phi_fourth_eq
plain-language theorem explainer
The golden ratio satisfies the fourth Fibonacci recurrence relation phi^4 = 3 phi + 2. Researchers constructing phi-ladder mass formulas or resonance conditions in Recognition Science cite this identity when extending the chain to higher powers. The proof is a calc block that substitutes the cubic identity then the quadratic identity with intervening ring steps.
Claim. The golden ratio satisfies the identity $phi^4 = 3 phi + 2$.
background
Recognition Science forces the golden ratio phi = (1 + sqrt(5))/2 as the self-similar fixed point of the J-cost function. The quadratic identity phi^2 = phi + 1 is the base recurrence from which all higher powers are generated. Upstream results establish the cubic phi^3 = 2 phi + 1 by direct multiplication and ring reduction. The module situates these relations among RS-native constants with the time quantum fixed at one tick.
proof idea
A calc tactic begins by writing phi^4 as phi times phi^3. It applies the cubic identity to obtain 2 phi^2 + phi. The quadratic identity then replaces phi^2 with phi + 1. A final ring step yields the target expression 3 phi + 2.
why it matters
This identity is invoked by the fifth-power lemma, the numerical bounds lemma, the Fibonacci universality theorem, and the Earth-brain resonance result that sets 3 phi^2 = phi^4 + 1. It supplies the fourth link in the phi recurrence chain used for the mass formula on the phi-ladder and for the eight-tick octave construction.
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