phi5_mul_phi5
plain-language theorem explainer
Researchers deriving the product of gravitational and quantum constants in Recognition Science cite this algebraic identity for fifth powers of the golden ratio. It confirms that φ raised to the fifth power multiplied by itself equals φ to the tenth. The derivation applies the real exponent addition rule under positivity of the base followed by numerical simplification of the resulting exponent.
Claim. $φ^5 ⋅ φ^5 = φ^{10}$ for the golden ratio φ > 0.
background
The Quantum Gravity Octave Duality module derives κ ⋅ ℏ = 8 from the single J-cost functional. J-cost is the AM-GM gap: J(x) = (x + x^{-1})/2 - 1, which is nonnegative and vanishes only at x = 1. This forces the constants κ = 8φ^5 and ℏ = φ^{-5} in RS-native units, with their product fixed at the octave factor 8. The lemma supplies elementary power arithmetic needed for these manipulations. Upstream structures establish J-cost convexity and the 8-tick local dynamics that generate the φ-ladder.
proof idea
The term proof is a one-line wrapper that rewrites the product via the real exponent addition rule Real.rpow_add under the positivity assumption on φ, then applies norm_num to resolve the exponent sum 5 + 5 = 10.
why it matters
The identity supports the central octave duality result κ ⋅ ℏ = 8 (QG-001) that locks quantum and gravitational scales by the eight-tick cycle. It fills a basic algebraic step in the module's derivation of Planck area 1/π and the Fibonacci mass ladder. Framework landmarks include T7 (eight-tick octave) and T5 (J-uniqueness). No downstream uses are recorded; the lemma closes a routine power-manipulation gap without touching open questions.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.