The Épure and Its Unfolding: Canonical Bridge to Euclidean Space

We establish a canonical, parameter-free bridge between the Minimal Discrete Geometry G■=(Z²,d■), which uniquely minimises the discrete π-invariant (π■=1), and Euclidean space in all dimensions n≥2. We call G■ the épure — a term from architectural drawing for the two-dimensional ground plan from which a three-dimensional structure is reconstructed without loss of information. The unfolding (déploiement) is defined via the Cauchy–Blaschke average of the unit ball over SO(n). Main results: exact closed forms for the deployment constants; triple optimality (uniqueness, isoperimetric, entropic); three equivalent constructive mechanisms; and the role of √5 as the sole surviving geometric invariant. All results are…