Quaternary Residue Coordinate Dimension
Abstract
The four-state residue carrier has three explicit pair collisions and statistical dimension three.
Theorem 1.1 (Every fixed coordinate pair is incomplete on the four-state carrier).
Proof. Machine-checked in Lean as D5/S3/Observer/ArithmeticTomography/QuaternaryResidueCoordinateDimension.quaternary_statistical_dimension_eq_three (✓ std3). ∎
Source. Repository-derived.
Commentary.
The theorem is stated directly on the finite residue-state carrier {0,10,15,21}; it has no ambient-state carrier parameter.
On this carrier, q2 with q3 merges 15 with 21, q2 with q5 merges 0 with 10, and q3 with q5 merges 0 with 15. These three public clauses force every two-coordinate selection to be incomplete.
All three coordinates are jointly injective on the same carrier. Together with the three collision clauses, this makes the least complete coordinate count exactly three.