AEGIS Algebra: A Unified Mathematical Framework for Executable Assurance of Agentic AI Artifacts

Agentic AI systems operate autonomously across digital and physical environments, rendering subjective trust, post-hoc audits, and scalar scores insufficient for safety, legality, and stability. This paper introduces AEGIS Algebra, a unified mathematical framework that defines assurance as a machine-executable, multi-dimensional state governing the admissible behavior of AI artifacts. The framework rests on five load-bearing formalisms, each governing a distinct modality the others cannot express: matroid theory for structural feasibility, lattice and order theory for dominance and absorbing failure, constraint satisfaction for admissibility, three-valued evaluation logic for soundness under partial evidence, and temporal logic for validity over time. These are integrated into a single assurance space rather than treated as parallel analyses. The result is non-compensatory by construction: strong performance on one dimension cannot offset failure on another, and structurally infeasible configurations do not exist in the assurance space. Regulatory text is translated through a strict pipeline — clause → predicate → Predicate Compute Unit (PCU) → gate — in which no layer substitutes for another. The paper presents formal definitions, five theorems (including a soundness-under-partial-evidence result for predicate evaluators), geometric intuition, and worked compliance examples deriving enforceable gate decisions end-to-end from HIPAA, GDPR, and SOC 2 source clauses. The framework is presented openly to support academic engagement, regulatory adoption, and long-term extensibility. The proprietary engineering corpus required to instantiate the framework against specific regulatory regimes is maintained as a separate asset and is not part of this deposit. How to cite: Miruke, D. AEGIS Algebra: A Unified Mathematical Framework for Executable Assurance of Agentic AI Artifacts. Preprint v2.2, June 2026.