Formal Execution Semantics and Governance Invariants for Hybrid Classical-Quantum Autonomous Systems

The increasing adoption of autonomous systems, artificial intelligence (AI), Large Quantitative Models (LQMs), high-performance computing (HPC) infrastructures, and quantum computing introduces new challenges for execution-layer governance. Existing governance models frequently assume deterministic execution environments in which proposed actions, execution outcomes, and resulting state transitions can be evaluated using deterministic semantics. Emerging computational systems increasingly violate these assumptions by producing probabilistic outputs, confidence intervals, uncertainty distributions, optimization result spaces, and quantum measurement distributions. This paper extends prior work on Artificial Intelligence Governance Control Planes (AGCPs), formal execution semantics, and Quantum Runtime Governance by introducing a formal execution model for hybrid probabilistic execution environments spanning classical systems, HPC infrastructures, AI systems, LQMs, and quantum computing resources. The paper formalizes the concepts of Probabilistic Admissibility and Replay-by-Distribution, demonstrating how deterministic governance authority may be preserved even when governed execution environments exhibit probabilistic behavior. Building upon previously proposed Quantum Runtime Governance Invariants, the paper develops a mathematical formalization of governance-preservation properties and establishes a collection of governance-preservation theorems addressing commit safety, governance continuity, hybrid execution graph safety, replayability, auditability, and non-bypass execution. The analysis demonstrates that governance authority, authorization integrity, admissibility integrity, provenance continuity, and commit-bound control can remain preserved across heterogeneous execution environments despite probabilistic computational behavior. The paper argues that quantum computing is not an isolated governance challenge but rather an instance of a broader class of probabilistic execution systems. Accordingly, the work provides a foundation for extending execution-layer governance into future computational environments that combine classical computing, artificial intelligence, quantitative modeling, quantum computing, and autonomous decision-making systems.