three_products
plain-language theorem explainer
The theorem proves the three pairwise products among the Einstein coupling κ, Newton's constant G, and reduced Planck constant ħ in RS units: κħ = 8, Għ = 1/π, and κG = 8φ¹⁰/π. Quantum-gravity researchers deriving scale relations from the J-cost functional would cite this to confirm octave locking between quantum and gravitational sectors. The proof is a term-mode packaging of three upstream lemmas on the individual products.
Claim. In RS-native units, the Einstein coupling κ, Newton's constant G, and reduced Planck constant ħ satisfy κħ = 8, Għ = 1/π, and κG = 8φ¹⁰/π, where φ is the golden ratio fixed point of the J-cost functional.
background
Recognition Science derives all constants from the J-cost functional J(x) = (x + x⁻¹)/2 - 1, which equals the AM-GM gap for the pair {x, x⁻¹}. In this framework ħ = φ⁻⁵, κ = 8φ⁵, and G = φ⁵/π, with the octave factor 8 arising from the eight-tick cycle. The module QuantumGravityOctaveDuality centers on the duality κħ = 8 as the first formal link between quantum action and gravitational coupling forced by a single functional equation.
proof idea
The proof is a term-mode wrapper that directly assembles the three lemmas kappa_hbar_octave, G_hbar_gauss_bonnet, and kappa_G_product into the required conjunction. No further rewriting or case analysis is performed.
why it matters
This result completes the set of pairwise products required for the module's QG-001 through QG-004 statements on octave duality and Gauss-Bonnet closure. It draws on the eight-tick octave (T7) and φ-fifth self-duality from the forcing chain, supplying the concrete numerical relations that connect the Recognition Composition Law to the Einstein equations in three dimensions.
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