Dimensional Gates and Arithmetic Constants

This paper isolates the role of the fundamental constants in the Shadow framework. The central distinction is elementary but decisive: a dimensionful constant is not a pure number. Its numerical value changes under a change of units and cannot therefore be derived, as a number, from a dimensionless mathematical structure without first choosing a dimensional calibration. Dimensionless constants, by contrast, are invariant under the unit group and are the legitimate targets of arithmetic derivation. We formalize this distinction by treating dimensions as a real vector space and unit changes as a scaling group action, whose invariant functions are precisely the dimensionless quantities of the Buckingham-Pi theorem. The three constants c, h-bar, and G are then identified as dimensional gates: c calibrates causal geometry, h-bar calibrates quantum phase and spectral action, and G calibrates the coupling of energy to curvature. Together they span the real dimension space of mass, length, and time, setting the rulers of the tower without being predictions of it. The pure predictions of the Shadow framework belong to the dimensionless sector: coupling constants, mass ratios, mixing angles, central charges, anomaly coefficients, normalized vacuum densities, and arithmetic spectral invariants. The conclusion is a rule of interpretation: units are gates, pure numbers are laws.