The Épure and Its Unfolding : Canonical Bridge from Minimal Discrete Geometry to Euclidean Space via Cauchy-Blaschke Unfolding

We establish a canonical, parameter-free bridge between the Minimal Discrete Geometry G■ = (Z², d■), where d■(P,Q) = |Δx| + 2|Δy| achieves the unique minimum π■ = 1 of the discrete π-invariant, and Euclidean space in all dimensions n ≥ 2. We call G■ the épure — a term from French architectural drawing designating the ground plan from which a three-dimensional structure is reconstructed without loss of information. The bridge process, called the déploiement (unfolding), is defined by the Cauchy-Blaschke operator Φ_n over SO(n) via Haar measure. MAIN RESULTS (1) Deployment constants. Φ_n(G■) = c_n B^n with exact closed form: c_n = (√5/π) · √π Γ(n/2) / (2Γ((n+1)/2)) giving c₂ = √5/π, c₃ = √5/4, c₃/c₂ = π/4 (exact). Verified for n = 2,...,8. (2) Triple optimality. The deployment is optimal in three independent senses: (a) Uniqueness — c_n B^n is the unique SO(n)-equivariant convex body extending h_B■ (b) Isoperimetric — unique surface-area minimizer at fixed volume (c) Entropic — unique maximum-entropy distribution at fixed second moment (3) Canonicity. The extension hB■,n is the unique support function satisfying axioms (E1)-(E3). (4) Three equivalent mechanisms. Cauchy-Blaschke rotation, geodesic torsion of the spherical fibration, and l=0 harmonic projection all compute c_n identically, confirming the canonical character. (5) Inverse deployment and discrimination. Direct deployment (Version A: Φ₃ in R³) vs projection of 4D deployment (Version B: Φ₄ in R⁴, temporal projection) yield distinct, experimentally discriminable predictions for Smin and γ (Barbero-Immirzi). All results are parameter-free and verified numerically in Python 3 (scipy, numpy). Code available upon request.