phi_plus_inv
plain-language theorem explainer
The golden ratio φ satisfies φ + φ^{-1} = √5. Researchers deriving J-costs or mass ladders in Recognition Science cite this algebraic identity to simplify expressions at the self-similar fixed point. The proof is a short tactic sequence that unfolds the definition of φ, applies field simplification and ring normalization, then invokes the square-root squaring identity to reach equality.
Claim. Let φ denote the golden ratio. Then φ + 1/φ = √5.
background
The Inequalities module collects fundamental algebraic lemmas that support the Recognition Science framework. The golden ratio φ is the positive fixed point of the self-similar forcing step (T6) and enters the J-cost definition J(x) = (x + x^{-1})/2 - 1. Upstream structures include the Constants bundle from CPM.LawOfExistence, which packages RS-native constants, and the cost functions from MultiplicativeRecognizerL4 and ObserverForcing that derive recognition costs from positive-ratio comparators.
proof idea
The proof unfolds the definition of φ from Constants. It introduces positivity of 5 and a non-zero denominator via norm_num and linarith. Field_simp and ring_nf reduce the expression, after which rw [Real.sq_sqrt] followed by ring closes the equality.
why it matters
This identity is applied directly by J_cost_phi to obtain J(φ) = (√5 - 2)/2 and by the YangMillsMassGap theorems Jcost_phi_exact and phi_plus_inv for the exact closed form. It supplies the algebraic step required for T5 J-uniqueness and the phi fixed point, enabling the eight-tick octave and the mass formula on the phi-ladder.
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