Acoustics Harmonic Distortion Rs Domain Cost At Eq

The cost at equality

Harmonic distortion in audio measures how much a device adds unwanted frequencies to a signal. The Recognition Science framework models such distortion with a cost function, a number that penalizes the gap between an input and an output. Its declaration domainCost_at_eq proves a single, narrow property: when the input and output are equal, the cost is zero. In symbols, for any nonzero real number r, the cost of r compared with r is exactly 0.

That fact is not special to acoustics. The framework defines the cost as J(m/e), where J is a universal function derived from five axioms about how recognition events must behave. The theorem domainCost_at_eq simply applies a general result, that J(1) = 0, to the ratio m/e when m = e. The proof is short: unfold the definition, cancel the division, and invoke the already-proved fact that J(1) = 0. A companion theorem shows the cost is never negative for positive inputs, and a separate constant, phi minus 1.5, is proved positive.

What the declaration does not claim is where the acoustics enters. The research note in the module suggests a threshold for audible distortion around 1 percent, computed as J(phi) squared, where phi is the golden ratio. That number is a research idea, not a proved result. The Lean code proves nothing about audibility, because the definition of domainCost never mentions sound, ears, or distortion. It is a general cost function wearing an acoustics label.

The honest summary: the machine-checked library proves three general facts about the cost function, and the acoustics interpretation remains an unverified hypothesis. A reader should not take the 1 percent threshold as established science. The framework's own documentation says the paragraph above the theorem is a research note recording where the idea was meant to go, not a result.

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

What this page does not claim

The 1 percent audibility threshold is not proved by the machine-checked library. The cost function's zero at equality says nothing about whether equal signals are distortion-free. The theorem does not depend on any acoustics-specific axiom or definition.

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/Acoustics/Harmonic_Distortion_RS.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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