kappa_normalized
plain-language theorem explainer
kappa_normalized defines the normalized curvature magnitude per vertex-sphere in the Q3 lattice by scaling the vertex count with the angular deficit and dividing by 4π. Researchers deriving the recognition length λ_rec from curvature costs in Recognition Science cite it when confirming Gauss-Bonnet normalization without prior reference to G. The definition is a direct algebraic combination of the vertex count and per-vertex deficit.
Claim. The normalized curvature magnitude per vertex-sphere is defined by $κ_{normalized} := (8 · δ) / (4π)$, where δ denotes the angular deficit at each vertex of the 3-cube.
background
The LambdaRecDerivation module derives λ_rec non-circularly from bit cost (=1) and curvature cost (=2λ²) via Q3 Gauss-Bonnet normalization, then defines G as a consequence. Q3_vertices counts the eight vertices of the 3-cube. angular_deficit_per_vertex equals 2π minus three times the dihedral angle, which evaluates to π/2 at each vertex where three faces meet.
proof idea
This is a one-line definition that multiplies the vertex count by the per-vertex angular deficit and divides by 4π.
why it matters
This supplies the normalized |κ| shown equal to 1 in the downstream theorem kappa_normalized_eq_one, which confirms the curvature cost for the λ_rec derivation. It closes the non-circular path to G := π λ_rec² c³ / ℏ and aligns with the eight-tick octave and D = 3 in the forcing chain.
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