euler_S2
euler_S2 supplies the integer value of the Euler characteristic for the 2-sphere in curvature identities. Workers on the non-circular λ_rec derivation cite it when normalizing the total curvature of the cube Q3 to 4π. The definition is a direct assignment of 2.
claimThe Euler characteristic of the 2-sphere $S^2$ equals 2.
background
The LambdaRecDerivation module derives the recognition length λ_rec from bit cost normalized to 1 and curvature cost 2λ² on the cube Q3. The module states that total curvature over the eight vertices equals 4π once the Euler characteristic of the bounding sphere is inserted, after which G is obtained as G := π λ_rec² c³ / ℏ. This keeps the derivation non-circular by treating G as a derived quantity rather than an input.
proof idea
This is a one-line definition that sets the Euler characteristic of S² to the integer 2. No lemmas are applied; the value is the standard topological invariant for the sphere.
why it matters in Recognition Science
The definition is referenced in total_curvature_gauss_bonnet to obtain total curvature = 2π × 2 = 4π and in kappa_normalized_eq_one to confirm |κ| = 1. It supports the GDerivationChain structure that links Q3 combinatorics through Gauss-Bonnet to the forcing chain step yielding G = λ_rec² c³ / (π ℏ). This step closes the curvature normalization required for the phi-ladder mass formula.
scope and limits
- Does not prove that the Euler characteristic of S² equals 2.
- Does not extend to higher-dimensional spheres or other manifolds.
- Does not incorporate the full statement of the Gauss-Bonnet theorem.
- Does not compute angular deficits or vertex counts.
formal statement (Lean)
150def euler_S2 : ℕ := 2
proof body
Definition body.
151
152/-- The dihedral angle at each cube edge is π/2 (right angle). -/