A Calculus of Execution: Collapse of Eschatological Nondeterminism

Le résumé fourni par la source

Highlights - A single-axiom calculus formalizes software execution from first principles. - Explicit eschatological criterion τ collapses nondeterminism within execution. - Layered architectures decompose eschatological purpose, not data. - Fractal epistemic sequences enable multi-dimensional convolution over K levels. - Selection is named as the sole primary source of nondeterminism in execution. Abstract This paper derives a calculus of software execution from one axiom: Execute(Aτ, Cⁿ) = e_{n+1}, C⁰ = ∅. Five prior formalizations each identified one structural component without achieving their synthesis — Tarski the execution lattice, Leibniz the eschatological condition, Turing the sequence, Mesarović the abstraction model, Cohen the constructive orthogonality — and Axiom 0 is their formal synthesis. The binary relation R ⊆ X×X is the foundational object; abstraction Aτ is its execution instance, governed by the eschatological criterion τ(C, ωτ) ∈ {true, false}, absent from every prior formalism. Selection — the atomic evaluation of τ at each admission step — is the sole primary source of nondeterminism in execution. Without selection, abstraction is only deterministic mapping; with implicit selection, nondeterminism is displaced outside execution into schedulers, guards, monitors, or governance layers; with explicit τ, selection becomes a governed admission criterion over C, making software evolutivity possible. Recursion and interaction are derived mechanisms. The execution context C is a strict subset of the ontological lattice E, growing from C⁰ = ∅; a three-axis ontological–eschatological–epistemic space arises constitutively from the rank structure of R (Appendix 1). C is a fractal epistemic sequence; multi-dimensional projection reduces to key-prefix lookup in O(log|C| + |result|). Two structural anomalies are resolved: the eschatological criterion left implicit, and nondeterminism located outside execution. Sixteen definitions conservatively introduce the formal objects of the calculus; four lemmas, one tautology, and fifteen corollaries are derived from Axiom 0 without additional postulates. The principal result (Corollary 11) establishes that layered architectures sharing one context perform decomposition of purpose, not decomposition of data. Governance, ethics, and execution are not separate layers but components of a single τ evaluated before every admission.

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