Temporal Algebra: Time Begins at δ(ψ) = 0

Temporal Algebra redefines time as a structural consequence of residual-driven leaps within directed acyclic graph (DAG) systems, rather than as a continuous external pa- rameter. Time emerges from non-zero structural residuals δ(ψ), obeying directional irreversibility and operating under noncommutative composition. This framework in- troduces five axioms that form a minimal generative system for structural temporality, supported by an algebraic structure and a temporal collapse category. The structural residual δ(ψ) originates from modal resonance dynamics in the Unified Resonant Struc- ture Framework (URSF) [1] and Resonance Grammar [2]. Unlike classical models that treat time as absolute, metric-curved, or passive, Temporal Algebra captures time as a discrete, structural process triggered by misalignment. This paper outlines the axioms, algebraic rules, categorical formulation, and empirical validation using gravitational wave data [3], emphasizing temporality as an emergent structural phenomenon.