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…