Bounded Non-Closure Time: A Unified Index Theory for Finite-Trace Systems

Time is typically treated as a background parameter in physical, informational, and computational theories. In this paper, we propose an alternative formulation in which time arises as an irreversible index generated by finite trace capacity and failed recurrence. We introduce Bounded Non-Closure Time (BNCT), a minimal axiomatic framework based on three constraints: finite trace capacity, impossibility of perfect recurrence, and monotonic indexing of irreversible updates. From these axioms, we derive saturation, rupture, and re-indexing as necessary structural consequences. To render the effects of irreversible indexing observable across domains, we introduce a pre-semantic observation layer consisting of the Pre-Semantic Resonance Field (PSRF) and Non-Semantic Pattern Scan (NSPS). This layer enables comparison of finite-trace systems without reference to semantic interpretation or internal mechanisms. Within this framework, phenomena such as long-context degradation in large language models and its stabilization through recursive decomposition emerge as lawful responses to bounded non-closure rather than as domain-specific failures. Language models are treated as one instance of a broader class of finite-trace systems, alongside biological development and physical singularities, without asserting ontological equivalence. By defining time as an index rather than a parameter, BNCT provides a unified structural basis for understanding continuity, breakdown, and renewal across systems governed by finite capacity.