Acoustics Harmonic Distortion Rs Domain Cost

The domain cost declaration

Harmonic distortion measures how much a device, such as an amplifier or loudspeaker, adds unwanted frequencies to a pure input tone. The standard metric is total harmonic distortion (THD), usually expressed as a percentage of the output signal. A common audibility threshold is roughly 0.5 to 1 percent THD, below which most listeners cannot reliably perceive the added distortion.

In the Recognition Science framework, the declaration domainCost defines a specific cost function for this acoustic setting. The framework models recognition as a discrete record of events, and the cost of recognition is forced by five plain conditions. The declaration sets the cost as J(m/e), where J is the framework's canonical cost function, m is a measured quantity, and e is an expected or reference quantity. When the measured value equals the expected value, the cost is zero. When both are positive, the cost is never negative. These two properties are proved in the machine-checked library of formal theorems, along with a third fact: the golden-ratio constant phi minus 1.5 is positive.

The library proves nothing specific to acoustics. The declaration defines domainCost generically, without tying m and e to any particular acoustic quantities. The docstring records a research note suggesting that J(phi) squared, about 1.39 percent, might match the audibility threshold, but this is explicitly labeled as an idea for where the work was meant to go, not a result. The formal theorems hold for any positive real numbers m and e, not for sound pressure levels or distortion percentages in particular.

What the declaration does establish is a small, clean mathematical skeleton. The cost function vanishes at equality, stays nonnegative for positive inputs, and the canonical threshold constant is positive. These facts are proved with no acoustic assumptions. The acoustic interpretation, the mapping from m and e to physical measurements, and the match to the 1 percent audibility threshold all remain open targets, not derived results.

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

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)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

What this page does not claim

The declaration does not prove that J(phi) squared equals any audibility threshold. The declaration does not define m and e in acoustic terms. The declaration does not establish that harmonic distortion in real devices follows the framework's cost function.

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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Derived articles

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