High-fidelity analytical torus generated via the PAPP v3.1.0 system (Phase 7.5). This model represents the positive branch (f⁺) of a "Bonnet Pair"—a rare geometric configuration where two non-congruent surfaces share identical metrics (isometry) and mean curvature. This specific iteration corresponds to the 1/3 harmonic ratio (Perfect Twelfth). Based on the 2025 research by Bobenko, Hoffmann, and Sageman-Furnas, the surface is a compact torus in ℝ³ with numerically verified isometry against its twin branch (f⁻) at machine precision (rel. error ≈ 10⁻¹⁴). Technical Specifications: • Topology: Closed Manifold Torus (Euler Characteristic χ = 0). • Resolution: 80×80 (6,400 vertices, 12,800 triangular faces). • Parameters: Optimized at the exact closure value τ = 0.231149*. • Material: PAPP Standard "Ocean/Teal" Kd (0.175, 0.582, 0.700). • Analysis: Validated via Laplacian Spectral Gap and Persistent Homology. Generated by PAPP Framework.