AIKernel: Formal Foundations for Trajectory Governance and the AIKernel.NET Runtime

**Abstract / Description** The AIKernel project introduces a formalized _Trajectory Governance Model_ designed to establish a deterministic safety boundary around the inherently stochastic reasoning processes of Large Language Models (LLMs) and AI agents. To address critical challenges such as context contamination, opacity of state transitions, and goal drift, this research models AI inference as a continuous dynamical trajectory rather than as isolated text-generation events. The system architecture incorporates a standard access control model (PDP / PEP / PIP) and an SGP (Structure, Generation, Polish) pipeline that structurally decouples logical reasoning from linguistic expression. By representing inference states geometrically as _Semantic Ellipsoids_ and calculating normalized Convergence (Ct) and Anomaly (At) scores, the framework dynamically enforces an auditable _Fail-Closed_ control perimeter that halts execution upon policy violation. This theoretical model is mapped to `AIKernel.NET`, an OS-like architecture that provides a governed AI runtime design focusing on semantic context, virtual file systems, provider abstraction, deterministic execution, and Fail-Closed governance. To empirically assess this framework, a comprehensive multi-phase experimental validation plan has been proposed, targeting candidate filtering, approval routing, and replay-based calibration. This research contributes the mathematical foundations required to advance verifiable context sovereignty and safety in enterprise-grade autonomous AI systems (DOI: 10.5281/zenodo.20223205).