phi_sq
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.
claimLet 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 in Recognition Science
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.
scope and limits
- Does not derive φ from the Recognition Composition Law.
- Does not address cosmological scaling or η values.
- Does not invoke the eight-tick octave or spatial dimension D = 3.
formal statement (Lean)
147private lemma phi_sq : phi^2 = phi + 1 := by
proof body
Tactic-mode proof.
148 have h : phi = (1 + Real.sqrt 5) / 2 := rfl
149 simp only [sq, h]
150 have h5 : Real.sqrt 5 ^ 2 = 5 := Real.sq_sqrt (by norm_num : (5 : ℝ) ≥ 0)
151 calc ((1 + Real.sqrt 5) / 2) * ((1 + Real.sqrt 5) / 2)
152 = (1 + Real.sqrt 5)^2 / 4 := by ring
153 _ = (1 + 2 * Real.sqrt 5 + (Real.sqrt 5)^2) / 4 := by ring
154 _ = (1 + 2 * Real.sqrt 5 + 5) / 4 := by rw [h5]
155 _ = (6 + 2 * Real.sqrt 5) / 4 := by ring
156 _ = (3 + Real.sqrt 5) / 2 := by ring
157 _ = (1 + Real.sqrt 5) / 2 + 1 := by ring
158
159/-- The Fibonacci-phi identity: φ^(n+1) = F_{n+1} × φ + F_n. -/