Acoustics Harmonic Distortion Rs Canonical Threshold

The audibility threshold

Harmonic distortion is the unwanted extra frequencies a loudspeaker or amplifier adds to a sound. Engineers measure it as a percentage of the total signal, and decades of listening tests place the threshold of audibility around 0.5 to 1 percent. A listener usually cannot hear distortion below that band, and can hear it clearly above it.

In Recognition Science, a ledger, a discrete record of recognition events, supplies a cost function that measures how far a ratio of two quantities is from unity. The framework's library, a machine-checked collection of formal theorems, defines a number called canonicalThreshold as phi minus 3/2, where phi is the golden ratio, about 1.618. The value is about 0.118, or 11.8 percent.

The library proves exactly one fact about this number: it is greater than zero. That is the entire content of the theorem attached to the definition. The library also proves that the underlying cost function vanishes when its two inputs are equal and is never negative for positive inputs. These are general properties of the cost function, not results specific to acoustics.

In Recognition Science, the number 11.8 percent is a candidate for an audibility threshold, but the framework does not claim it is the measured one. The research note in the source file records an intention: raising the cost to a power related to spatial dimension would bring it near 1.39 percent, closer to the conventional 1 percent figure. That calculation is a research note, not a proved result. The definition of the threshold does not connect the mass and energy variables in the cost function to any physical quantity in acoustics.

What the declaration does establish is a precise, named constant inside a formal system, with a proof that it is positive. What it does not establish is that this constant corresponds to any measurable property of sound. The gap is not a failure of the proof; it is a missing definition of what mass and energy mean for a loudspeaker. Until that definition exists, the threshold remains a number in search of a physical interpretation.

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

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 canonicalThreshold is not claimed to match the measured audibility threshold of 0.5 to 1 percent. The framework does not claim that harmonic distortion is a recognition event in any physical sense. No theorem connects the canonicalThreshold to any specific acoustic measurement.

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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