kappa_normalized_eq_one
The normalized curvature magnitude per vertex-sphere on the recognition cube equals one, as required by Gauss-Bonnet normalization on the bounding sphere. Researchers deriving the recognition length lambda_rec non-circularly from unit bit cost and curvature cost J_curv = 2 lambda^2 cite this to fix the normalization before defining G. The proof is a short algebraic reduction that unfolds the definition, rewrites via the total curvature identity, simplifies the Euler characteristic, and finishes with ring.
claimThe normalized curvature magnitude per vertex-sphere satisfies $kappa_normalized = 1$, where $kappa_normalized := Q_3 vertices * angular_deficit_per_vertex / (4 pi)$ and the total curvature identity gives $Q_3 vertices * angular_deficit_per_vertex = 4 pi$.
background
The LambdaRecDerivation module derives lambda_rec non-circularly by balancing the normalized bit cost of one against the curvature cost J_curv(lambda) = 2 lambda^2. This J_curv follows from a curvature packet of magnitude 4 distributed over the eight vertices of Q3, normalized via Gauss-Bonnet on the bounding sphere S^2 whose Euler characteristic is 2. The definition kappa_normalized := Q3_vertices * angular_deficit_per_vertex / (4 * Real.pi) therefore measures the curvature per vertex-sphere in units that make the total equal to 4 pi. The upstream theorem total_curvature_gauss_bonnet states that Q3_vertices * angular_deficit_per_vertex = 2 * Real.pi * euler_S2, with euler_S2 defined as 2.
proof idea
The proof unfolds the definition of kappa_normalized, rewrites the product of vertex count and deficit using the total_curvature_gauss_bonnet theorem, simplifies euler_S2 to the constant 2, and reduces the resulting expression to 1 by ring.
why it matters in Recognition Science
This result fixes the curvature normalization inside the non-circular lambda_rec derivation of the Constants.LambdaRecDerivation module, allowing G to be defined afterward as pi * lambda_rec^2 * c^3 / hbar. It rests on the curvature packet axiom from PlanckScaleMatching and the Gauss-Bonnet identity for the cube, consistent with the recognition composition law and the eight-tick octave structure. No downstream uses are recorded yet.
scope and limits
- Does not derive the numerical value of lambda_rec.
- Does not reference or presuppose Newton's constant G.
- Does not extend the normalization to dimensions other than D=3.
- Does not supply error bounds or numerical approximations.
formal statement (Lean)
172theorem kappa_normalized_eq_one : kappa_normalized = 1 := by
proof body
Term-mode proof.
173 unfold kappa_normalized
174 rw [total_curvature_gauss_bonnet]
175 simp [euler_S2]; ring
176
177/-- J_curv = 2λ² is the curvature cost per recognition token.
178 Derivation: |κ_normalized| × (4πλ²) / (2π × χ(S²))
179 = 1 × (4πλ²) / (2π × 2) = 2λ² / 2 ... wait, let's be precise:
180 J_curv = (|κ|/(2χ)) × (A/(2π)) where |κ| = 4, χ = 2, A = 4πλ²
181 = (4/4) × (4πλ²/(2π)) = 1 × 2λ² = 2λ². -/