total_curvature_gauss_bonnet
plain-language theorem explainer
The total curvature summed over the eight vertices of the 3-cube equals 4π, which equals 2π times the Euler characteristic of the sphere. Researchers deriving the gravitational constant from the recognition length in the Recognition Science framework cite this identity as the Gauss-Bonnet step for the cube. The proof is a one-line algebraic reduction that unfolds the vertex count, per-vertex deficit, and Euler number before applying ring simplification.
Claim. The product of the 3-cube vertex count and the angular deficit per vertex equals twice $π$ times the Euler characteristic of the 2-sphere: $8 · (π/2) = 2π · 2$.
background
The LambdaRecDerivation module derives the recognition length λ_rec non-circularly from the normalized bit cost of 1 and the curvature cost 2λ² obtained via Q₃ Gauss-Bonnet normalization, without prior reference to G. Q3_vertices is the constant 8. The angular deficit per vertex is defined as 2π minus three times the dihedral angle, and the sibling theorem angular_deficit_value establishes that this equals π/2. euler_S2 is the constant 2, the Euler characteristic of the bounding sphere S².
proof idea
This is a one-line term proof. It applies simp to unfold Q3_vertices, euler_S2, and angular_deficit_value, then uses ring to verify the numerical identity.
why it matters
It supplies step2_gauss_bonnet inside G_derivation_chain_complete, which assembles the full non-circular derivation of G as π λ_rec² c³ / ℏ. The result realizes the Gauss-Bonnet theorem on the discrete cube, matching the topology of S² and supporting the eight-tick octave together with three spatial dimensions from the forcing chain. It also feeds kappa_normalized_eq_one, which sets the normalized curvature magnitude to 1 for the balance equation that fixes λ_rec.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.