Cost Frequency Ladder

The cost frequency ladder

The golden ratio, usually written φ, is the number that satisfies r² = r + 1. Its decimal value is about 1.618. It appears in geometry, in the Fibonacci sequence, and in many natural patterns. In this framework, it appears again, but for a different reason: it is the cheapest non-trivial step up from any frequency.

Recognition Science starts with a ledger, a discrete record of events. For any two frequencies f₁ and f₂, the framework defines a cost, a number that measures how expensive it is to recognize the ratio r = f₂/f₁. The cost function is J(r) = ½(r + r⁻¹) − 1. It has a simple property: the cost of a ratio equal to 1 is 0, and the cost is never negative for positive ratios. Equal frequencies cost nothing; any difference costs something.

The key question is: what is the least expensive way to go up from a given frequency f? The framework calls a ratio self-similar if it satisfies r² = r + 1. That equation has two roots, but only one is positive: the golden ratio φ. The framework proves that φ is the unique positive self-similar ratio. Therefore, for any positive frequency f, the first φ-harmonic, the next frequency up that is self-similar, is f × φ. This is the minimal-cost non-trivial resonance above f.

In Recognition Science, this result is called the φ-harmonic theorem. It is not an assumption; it follows from the cost function and the definition of self-similarity. The library proves it formally: for any positive f, the structure PhiHarmonicForced f exists, with harmonic = f × φ, and it is unique. The proof relies on the fact that φ is the only positive solution to r² = r + 1.

This ladder is a building block. It justifies defining the first rung of a resonance ladder as Mode1 × φ, as used in the BodyCosmosResonance module. The cost function forces a specific next frequency, not an arbitrary choice. The framework sees this as an example of how a simple cost rule can determine structure.

/-- 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

/-- J-cost of unit ratio is zero: equal frequencies have no cost. -/
theorem frequencyRatioCost_unit : frequencyRatioCost 1 = 0 := by
  unfold frequencyRatioCost Jcost; simp

/-- J-cost is non-negative for positive ratios. -/
theorem frequencyRatioCost_nonneg {r : ℝ} (hr : 0 < r) :
    0 ≤ frequencyRatioCost r :=
  Jcost_nonneg hr

What this page does not claim

The cost function J(r) is not derived in this module; it is taken as given. The φ-harmonic theorem does not claim that any physical oscillator actually exhibits this resonance, only that it is the minimal-cost step within the model. The phrase 'minimal-cost non-trivial ratio' applies only to self-similar ratios, not to all possible ratios.

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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