Finite-Distinguishability Closure and a Distinguished Dimensionless Residue

This work presents a mathematical closure model in which a distinguished dimensionless residue emerges from structural constraints under finite distinguishability. The model is formulated as an operational closure system in structural dimension d=3 and is based on the following axioms: • Binary hypercubic refinement • Non-privileged representation • Full closure self-audit • Scalar audit readout • Single-valued rotational history audit • Refinement-invariant anchor Within this framework, the paper derives the refinement counting law, audit capacity, locking depth, audit covariance dimension, and geometric projection constants. These ingredients combine additively to determine a distinguished dimensionless residue selected by the closure structure of the model. The analysis is presented purely at the mathematical level and does not assign a physical interpretation to the resulting constant. All derivations are self-contained and reproducible from the included LaTeX source. The resulting residue numerically evaluates to approximately 137.035999…