Metrological Non-Invertibility and Projection Constants in Historical Scale-Based Construction

Why do stable ratios appear in ancient Egyptian metrology when absolute measuresclearly disperse? And what kind of “constants” are these, if they cannot betraced back to fixed standards, exact geometry, or numerical approximation? This paper argues that the Short Cubit ratio—and its angular counterpart, theseked—are not empirical coincidences. They arise as structural residues of anon-invertible projection from scale dynamics to observable construction. Withinthis framework, a projection constant is identified: a dimensionless scalarthat survives realization while the underlying scale action is irreversiblydiscarded. Linear and angular measures are shown to organize into a coupledmetric–angular lattice within a restricted observable domain, where irrationaland multiplicative scale relations cannot be implemented directly. The analysisreframes the Short Cubit problem and invites a reconsideration of what“constants” mean in historical construction: not encoded numbers, but inevitableinvariants imposed by projection, constraint, and practice. This is a preprint. The manuscript may undergo further revision prior to journal submission or publication.