phi_cost_fixed_point
The golden ratio satisfies the fixed-point equation φ = 1 + 1/φ. Researchers constructing the φ-ladder of minimal-cost resonances cite this identity when defining the first harmonic above a base frequency. The proof is a direct algebraic reduction from the quadratic relation φ² = φ + 1 using field simplification and linear arithmetic.
claimThe golden ratio satisfies the equation $φ = 1 + 1/φ$.
background
The Cost.FrequencyLadder module defines the J-cost of a positive ratio r by J(r) = ½(r + r⁻¹) − 1. The golden ratio φ is introduced as the unique positive fixed point of the self-similar recursion r = 1 + 1/r, which rearranges to the quadratic φ² = φ + 1. This supplies the algebraic relation needed to identify f × φ as the minimal-cost non-trivial resonance above any frequency f.
proof idea
The proof applies the upstream lemma phi_sq_eq (φ² = φ + 1) together with phi_ne_zero. It then uses field_simp to clear the denominator and linarith to obtain the fixed-point identity.
why it matters in Recognition Science
This supplies the fixed-point identity required by the φ-harmonic construction in the same module. It realizes the T6 step of the forcing chain in which φ is forced as the self-similar fixed point, and it underpins the claim that f × φ is the first minimal-cost resonance on the ladder.
scope and limits
- Does not prove uniqueness of the positive root.
- Does not evaluate the J-cost at this fixed point.
- Does not extend the identity beyond the reals.
- Does not connect the fixed point to explicit physical frequencies.
formal statement (Lean)
69theorem phi_cost_fixed_point : phi = 1 + 1 / phi := by
proof body
Term-mode proof.
70 have hsq := phi_sq_eq
71 have hne := phi_ne_zero
72 field_simp at hsq ⊢; linarith
73
74/-! ## The φ-Harmonic Theorem -/
75
76/-- For any positive frequency f, the first φ-harmonic is f × φ.
77 This is the minimal-cost non-trivial resonance above f.
78
79 The forcing chain:
80 1. J(r) is the cost of ratio r (from T5)
81 2. Self-similar ratios (r² = r + 1) are the scale-invariant resonances
82 3. φ is the unique positive self-similar ratio (from T6)
83 4. Therefore f × φ is the unique first φ-harmonic of f -/