J_curv_zero
plain-language theorem explainer
The curvature cost functional vanishes at zero scale. Researchers deriving the recognition wavelength from the J_bit equals J_curv extremum condition cite this as the base case that anchors the equilibrium argument. The proof is a direct simplification that substitutes the explicit definition J_curv(λ) = 2λ².
Claim. $J(0)=0$ where the curvature cost is defined by $J(λ)=2λ^2$.
background
J_curv is the curvature cost functional introduced in the PlanckScaleMatching module. It encodes the physical hypothesis that a ±4 curvature packet distributed over the eight vertices of the Q3 hypercube yields a total cost of 2λ². The module derives λ_rec ≈ 0.564 ℓ_P from the equilibrium condition J_bit = J_curv(λ_rec) followed by face averaging that inserts the factor 1/π. An upstream definition in LambdaRecDerivation supplies the same functional J_curv(λ) = 2λ² obtained from Gauss-Bonnet normalization on S² with χ=2 and bounding area 4πλ².
proof idea
One-line wrapper that applies the simp tactic to the definition of J_curv.
why it matters
This supplies the zero point required for the extremum argument that determines λ_rec in Conjecture C8. It closes the base case of the curvature cost before the module proceeds to equate J_bit with J_curv and restore SI dimensions via c³λ²/(ℏG). The result sits inside the Recognition Science derivation chain that begins from the unique cost functional J(x) and reaches D=3 spatial dimensions through the eight-tick octave.
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