The Epsilon Conjecture: Minimal Non-Neutrality of Cycles and the Emergence of Metric Time

The Epsilon Conjecture proposes a structural hypothesis in the foundations of physics concerning the emergence of temporal orientation, commensurable duration, and effective metric geometry. It postulates that admissible cycles of evolution may leave irreducible compositional residues, implying a minimal non-neutrality of closure. The non-commutative composition of residues introduces an order-memory that yields a primitive arrow of history independent of thermodynamics. Metric time does not exist at the fundamental level but emerges in coherence domains where a duration gauge becomes quasi-additive and memory cost stabilizes. Geometry is interpreted as the minimal envelope of accumulated compositional cost. The framework introduces no new particles or fields; it defines a structural program linking non-commutative composition, subadditive weights, quasi-metrics, and large-scale coherence regimes. The conjecture outlines a falsifiable research agenda at the interface of the problem of time, emergent geometry, and large-scale structure.