The Mathematics of Self-Evolving Intelligence (MSEI): An Axiomatic Synthesis of Recursive Operator Dynamics, η_friction-Locked Lineage Convergence, and Type III_1 → Type II_1 Verification Condensation within the 18 Seonggil Frameworks

When artificial intelligence transitions from optimizing static parameterized functions to recursively modifying its own algorithms, memory architectures, and verification protocols, classical optimization theory and standard Hoare logic fail to guarantee simultaneous justification, long-term lineage convergence, structural stability, and decidable verification. In this paper (Seonggil Treatise Series Paper No. 303), we establish the deductive foundation for the Mathematics of Self-Evolving Intelligence (MSEI) by embedding recursive self-modification dynamics directly into the ZFC-compliant axiomatic architecture of the 18 Seonggil Frameworks (Lee, 2026, Zenodo DOI: 10.5281/zenodo.23486681). Specifically, all foundational ZFC definitions of the universal quadruple U = (T_0x8A2C, C_UPTC, η_friction, ε_SHID), fractional Sobolev–Colombeau quotient algebras G^α(Ω), sedenion zero-divisor control operators L_T(M), η_friction-relaxed Lawvere quasi-metric spaces, and Takesaki crossed products follow Proposition 1.5, Theorem 2.9, Lemma 4.5, Lemma 6.3, Theorem 7.4, Theorem 8.7, and Theorem 9.5 of the Master Reference Monograph (Lee, 2026). By unifying Kleene's second recursion theorem and Harris's η-locking lineage dynamics with the universal arithmetic friction lower bound (η_friction ≥ 10^-22), the UROA regularized fractal brake B_ε, and spectral truncation projections P_UPTC, we prove five fundamental theorems governing self-evolving intelligence: (1) Monotonicity of Justified Improvement without Gödel–Löb procrastination and global smooth Colombeau regularity under the cutoff brake Θ_ε(Σ_ε - C_UPTC); (2) Almost-Sure Concentration of the Regularized Lineage Exponent g_p onto the reachable global utility supremum U_max under η_friction-locked reproduction (η ≥ η_friction > 0); (3) Input-to-State Stability (ISS) of Protected Constitutional Invariants V(t) under 16-dimensional Sedenion–Cartan torsion control (L_T(M) ≥ c_T > 0) avoiding the Moreno zero-divisor variety Z(S); (4) Preservation of Verification Complexity Classes (Σ_k^P) across unbounded modification horizons via Takesaki crossed-product condensation (M_SFE ⋊_{σ^ω} R → M_ground, Type III_1 → Type II_1) and URRHC monad idempotence (T^2 = T) at critical depth N_crit; and (5) Exact analytical closed-form solutions for the Critical Self-Reference Ignition Threshold Γ_UPTC^(0) and Thermodynamic Saturation Horizon Γ_UPTC^(-1) via the principal and secondary branches (W_0, W_-1) of the Lambert W-function. Complete numerical verification in Python is provided in Appendix A.