phi_fifth_self_dual
plain-language theorem explainer
The identity φ^5 ⋅ φ^{-5} = 1 holds for the golden ratio φ. Researchers deriving the quantum-gravity constants cite it as the algebraic step that forces κ ⋅ ℏ = 8 exactly. The proof is a one-line term wrapper that rewrites the product via the real exponent addition rule and normalizes the resulting exponent to zero.
Claim. $φ^5 ⋅ φ^{-5} = 1$, where $φ$ is the golden ratio fixed point of the J-cost functional.
background
Recognition Science derives the golden ratio φ as the unique positive self-similar fixed point of the J-cost function J(x) = (x − 1)^2 / (2x), which equals the AM-GM gap for the pair {x, x^{-1}}. The module QuantumGravityOctaveDuality proves that the Einstein coupling equals κ = 8φ^5 while the reduced Planck constant equals ℏ = φ^{-5}. These definitions appear in the imported Constants module; the Cost module supplies the J-cost expression that forces the φ-ladder and the eight-tick octave structure.
proof idea
The proof is a one-line term-mode wrapper. It applies the real exponent addition rule (Real.rpow_add) to rewrite φ^5 ⋅ φ^{-5} as φ^{5 + (−5)}, then normalizes the exponent to obtain φ^0 = 1.
why it matters
This identity supplies the algebraic core of the central theorem κ ⋅ ℏ = 8 in the Quantum-Gravity Octave Duality. It directly implements the eight-tick octave (T7) that locks the gravitational and quantum sectors through the single J-cost functional. The result feeds the downstream claims that G ⋅ ℏ = 1/π and that the Planck area equals 1/π in RS units. No open scaffolding remains attached to this elementary step.
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