hbar_fibonacci_form
plain-language theorem explainer
The reduced Planck constant in RS-native units equals 1/(5φ + 3) with φ the golden ratio. Quantum-gravity unification work cites this when establishing the octave duality κ ħ = 8. The proof is a direct algebraic substitution from ħ = φ^{-5} combined with the Fibonacci identity φ^5 = 5φ + 3.
Claim. In RS-native units the reduced Planck constant satisfies $ħ = 1/(5φ + 3)$, where φ is the golden ratio.
background
The Quantum-Gravity Octave Duality module fixes ħ via the J-cost functional and the forcing chain. Upstream, hbar_eq_phi_inv_fifth states THEOREM C-004.1: ħ = φ^{-5} in RS-native units. The companion lemma phi_fifth_eq records the key identity φ^5 = 5φ + 3 obtained from the Fibonacci recurrence φ^{n+2} = φ^{n+1} + φ^n.
proof idea
The proof rewrites ħ via hbar_eq_phi_inv_fifth, invokes phi_fifth_eq to replace φ^5 by 5φ + 3, applies rpow_neg, and closes with the ring tactic.
why it matters
This supplies the explicit Fibonacci form of ħ required by the downstream QG Octave Certificate qg_octave_cert, which packages the central result κ ħ = 8. It fills QG-001 in the module and connects directly to the eight-tick octave (T7) in the Recognition Science forcing chain.
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