G_hbar_gauss_bonnet
plain-language theorem explainer
Recognition Science derives the product of the gravitational constant and reduced Planck constant as one over pi in native units. Unification researchers cite the result when closing the Gauss-Bonnet factor inside the octave duality certificate. The term proof rewrites the product via the definition of G then cancels terms with field simplification under the nonzero conditions on pi and hbar.
Claim. In Recognition Science native units with the speed of light set to one, the gravitational constant and reduced Planck constant satisfy $G · ℏ = 1/π$.
background
The Quantum-Gravity Octave Duality module fixes constants via the J-cost functional equation, where J(x) equals the AM-GM gap (x-1)²/(2x) for x>0. The reduced Planck constant is defined as hbar = phi^{-5} and proved positive. The gravitational constant is introduced as G = lambda_rec² c³ / (π hbar), which reduces exactly to 1/(π hbar) once c=1. This places the theorem immediately after the Constants definitions of G and hbar_pos.
proof idea
One-line term proof that rewrites the left-hand side with G_eq_inv_pi_hbar then applies field_simp, discharging the side conditions Real.pi_ne_zero and ne_of_gt hbar_pos.
why it matters
The result supplies the middle conjunct of three_products and is invoked directly by planck_area_eq_inv_pi to obtain the Planck area as 1/π. It completes the QG-002 slot in the module, identifying 1/π as the minimal Gauss-Bonnet curvature quantum for three-dimensional space and reinforcing the eight-tick octave lock between kappa and hbar.
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