G_eq_inv_pi_hbar
plain-language theorem explainer
The lemma states that in recognition-science native units with recognition length and speed of light both unity, Newton's gravitational constant equals the reciprocal of pi times the reduced Planck constant. Researchers deriving the Gauss-Bonnet curvature quantum or Planck area from the J-cost functional would cite this identity when closing the quantum-gravity relations. The proof is a one-line term-mode simplification that substitutes the definitions of G, lambda_rec, ell0, and c.
Claim. In RS-native units with $λ_{rec}=c=1$, Newton's gravitational constant satisfies $G=1/(π ħ)$.
background
The Quantum-Gravity Octave Duality module derives all constants from the single J-cost functional, defined as the AM-GM gap J(x)=(x+x^{-1})/2-1. The gravitational constant is introduced via G:=λ_rec² c³/(π ħ), with λ_rec the recognition length from Bridge.DataCore and ħ=φ^{-5} from the phi-ladder fixed point. In native units the module sets ell0=1 and c=1, collapsing the expression for G directly to 1/(π ħ). Upstream, lambda_rec is shown to equal ell0 under the 8-tick cycle, and hbar is fixed by the coherence energy scale.
proof idea
The proof is a term-mode one-line wrapper that applies simp to the definitions of G, lambda_rec, ell0, c, one_pow, and mul_one, reducing the right-hand side immediately to 1/(π ħ).
why it matters
This lemma supplies the intermediate step for the downstream theorem G_hbar_gauss_bonnet, which states the Gauss-Bonnet closure G·ħ=1/π and identifies the factor 1/π as the minimal curvature quantum of Q₃. It also feeds G_eq_phi_fifth_over_pi, recovering G=φ⁵/π. Within the Recognition Science chain the result connects the J-cost AM-GM gap (T5) to the eight-tick octave (T7) and the emergence of D=3, confirming that the Planck area equals 1/π in native units.
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