G_fibonacci_form
plain-language theorem explainer
The RS-native gravitational constant equals (5φ + 3)/π. Researchers closing the quantum-gravity octave or building the fermion mass ladder cite this to replace the fifth-power expression with its linear Fibonacci form. The proof is a term-mode reduction that rewrites G via its φ^5/π definition and substitutes the algebraic identity φ^5 = 5φ + 3.
Claim. $G = (5φ + 3)/π$, where $G$ is the gravitational constant in RS-native units and $φ$ is the golden ratio satisfying $φ^2 = φ + 1$.
background
The Quantum-Gravity Octave Duality module derives all constants from the single J-cost functional equation. In RS units (c = 1, ħ = φ^{-5}), the gravitational constant is defined as G = φ^5 / π. The upstream lemma phi_fifth_eq states: 'Key identity: φ⁵ = 5φ + 3 (Fibonacci recurrence). φ⁵ = φ × φ⁴ = φ(3φ + 2) = 3φ² + 2φ = 3(φ + 1) + 2φ = 5φ + 3.' This identity follows directly from the recurrence relations for powers of φ.
proof idea
The term proof first rewrites G using its φ-fifth-over-pi definition, applies congr to align the expressions, casts the integer 5 to reals via norm_cast and rpow_natCast, then invokes the exact lemma phi_fifth_eq to finish the equality.
why it matters
This supplies the explicit Fibonacci form of G needed for the mass-ladder claim in the same module, where m_r = y · φ^r obeys the recurrence m_{r+2} = m_{r+1} + m_r. It directly supports QG-005 in the octave duality certificate by expressing the gravitational constant itself as a linear combination of φ and 1, consistent with the self-similar fixed point of the J-cost equation. The result sits inside the chain from T5 (J-uniqueness) through the eight-tick octave to D = 3.
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