G_pi_eq_phi5
plain-language theorem explainer
Recognition Science derives G π = φ^5 in native units from the J-cost functional. Unification researchers cite the result to show the gravitational coupling and inverse Planck action share the identical φ^5 factor without independent tuning. The proof is a one-line rewrite from the explicit form of G followed by field simplification over the reals.
Claim. In RS-native units with c = 1, Newton's gravitational constant satisfies $G π = φ^5$, where φ is the golden-ratio fixed point of the J-cost functional equation.
background
The module establishes quantum-gravity octave duality with the central result κ ℏ = 8. J-cost is defined as the AM-GM gap: for x > 0, J(x) = (x − 1)^2 / (2x), which is nonnegative and zero only at x = 1. G is introduced via the RS-native expression G = λ_rec² c³ / (π ℏ) and reduces to φ^5 / π once the phi-ladder and octave relations are substituted. Upstream results supply the explicit G definition from Constants and the phi-power scale function from Cosmology.
proof idea
The term proof rewrites via the lemma G_eq_phi_fifth_over_pi that encodes G = φ^5 / π, then applies field_simp with Real.pi_ne_zero to cancel the denominator and obtain the target equality.
why it matters
The declaration feeds the parent theorems G_pi_eq_inv_hbar and phi5_is_both_quantum_and_gravitational, which equate G π with 1/ℏ and confirm φ^5 appears in both quantum and gravitational presentations. It realizes the QG-002 Gauss-Bonnet closure step inside the module and the T8 octave structure of the forcing chain. The result closes the zero-free-parameter claim for the gravitational sector.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.