G_derivation_chain_complete
plain-language theorem explainer
The theorem certifies a complete non-circular chain deriving Newton's constant G and the Einstein kappa from Q3 geometry, Gauss-Bonnet curvature, and the bit-curvature balance condition. Researchers tracing fundamental constants to recognition costs would cite it to confirm G equals lambda_rec squared times c cubed over pi hbar with hbar equal to phi to the minus five. The proof is a term-mode record constructor that supplies each field of the GDerivationChain structure by reflexivity or direct appeal to prior lemmas.
Claim. The derivation chain holds: the cube Q_3 has eight vertices; total curvature equals 4π by Gauss-Bonnet; curvature cost satisfies J_curv(λ) = 2λ² for all λ; there exists a unique positive λ such that J_curv(λ) equals the normalized bit cost; G equals λ_rec² c³ / (π ℏ) with ℏ = φ^{-5}; and κ equals 8φ^5.
background
The module derives the recognition length λ_rec without circular reference to G. It starts from the normalized bit cost of one and the curvature cost 2λ² obtained from Q3 Gauss-Bonnet normalization, then defines G as the consequence G := π λ_rec² c³ / ℏ. The structure GDerivationChain packages the six steps that connect Q3 combinatorics through curvature balance to the final value of kappa. Upstream lemmas supply the individual links: total_curvature_gauss_bonnet states that eight vertices times the angular deficit per vertex equals 2π times the Euler characteristic of the sphere; J_curv_derivation shows the curvature cost per token is exactly 2λ²; balance_determines_lambda proves uniqueness of the positive root from the cost-balance equation; and kappa_einstein_eq recovers κ = 8φ^5 after unfolding the definitions of G, ℏ, and λ_rec.
proof idea
The term proof constructs the GDerivationChain record directly. It sets the vertex-count field by reflexivity, the Gauss-Bonnet step by total_curvature_gauss_bonnet, the curvature-cost formula by J_curv_derivation, the uniqueness step by balance_determines_lambda, the G-formula field by reflexivity, and the kappa field by kappa_einstein_eq. No additional tactics are required beyond these direct assignments.
why it matters
This theorem closes the non-circular derivation of G inside the Recognition Science framework, linking Q3 geometry to the forcing-chain constants (T5 uniqueness, T8 three-dimensionality, ℏ = φ^{-5}). It supplies the explicit certificate that kappa equals 8φ^5 without presupposing Newton's law. No downstream uses are recorded yet, but the chain supports later derivations of the fine-structure interval (137.030, 137.039) and the mass ladder. The result touches the open question of whether the same balance condition extends consistently to the full eight-tick octave.
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