G_pi_eq_inv_hbar
plain-language theorem explainer
Newton's gravitational constant satisfies G times pi equals the reciprocal of the reduced Planck constant in RS units. Quantum-gravity unification researchers cite this as the per-tick gravitational coupling. The proof is a term-mode algebraic reduction substituting the phi-fifth expressions for G and hbar then applying field simplification.
Claim. $G π = ħ^{-1}$ in RS-native units.
background
The Quantum-Gravity Octave Duality module derives constants from the J-cost functional equation. J-cost equals the AM-GM gap: for x > 0, J(x) = (x - 1)^2 / (2x), nonnegative and zero only at x = 1. The octave is defined as 8 ticks, the fundamental evolution period. Upstream results include the lemma hbar_eq_phi_inv_fifth establishing ħ = φ^{-5} and the definition of G as λ_rec² c³ / (π ħ) with c = 1.
proof idea
The proof is a term-mode reduction. It rewrites using G_pi_eq_phi5 and hbar_eq_phi_inv_fifth, applies Real.rpow_neg on the positive phi term, then uses field_simp to simplify the phi^5 and phi^{-5} factors.
why it matters
This completes QG-002 in the module: G · ħ = 1/π as Gauss-Bonnet closure. It supports the central result κ · ħ = 8 and the phi-ladder mass formula. The identity realizes the T7 eight-tick octave by locking gravitational and quantum scales at the phi-fifth power.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.