The Scale-Invariance Boundary of the Minimal Discrete Épure

The Minimal Discrete Épure G■=(Z²,d■), with d■(P,Q)=|Δx|+2|Δy|, is the unique discrete geometric space with π■=1. We prove that d■ is positively homogeneous of degree one, and derive its central structural consequence: the scale-invariance boundary. Every quantity built from d■ alone is either dimensionless and scale-invariant (a pure number the geometry fixes) or dimensionful and scale-covariant (carrying no absolute value, the lattice unit being free). The framework therefore derives dimensionless ratios (angles, gauge structure, coupling ratios, hierarchies) but not absolute scales (Newton's constant, masses, the cosmological scale). We confirm the boundary by six mutually independent arguments and prove that the sign of the one-loop β-function — the one rescaling-invariant datum — reproduces the Standard-Model running pattern under the division-chain identification, so that scale ratios are derivable by dimensional transmutation while a single reference unit remains posited. This is the normal epistemic status of any physical theory. Every quantitative claim is accompanied by a reproducible numerical protocol.