G_eq_phi_fifth_over_pi
plain-language theorem explainer
The declaration proves that in recognition science native units Newton's gravitational constant equals phi to the fifth power divided by pi. Researchers deriving quantum-gravity relations from the single J-cost functional would cite it to obtain the Gauss-Bonnet closure G hbar = 1/pi. The proof is a short term-mode algebraic reduction that substitutes the prior hbar identity and applies field simplification.
Claim. In RS-native units, Newton's gravitational constant satisfies $G = phi^5 / pi$, where $phi$ is the unique positive fixed point of the J-cost map $J(x) = (x + x^{-1})/2 - 1$.
background
Recognition Science derives all constants from the J-cost functional equation whose self-similar fixed point is the golden ratio phi. The module QuantumGravityOctaveDuality proves that the Einstein coupling kappa and the reduced Planck constant hbar are dual under the eight-tick octave, with product exactly 8. This supplies the explicit form for G that closes the Gauss-Bonnet relation G hbar = 1/pi. The upstream lemma hbar_eq_phi_inv_fifth states that hbar equals phi to the negative fifth power, the fundamental identity for the coherent energy scale. The definition of G in Constants reduces to 1 over pi hbar once c equals 1.
proof idea
The proof is a term-mode reduction. It first rewrites G via the inverse relation to hbar, substitutes hbar = phi^{-5}, invokes positivity of phi^5, and finishes with field_simp to clear the denominator.
why it matters
This result is invoked by seven downstream theorems in the same module, including G_fibonacci_form, G_over_hbar_phi_tenth, kappa_G_product, and qg_octave_cert. It realizes the QG-002 statement in the module documentation that G hbar equals 1/pi, supplying the minimal Gauss-Bonnet curvature quantum of Q3. The identity anchors the phi-ladder mass formula and the alpha band within the T5-T8 forcing chain.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.