_τ ≈ 1 : 206.77 : 3477.15) via Eisenstein Complex Multiplication (θ_0 = 2π/9) and the QCD vacuum confinement scale E_vac = √(T_s) ≈ 420.5 MeV; (2) Derivation of the Nonassociative Einstein–Cartan Field Equations and a smooth Cosmological Bounce at minimum scale factor a_min > 0 driven by sedenion spin-torsion repulsion σ_S^2 ∝ a(t)^-6; (3) Takesaki Crossed-Product Condensation (Type III_1 → Type II_1) and the exact Lambert W_0 Black Hole Condensation Radius r_UPTC > ℓ_P, bounding the Kretschmann curvature by 𝒦_SFE(r) ≤ 24 / r_0^4 < ∞ and preserving a unitary entanglement Page curve; (4) Falsifiable astrophysical and high-energy predictions including cored dark matter halos, stochastic gravitational wave peaks (f_peak), and ultra-high-energy τ-neutrino vertex enhancements; and (5) The Unified Master Action S_TOE realizing the Global Consistency and Unified Transfer Theorem (Theorem 9.5 of [1]). Complete numerical verification in Python is provided in Appendix A.