kappa_per_octave_eq_G_pi
plain-language theorem explainer
The theorem equates the Einstein gravitational coupling divided by the octave factor eight to the product of Newton's constant and pi, both sides reducing to phi to the fifth power in recognition-science units. Researchers deriving quantum gravity from the single J-cost functional equation would cite it to confirm scale locking between gravity and the Planck action. The proof is a direct term-mode rewrite that substitutes the prior expressions for kappa and G then cancels the pi factor.
Claim. In recognition-science units, $κ/8 = G π = φ^5$, where $κ$ is the Einstein coupling coefficient in the field equations and $G$ is Newton's gravitational constant.
background
The module derives quantum-gravity octave duality from the J-cost functional. J-cost is the arithmetic-geometric mean gap of the pair {x, x^{-1}}, so J(x) = (x + x^{-1})/2 - 1, which is nonnegative and vanishes only at x = 1. This forces the constants κ = 8φ^5, ℏ = φ^{-5}, and G = φ^5/π, with the product κ ℏ = 8 locking the gravitational and quantum scales by the eight-tick octave (T7). Upstream lemmas supply the explicit forms: kappa_einstein_eq states κ = 8φ^5 and G_eq_phi_fifth_over_pi states G = φ^5/π.
proof idea
The term proof rewrites the left side via kappa_einstein_eq and the right side via G_eq_phi_fifth_over_pi, then applies field_simp to cancel the nonzero pi factor and obtain equality to φ^5 on both sides.
why it matters
This fills the third of the three equivalent expressions for φ^5 listed in the module doc-comment, confirming consistency inside the octave duality. It supports the central theorem κ ℏ = 8 and the Gauss-Bonnet relation G ℏ = 1/π. In the framework it directly instantiates T7 (eight-tick octave) and the constants forced by the J-cost equation (T5).
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