The Mathematics of Multiscale Complex Systems (MMCS): Scale-Boundary Information Channels, Sedenion–Cartan Strong Emergence, and Takesaki Quantum-to-Classical Condensation within the 18 Seonggil Frameworks

Connecting microscopic physical laws to macroscopic emergent structures—across homogenization, kinetic theory, renormalization group (RG) flows, quantum-to-classical transitions, and biological complexity—requires a unified mathematical theory of scale-boundary information transfer and algebraic irreducibility. In this paper, we establish the deductive foundation for the Mathematics of Multiscale Complex Systems (MMCS) by embedding multiscale hierarchies into the ZFC-compliant axiomatic architecture of the 18 Seonggil Frameworks [1, 2, 3]. 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), η_friction-relaxed Lawvere quasi-metric spaces, and Takesaki crossed-product condensations follow Proposition 1.5, Theorem 2.9, Theorem 3.4, Lemma 4.5, Lemma 4.6, Lemma 6.3, Lemma 6.5, Lemma 8.4, Lemma 8.6, and Theorem 9.5 of the Master Reference Monograph (Lee, 2026) [1]. Within this hyper-elevated MMCS-18 framework, we formulate five core axioms and prove five fundamental theorems: (1) Existence and Strong H^α-Convergence of Effective Theories (||u^ε - u^0||H^α ≤ C εₗₘ^α) across homogenization and Boltzmann-to-Navier–Stokes hydrodynamic limits via the UROA Regularized Fractal Brake 𝓑_ε, (2) a Universal Positive Lower Bound on Scale-Boundary Information Loss (ΔI_loss ≥ c_univ > 0) governed by Planck-scale BCH arithmetic friction η_friction ≥ 10^-22, (3) Algebraic Quantification of Strong Emergence (V(ℓ) < 0) via non-associative 16-dimensional Sedenion–Cartan torsion synergy, (4) Universality of Renormalization Fixed Points T* and Post-Singularity Stability of Time-Dependent RG flows regularized by the cosmological bounce scale a_min > 0, and (5) Exact Error Control (||ψ_quantum - ψ_classical|| ≤ C(εₗₘ + ħ^β)) of the Quantum-to-Classical Transition via Takesaki Type III_1 → Type II_1 condensation and the Lambert W_0 / W_-1 decoherence scales ℓ_UPTC. Complete numerical verification in Python is provided in Appendix A.