J_curv_derivation
plain-language theorem explainer
The curvature cost per recognition token at scale lambda equals twice lambda squared. Researchers deriving Newton's constant non-circularly from normalized bit and curvature costs cite this equality. The proof is a single reflexivity step that matches the definition of the curvature cost functional.
Claim. $J_ {rm curv} (lambda) = 2 lambda^2$, where $J_{rm curv}$ is the curvature cost of embedding one recognition token derived from four curvature quanta on $Q_3$, Gauss-Bonnet normalization on $S^2$ with Euler characteristic 2, and bounding area $4 pi lambda^2$.
background
The module formalizes a non-circular derivation of the recognition length lambda_rec. It uses only the bit cost normalized to 1 and the curvature cost 2 lambda^2 from Q3 Gauss-Bonnet normalization, without reference to Newton's G. G is defined afterward as a consequence: G := pi lambda_rec^2 c^3 / hbar. J_curv(lambda) is the curvature cost of embedding one recognition token at scale lambda, obtained from |kappa| = 4 curvature quanta on Q3, Gauss-Bonnet on S2 with chi = 2, and area 4 pi lambda^2, which simplifies directly to 2 lambda^2. Upstream results include the identical definition of J_curv in PlanckScaleMatching (curvature packet axiom distributing a plus-or-minus-4 packet over eight Q3 vertices) and the lambda normalization constant lambda = ln(phi) from RGTransport.
proof idea
The proof is a one-line wrapper that applies reflexivity to the definition of J_curv, which is already stated as 2 * lambda ^ 2.
why it matters
This theorem supplies the explicit curvature cost formula used as step3_J_curv_formula in G_derivation_chain_complete. It closes the curvature packet axiom from Q3 geometry and Gauss-Bonnet normalization, feeding the balance condition that determines lambda_rec without circularity. The result aligns with Recognition Science landmarks by providing the J-cost term for the phi-ladder scaling and the eight-tick octave structure.
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