What question did this study set out to answer?

This research aims to establish a parameter-free bridge between minimal discrete geometry and Euclidean space using the Cauchy-Blaschke unfolding process.

April 19, 2026Open Access

The Épure and Its Unfolding: Canonical Bridge from Minimal Discrete Geometry to Euclidean Space via Cauchy-Blaschke Unfolding₃/6

Key Points

This research aims to establish a parameter-free bridge between minimal discrete geometry and Euclidean space using the Cauchy-Blaschke unfolding process.
Defined the épure and its deployment using the Cauchy-Blaschke operator over SO(n).
Derived deployment constants for various dimensions (n ≥ 2).
Analyzed uniqueness, isoperimetric properties, and entropic characteristics of the deployment.
Verified findings for dimensions n from 2 to 8.
Deployment constants were expressed in closed form and shown to hold for multiple dimensions.
The deployment was proven optimal in three distinct senses: uniqueness, isoperimetric, and entropic.
Validated computationally in Python for numerical accuracy.

Abstract

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 Φₙ over SO (n) via Haar measure. MAIN RESULTS (1) Deployment constants. Φₙ (G■) = cₙ Bⁿ with exact closed form: cₙ = (√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 — cₙ Bⁿ is the unique SO (n) -equivariant convex body extending hB■ (b) Isoperimetric — unique surface-area minimizer at fixed volume (c) Entropic — unique maximum-entropy distribution at fixed second moment (3) Canonicity. The extension h₁■, ₍ 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 cₙ 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.

The Épure and Its Unfolding: Canonical Bridge from Minimal Discrete Geometry to Euclidean Space via Cauchy-Blaschke Unfolding₃/6

Key Points

Abstract

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