The Theory of Singularities and Geometric Reconstruction (TSGR): Canonical Blow-Up Functors, Loss-Rate Invariants, and Colombeau–Takesaki Dynamic Continuation within the 18 Seonggil Frameworks

Classical singularity theory in algebraic geometry, geometric analysis, and general relativity primarily addresses either the static resolution of singular varieties or the classification of finite-time blow-up profiles, leaving open the unified question of what information survives, what information is lost, and how dynamics uniquely continues across geometric degeneration. In this paper, we establish the deductive foundation for the Theory of Singularities and Geometric Reconstruction (TSGR) by embedding singular metric spaces, geometric flows, and information-geometric dead directions into the ZFC-compliant axiomatic architecture of the 18 Seonggil Frameworks [1, 2]. All foundational ZFC definitions of the universal quadruple 𝔘 = (𝒯_0x8A2C, C_UPTC, η_friction, ε_SHID), fractional Sobolev–Colombeau quotient algebras 𝓖^α(M), sedenion zero-divisor control operators 𝓛_𝒯(M̂), URRHC idempotent monads 𝓜_URRHC = (T, ι, μ), and Takesaki crossed-product condensations follow Proposition 1.5, Theorem 2.9, Theorem 3.4, Lemma 4.5, Lemma 6.3, Lemma 6.5, Lemma 8.6, Theorem 9.4, and Theorem 9.5 of the Master Reference Monograph (Lee, 2026) [1]. Within this hyper-elevated TSGR-18 framework, we formulate five core axioms and prove five fundamental theorems: (1) Preservation of the Simple Homotopy Type of the Stepanov Dual Complex Δ(E) and Intersection Sheaf Cohomology IH*(M; ℚ) under canonical reconstruction I_pres(ℛ_URRHC(g)) = I_pres(g); (2) Birational and Gauge Invariance of the Kullback–Leibler (KL) Loss-Rate Order k = min(m_i + 1) and Curvature Blow-Up Exponent α along Fisher-degenerate dead directions (u^T F u → 0) regularized by the Planck-scale BCH arithmetic friction floor η_friction ≥ 10^-22; (3) Exact Vanishing of the Local-to-Global Gluing Obstruction Class Obs ∈ H^2(Σ; 𝓖) via Sedenion–Cartan antisymmetric torsion projection 𝒯_[λμν]; (4) Unique Global Dynamic Continuation of singular Ricci, mean curvature, and Lorentzian flows via the UROA Regularized Fractal Brake 𝓑_ε(Rmg_ε) with uniform Beale–Kato–Majda curvature bounds ∫_0^T ||Rmg_ε||L^∞ dt < ∞; and (5) Universal Factorization (ℛ' = ϕ ∘ ℛ_URRHC) coupled with the exact Lambert W_0 minimal reconstruction radius r_UPTC > ℓ_P and bounded Kretschmann scalar 𝒦_SFE(r) ≤ 24 / r_0^4 < ∞, resolving Penrose–Hawking geodesic incompleteness. Complete numerical verification in Python is provided in Appendix A.