Cost Frequency Ladder Phi Unique Self Similar

The unique self-similar ratio

The golden ratio, usually written φ, is the number that satisfies a simple self-referential rule: it is equal to one plus its own reciprocal. In symbols, φ = 1 + 1/φ, which is the same as the quadratic equation φ² = φ + 1. Solving that equation gives φ = (1 + √5)/2, approximately 1.6180339887. This is the classical definition, and it appears throughout mathematics: in the geometry of the regular pentagon, in the Fibonacci sequence, and in the continued fraction [1; 1, 1, 1, ...], which is the slowest-converging continued fraction of any irrational number.

The equation r² = r + 1 has two real solutions: φ and its negative reciprocal, −1/φ. If you restrict attention to positive numbers, however, the situation is starkly cleaner. The positive numbers that satisfy the self-similarity equation r² = r + 1 are exactly one: φ itself. This is a simple algebraic fact that follows from factoring the difference between the two roots, and it is the content of the theorem named phi_unique_self_similar in the framework's machine-checked library of formal theorems. The theorem states that if r is a positive real number and r² = r + 1, then r = φ.

In Recognition Science, this uniqueness result earns a specific interpretation. The framework models a discrete record of events, called a ledger, and assigns a cost to any positive ratio between two quantities. The cost function J(r) = ½(r + r⁻¹) − 1 evaluates the price of a ratio r, with equal quantities costing zero and larger deviations costing more. The framework then considers ratios that are self-similar, meaning the ratio r satisfies r² = r + 1. The theorem phi_unique_self_similar proves that among all positive self-similar ratios, φ is the unique one, and therefore it is the minimal-cost non-trivial ratio in this sense. For any positive frequency f, the first φ-harmonic f × φ is the unique minimal-cost resonance above f, as encoded in the structure PhiHarmonicForced.

This is a statement about a specific algebraic equation and its consequences within a formal system. It does not claim that φ is the most harmonious number in all of music theory, nor that physical systems must resonate at φ multiples. It does not assert that the golden ratio appears everywhere in nature. The theorem is a precise, narrow result: given the framework's cost model and the definition of self-similarity, the positive solution to r² = r + 1 is unique. The broader step, connecting this uniqueness to a full theory of resonance in physical systems, is a separate claim that the framework's library treats as a model choice, not as a consequence of this theorem alone.

/-- φ is the UNIQUE positive self-similar ratio.
    Proof: r² = r + 1 and φ² = φ + 1 give (r−φ)(r+φ) = r−φ,
    so (r−φ)(r+φ−1) = 0. Since r > 0 and φ > 1, r+φ−1 > 0,
    so r = φ. -/
theorem phi_unique_self_similar {r : ℝ} (hr_pos : 0 < r)
    (hr_ss : IsSelfSimilarRatio r) : r = phi := by
  unfold IsSelfSimilarRatio at hr_ss
  have hphi_sq := phi_sq_eq
  have hphi_pos := phi_pos
  have hphi_gt1 := one_lt_phi
  have hdiff : (r - phi) * (r + phi - 1) = 0 := by nlinarith
  rcases mul_eq_zero.mp hdiff with h | h
  · linarith
  · exfalso; nlinarith

/-- The φ-harmonic is forced for any positive frequency. -/
noncomputable def phi_harmonic_forced {f : ℝ} (hf : 0 < f) : PhiHarmonicForced f where
  harmonic := f * phi
  harmonic_eq := rfl
  ratio_is_phi := by rw [mul_div_cancel_left₀ _ (ne_of_gt hf)]
  ratio_self_similar := by
    rw [mul_div_cancel_left₀ _ (ne_of_gt hf)]
    exact phi_is_self_similar
  ratio_unique := fun r hr_pos hr_ss => phi_unique_self_similar hr_pos hr_ss

/-- The J-cost of a frequency ratio r = f₂/f₁. -/
noncomputable def frequencyRatioCost (r : ℝ) : ℝ := Jcost r

What this page does not claim

The theorem does not claim that physical systems must resonate at φ multiples. The theorem does not claim that φ is the most harmonious number in music theory. The theorem does not assert that the golden ratio appears throughout nature as a universal constant.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Cost/FrequencyLadder.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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