phi5_fibonacci
plain-language theorem explainer
The golden ratio satisfies φ^5 = 5φ + 3. Cosmologists building Recognition Science certificates for dark energy, Hubble tension, and inflaton models cite this identity when populating their certification structures. The proof imports the base quadratic relation φ² = φ + 1, derives the cubic and quartic powers by linear arithmetic, and closes the quintic case with the same tactic.
Claim. Let φ = (1 + √5)/2 be the golden ratio. Then φ^5 = 5φ + 3.
background
The module catalogues five canonical RS number-theoretic identities, with this one listed as the Fibonacci case for exponent 5. The golden ratio φ is the self-similar fixed point forced in the T6 step of the unified forcing chain. The base identity φ² = φ + 1 is supplied by the upstream lemma phi_sq_eq, which is proved in both Constants and PhiLadderLattice by unfolding the definition of φ and applying ring or field simplification.
proof idea
The tactic proof first obtains the quadratic identity via phi_sq_eq. It then derives the cubic relation φ^3 = 2φ + 1 and the quartic φ^4 = 3φ + 2 by nlinarith, after which a final nlinarith step assembles the target quintic equality.
why it matters
This identity is referenced directly in the certification structures darkEnergyEoSDepthCert, HubbleTensionCert, and InflatonCert, where it populates the phi5_fibonacci field. It forms one of the five key identities enumerated in the module documentation and supports the phi-ladder rung calculations used in mass formulas and gap45 derivations. The result closes a scaffolding step in the NumberTheoryFromRS catalogue with zero sorrys.
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