planck_area_eq_inv_pi
plain-language theorem explainer
In Recognition Science native units the Planck area equals 1/π. Quantum-gravity unification work cites this result when connecting the recognition scale to the Gauss-Bonnet curvature quantum of the three-dimensional manifold. The proof is a one-line wrapper that reduces the left-hand side to the prior identity G · ℏ = 1/π after setting c = 1.
Claim. $G ħ / c³ = 1/π$ in RS-native units, where $G$ is the gravitational constant derived as $λ_{rec}² c³/(π ħ)$, $ħ = φ^{-5}$, and $c = 1$.
background
Recognition Science obtains all constants from the single J-cost functional equation. The J-cost is the arithmetic-geometric mean gap: for x > 0, J(x) = (x − 1)²/(2x), which is exactly AM(x, x⁻¹) − GM(x, x⁻¹) with equality only at x = 1. The module establishes the octave duality κ · ħ = 8, with κ = 8φ⁵ and ħ = φ⁻⁵, and derives G = λ_rec² c³/(π ħ) from Gauss-Bonnet closure on the recognition manifold Q₃. Upstream, the definitions of G and ħ appear in Constants, while the functional equation and AM-GM identities are proved in Cost.FunctionalEquation.
proof idea
The proof is a one-line wrapper. It first applies simp to replace c with 1 (using one_pow and div_one), then invokes the lemma G_hbar_gauss_bonnet that already establishes G · ħ = 1/π.
why it matters
This is QG-003 inside the Quantum Gravity Octave Duality module. It is invoked by planck_area_pos to obtain positivity and by qg_octave_cert to assemble the full certificate. The result closes the chain from the J-cost AM-GM gap through the phi-ladder constants (T5–T6) and the eight-tick octave (T7) to the Planck scale being the recognition scale divided by √π, consistent with D = 3 (T8) and the alpha band.
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