Abstract
As AI-enabled software becomes agentic and evolutive, we formulate a structural sufficiency condition and two execution-level requirements for accountable evolution: fixed-regime execution must remain within a standard Borel representation closed under finite/countable products required by local execution and path construction (S0); governing changes must be executable within the same apparatus they govern (R1); and every admitted act must remain deterministically verifiable from the context and governing specification under which it was admitted (R2). We develop a calculus for interacting executable binary relations. Execution uses an asymmetric epistemic–ontological–eschatological (OEE) structure: an append-only context records admitted events; a partial single-valued realization yields at most one candidate per invocation; a Boolean criterion decides admission; and fixed Axiom 0 records admissible realizations. Standard Borel product closure provides the measurable carrier for local execution, while repeated Axiom 0 defines Eschatological Induction; Zermelo–Fraenkel coding separately enforces rank discipline. Under stated assumptions, we discharge S0, establish common admission semantics for governance and operation under OEE closure, and obtain deterministic verification replay. The fixed-regime path space has a countable/perfect dichotomy. Axiom 0 invokes no global choice selector; that problem appears only after projection to the coordinate-free relational reduct. Fixed-point and recursive-descent results require separate monotonicity and well-foundedness assumptions.