Self-Verifying Knowledge Evolution Systems (SVKES): A Deductive Framework Unifying Conjecture–Counterexample Dynamics, Compositional Knowledge Synthesis, and Kernel-Protected Self-Modification within the ZFC Axiomatic Bridge

As automated reasoning and machine intelligence transition from static theorem proving to autonomous conjecture generation, modular theory synthesis, and algorithmic self-modification, existing optimization and formal verification theories face a foundational integration challenge. In this paper, we present a unified deductive framework, the Self-Verifying Knowledge Evolution System (SVKES), which synthesizes three emerging mathematical domains—Conjecture–Counterexample Dynamics (CCD), Mathematics of Compositional Knowledge (MCK), and Mathematics of Verified Self-Modification (MVSM)—anchored directly in our 71-page ZFC Master Reference Monograph (Lee, 2026, DOI: 10.5281/zenodo.23486681) [1]. All foundational spaces and universal structures 𝔘 = (𝒯_0x8A2C, C_UPTC, η_friction, ε_SHID) follow Proposition 1.5, Theorem 2.9, Lemma 4.5, Lemma 6.3, Lemma 6.5, Theorem 6.6, Lemma 7.2, Lemma 7.3, Lemma 8.4, Lemma 8.6, Lemmas 9.1–9.3, and Theorem 9.5 of [1]. Formulating the global state space Σ_t = (H_t, C_t, I_t, 𝓜_t, S_t) as a symmetric monoidal 2-category enriched over the η_friction-relaxed Lawvere metric poset 𝒱_η inside the higher ∞-topos 𝒮_SMA, we establish six core axioms (SVKES-1 through SVKES-6) and prove six fundamental theorems: (1) Soundness and Lifetime Statistical Safety of Closed Knowledge Evolution via the URRHC idempotent monad 𝕄_URRHC (T^2 = T) and Basel martingale e-values on the condensed Takesaki Type II_1 factor 𝓜_ground; (2) Pareto Frontier Structure of dual discovery-falsification objectives (J_thm vs. J_refute); (3) Sufficiency and Minimality of the 3-Component Trusted Kernel K* = (Check_𝓘, VerifyC,π, Rollback); (4) Safe Commutativity of Module Composition and System Self-Modification (Φ̂ ∘ ⊗ ≃Safe ⊗ ∘ Φ̂) under vanishing Čech interface obstruction Obs_⨂ ∈ H^1 and Sedenion Modulus Shield control 𝓛_𝒯(M̂) ≥ c_𝒯 > 0; (5) UPTC Channel-Capacity Lower Bounds on Abstraction Information Loss and Quotient Invariant Emergence; and (6) Necessity of a Tarski–Löb Stratified Verification Hierarchy K* = V_0 subsetneq V_1 subsetneq V_2 subsetneq ⋯. Complete Python numerical and algorithmic verification is provided in Appendix A.