Bounded Non-Closure Time in Finite-Trace Systems Slip Accumulation, Index Formation, and a Qubit-Level Mapping

Bounded Non-Closure Time (BNCT) is introduced as a structural account of temporal ordering that does not presuppose spacetime background, observers, or thermodynamic entropy. Time is defined as the monotonic index of irreversible updates produced when finite trace capacity fails to perfectly encode recurrent motion. This failure leaves a deterministic residual—Slip (\sigma)—whose accumulation induces directionality. Key contributions: • Minimal formalization: Slip is defined in finite-state systems under a bounded trace register of capacity K, yielding a monotone index \tau without stochastic or thermodynamic postulates. • Index singularity: A purely combinatorial threshold K_c = |\mathcal{H}^{(n)}| is identified (canonically K_c = |S|^{n+1}), across which the temporal index becomes structurally guaranteed. • Physical mapping: Landauer’s principle is reframed as an implementation boundary (cost condition) rather than the origin of irreversibility, and a qubit-level mapping connects Slip accumulation to decoherence and measurement as record-capacity effects. Conclusion: Time is not a background parameter but an index emerging from the structural mismatch in which a finite capacity cannot fully represent an unbounded (or expanding) history.