Cost Frequency Ladder Frequency Ratio Cost Unit

The zero-cost baseline

In the Recognition Science framework, a ledger, a discrete record of events, assigns a cost to every positive ratio between two frequencies. The cost function J(r) = ½(r + r⁻¹) − 1 evaluates that cost. The theorem named frequencyRatioCost_unit states that when the ratio is exactly one, meaning the two frequencies are equal, the cost is zero. This is the baseline: identical frequencies cost nothing to recognize.

The result is proved in the framework's machine-checked library of formal theorems. It is a direct consequence of the definition of the cost function and the arithmetic fact that ½(1 + 1) − 1 = 0. The proof is a single simplification step, not a chain of deep lemmas. It establishes a boundary condition, not a law of motion.

What the theorem does not claim is just as important. It does not say that any other ratio has zero cost, nor does it say anything about which ratios are preferred or forbidden. It does not establish that the golden ratio φ, the unique positive solution to r² = r + 1, is special. That separate result, proved elsewhere in the same library, shows φ is the unique positive self-similar ratio and therefore the minimal-cost non-trivial resonance above any frequency. The unit-ratio theorem is a prerequisite for that story, but it is not the story.

The practical meaning is a starting point. When an oscillating system sits at frequency f, the framework asks what the cheapest step up is. The answer, f × φ, comes from the uniqueness of φ, not from the zero-cost baseline. The unit theorem simply clears the ground: it says the ledger charges nothing for recognizing a frequency as itself. Everything else about the cost ladder builds from there.

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

/-- 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

What this page does not claim

The theorem does not assign zero cost to any ratio other than exactly one. The theorem does not state that φ is the minimal-cost ratio; that is a separate result about self-similar ratios. The theorem does not describe how costs combine for multiple frequency steps.

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)

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Derived articles

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