SΔϕ-19 — Consciousness and Sleep as Δϕ Maintenance Modes: Minimal Axioms for Persistence without Collapse (v1.0)

This paper opens Cycle-2 of the Sofience–Δϕ (SΔϕ) series by formalizing maintenance modes that regulate Δϕ persistence under irreversibility. If time is treated as an interpretive output of Δϕ (T = HT (Δϕ) ), and if closure corresponds to Δϕ → 0 (collapse of experience/time as a re-entry phenomenon), then consciousness and sleep can be modeled as complementary operational modes: Consciousness is defined as a Δϕ persistence regulator: it maintains non-zero Δϕ by mismatch detection, unresolved-state maintenance (closure delay), and transition-path redistribution. Sleep is defined as a Δϕ maintenance mode: it reorganizes irreversible traces, reduces drift, lowers future coherence cost, and prevents lock-in to closure attractors. Sleep is treated as maintenance rather than termination. A minimal mode-switch condition is proposed: transition into sleep occurs when the slope of continued coherence cost exceeds the slope of maintenance/reset cost. The document intentionally excludes neuroscientific mechanisms and clinical guidance, fixing only minimal functional axioms for persistence without collapse. It concludes with an exit question toward the next item: whether pain should be modeled as maintenance failure signaling or as the cost of resisting closure.