Bounded Non-Closure Time (BNCT): A Unified Temporal Index for Finite-Trace Systems

Bounded Non-Closure Time (BNCT): A Unified Temporal Index for Finite-Trace Systems Time is traditionally treated as an external background parameter or as a thermodynamic consequence of entropy. This paper formalizes Bounded Non-Closure Time (BNCT), defining time as an irreversible update index (n) inherent to any system with finite structural retention capacity. The BNCT framework rests on three fundamental axioms: • BN-1 (Boundedness Principle): Every physical or informational system possesses a finite capacity C to retain structural states. • VC-1 (Center-Crossing Prohibition): Systems are structurally forbidden from reoccupying identical prior states, ensuring non-closure. • TP-1 (Index Principle): Time is the natural-number index of irreversible updates; if no update occurs, no temporal index exists. We demonstrate that continuous temporal evolution emerges from the accumulation of micro-structural residues (Slips), while discrete events—such as hatching, differentiation, or system collapse—emerge as Index Singularities when the finite capacity C is saturated. This theory provides an observer-free derivation of the arrow of time. It establishes structural irreversibility as the precondition of thermodynamic dissipation, rather than its result, providing a novel interface to Landauer’s Principle. BNCT is applicable across diverse substrates, including biological double-helix dynamics, large language models (context saturation), and fundamental physics.