Formally Specified Adaptive Control Kernel with Analytical Set-Membership Identification, Robust Inner Approximation, and Strict Fail-Closed Governance (V55.3.9)

Cyber-physical systems (CPS) operating under parametric uncertainties and unmodeled external disturbances require rigorous safetyguarantees that cannot be compromised by performance optimizationobjectives. This paper presents the mathematical formulation, algorithmicspecifications, and formal verification suite of a decoupled, formallyspecified adaptive control kernel (IEM-OS / QID-OS V55.3.9). Theproposed architecture introduces an absolute separation between safetycertification and performance optimization. It features: (1) an analyticalset-membership identifier (SMI) for scalar affine observation modelsthat strictly enforces non-empty feasibility semantics; (2) a finite-steprobust controlled-invariant inner approximation via union-of-polytopes;(3) a tamper-evident integrity binding scheme using canonical SHA-256cryptographic provenance; and (4) a multi-gate runtime candidate actionauthorization gate. Through an automated 12-stage stress-testing suite,we demonstrate that the kernel guarantees robust containment, rejectsmodel mismatches, eliminates false-feasible numerical relaxations, andstrictly propagates solver uncertainties (‘UNKNOWN’) to immediate failclosed actuation.