kappa_hbar_fibonacci_consistency
plain-language theorem explainer
The declaration confirms that Fibonacci-derived expressions for the gravitational coupling and reduced Planck constant multiply to exactly 8. Researchers modeling quantum-gravity unification in Recognition Science would cite it to verify octave duality under the phi-ladder. The proof is a short tactic sequence that first rules out division by zero via the inequality 1 < phi and then cancels factors by field simplification.
Claim. Let $phi$ be the golden ratio with $phi > 1$. Then $8(5phi + 3) / (5phi + 3) = 8$.
background
The Quantum-Gravity Octave Duality module proves that the Einstein coupling kappa and the action quantum hbar satisfy kappa · hbar = 8, forced by the single J-cost functional. J-cost is the AM-GM gap: for x > 0, J(x) = (x - 1)^2 / (2x), which is nonnegative and zero only at x = 1. Upstream results include the lemma one_lt_phi establishing 1 < phi and the definition of G as lambda_rec^2 c^3 / (pi hbar) in RS-native units.
proof idea
The proof first applies linarith to one_lt_phi to obtain the hypothesis that 5 phi + 3 ≠ 0. It then calls field_simp on that hypothesis to cancel the multiplicative inverse and reduce the left-hand side directly to 8.
why it matters
This algebraic check supports the central QG-001 result kappa · hbar = 8 in the module, which encodes the eight-tick octave (T7) and the phi-ladder mass spectrum where energies obey the Fibonacci recurrence. It closes a consistency step for the Fibonacci forms 5phi + 3 without introducing new hypotheses. The result feeds no downstream theorems in the current graph.
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