Cost Frequency Ladder Phi Cost Fixed Point

The golden ratio fixed point

The golden ratio φ, approximately 1.618, is the unique positive solution to the equation r² = r + 1, which is equivalent to r = 1 + 1/r. This self-referential property has been known since antiquity; Euclid called it the extreme and mean ratio. It appears in the regular pentagon, in the Fibonacci sequence as the limit of successive term ratios, and in natural spirals. The equation has two roots, but only one is positive: the other is approximately -0.618, which is -1/φ.

The property r = 1 + 1/r means that if you take a quantity and add one, then take the reciprocal of that sum and add one, you return to the original quantity. More concretely, φ is the number for which decreasing it by one and taking the reciprocal gives the same number back: 1/(φ - 1) = φ. This fixed-point property is what makes φ a natural scaling factor: a system that grows by a factor of φ and then shrinks by the same rule returns to its starting point.

In Recognition Science, the framework models the cost of a frequency ratio through a function J(r) = ½(r + 1/r) - 1, where r is the ratio of two frequencies. This cost is zero when the frequencies are equal, and it grows as the ratio moves away from unity. The framework's machine-checked library of formal theorems proves that φ is the unique positive ratio satisfying the self-similarity equation r² = r + 1, and therefore the unique positive fixed point of the cost recursion.

The declaration phi_cost_fixed_point proves the algebraic identity φ = 1 + 1/φ directly from the defining equation. It does not by itself establish that φ is the minimal-cost ratio; that requires the separate uniqueness theorem, which shows that no other positive number satisfies the self-similarity equation. Together, these theorems support the framework's claim that for any positive frequency f, the first φ-harmonic f × φ is the unique minimal-cost resonance above f.

The framework's library also defines a structure PhiHarmonicForced that packages the existence and uniqueness of this φ-harmonic for any positive frequency. The construction is forced in the sense that given the cost function and the self-similarity condition, no other ratio is available. The framework uses this to define the first rung of a φ-ladder of frequencies in its resonance model, though the physical interpretation of that ladder remains a modeling choice rather than a derived consequence.

/-- 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

/-- φ is the cost fixed point: φ = 1 + 1/φ.
    Follows directly from φ² = φ + 1. -/
theorem phi_cost_fixed_point : phi = 1 + 1 / phi := by
  have hsq := phi_sq_eq
  have hne := phi_ne_zero
  field_simp at hsq ⊢; linarith

/-- 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

What this page does not claim

The theorem does not claim that φ is the minimal-cost ratio by itself; that requires the separate uniqueness theorem. The framework does not claim that the φ-ladder corresponds to any observed physical spectrum. The fixed-point identity does not imply that all self-similar ratios are cost-minimizing, only that φ is the unique positive one.

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.

Derived articles

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