phi_sq
plain-language theorem explainer
The lemma states that the golden ratio φ satisfies φ² = φ + 1. Researchers deriving the baryon-to-photon ratio η from Recognition Science's 8-tick CP asymmetry would invoke this algebraic relation when scaling phases on the φ-ladder. The proof is a direct tactic verification that unfolds the closed-form definition of φ and reduces the square via ring rewriting and the identity (√5)² = 5.
Claim. Let φ = (1 + √5)/2 be the golden ratio. Then φ² = φ + 1.
background
The module COS-007 derives the observed matter abundance η ≈ 6.1 × 10^{-10} from CP violation inside the 8-tick phase structure of Recognition Science, where φ supplies the self-similar scaling for asymmetry ε_CP. The golden-ratio relation is the algebraic engine behind the φ-ladder mass formula and the T6 fixed-point construction. Upstream, the identity event in ObserverForcing is defined as the J-cost minimum at state 1, while the VantageCategory identity functor is the structure-preserving map that leaves states and transitions unchanged.
proof idea
The tactic proof unfolds phi to its explicit form (1 + Real.sqrt 5)/2, expands the square with ring, substitutes (sqrt 5)² = 5 via Real.sq_sqrt, and reduces the resulting linear expression back to phi + 1 by further ring steps.
why it matters
This supplies the defining algebraic identity for φ that enters every downstream scaling in the matter-antimatter module, including eta_from_epsilon and the Sakharov-condition lemmas. It realizes the T5 J-uniqueness and T6 phi fixed-point steps of the UnifiedForcingChain inside the cosmology setting. The relation is standard and closed; no scaffolding remains.
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