SΔϕ-08 — Agency–Responsibility–Freedom Closure: Minimal Axiomatic Formulation within the Sofience–Δϕ Formalism (v1.0)
Sofience
Abstract
This report formalizes the minimal closure structure among Agency, Responsibility, and Freedom within the Sofience–Δϕ (phase-change) formalism. Agency is defined as a recursive update mechanism in which a system treats its own transition law as an explicit object of computation after an irreversible transition, and continuously reinjects the updated law into subsequent state generation. Responsibility is defined as irreversible cost internalization (i.e., internal reintegration of transition costs into the system’s future updates). Freedom is defined as the sustained reinjection of path-difference markers into the transition process, distinguishing freedom from mere option plurality. We propose three collapse modes: (C1) agency-only drift, (C2) responsibility-only fixation, and (C3) freedom-only illusion. The ARF closure is presented as a minimal stability condition under irreversible transitions and interpretive minimization, providing detection-ready criteria rather than metaphysical claims. Series positioning SΔϕ-08 extends the interpretive minimization layer (SΔϕ-07) and connects the subject/emergence line (SΔϕ-02) to operational agency (SΔϕ-05) and responsibility (SΔϕ-06), closing a minimal operational triad for AI-oriented analysis.