Acoustics Harmonic Distortion Rs Canonical Threshold Pos

The audibility threshold

Harmonic distortion is the unwanted change a device adds to a sound by creating new frequencies that were not in the original signal. Audio engineers measure it as a percentage, and the threshold at which listeners notice it has been studied for decades. A common rule of thumb puts the audible limit near one percent total harmonic distortion, with some listeners detecting as little as half a percent on pure tones.

In the Recognition Science framework, the ledger (a discrete record of recognition events) gives rise to a cost function J(x) = (x + 1/x)/2 - 1, and the golden ratio φ ≈ 1.618 appears throughout the framework's structure. The framework's library of machine-checked formal theorems contains a file that connects this cost function to harmonic distortion. In that file, a proposed audibility threshold is defined as canonicalThreshold = φ - 3/2, which equals about 0.118, or 11.8 percent. The declaration canonicalThreshold_pos proves that this number is greater than zero, using the fact that φ exceeds 1.5.

The proof is genuine and machine-checked: it derives the positivity of the threshold from the definition of φ. But the file's own documentation is explicit about what this does not establish. The threshold 11.8 percent is far above the roughly one percent audibility limit that audio engineers actually measure. The file notes that squaring the threshold, J(φ)², gives about 1.39 percent, which is closer to the empirical value, but this is recorded as a research note, not a theorem. The formal proof establishes only three general facts about the cost function: it is zero when the input equals one, it is nonnegative for positive inputs, and φ - 3/2 is positive. Nothing in the proof connects these facts to the physics of hearing or to any definition of what makes a sound audibly distorted.

The reason is that the file defines its cost as J(m/e) for two real numbers m and e, but never defines what m and e mean in acoustic terms. The declaration proves a fact about a formula, not a fact about sound. What would turn this into a theorem about harmonic distortion is a definition of m and e in the subject's own terms, for example as measured distortion energy and original signal energy. Until that definition exists, the positivity proof remains a mathematical curiosity, and the audibility claim remains a hypothesis with a named falsifier: a listening test showing that listeners can detect distortion below 1.39 percent would contradict it.

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)

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

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

What this page does not claim

The proof does not establish that 11.8 percent is the actual audibility threshold for harmonic distortion. The proof does not derive the one percent threshold from the framework's axioms. The squared value 1.39 percent is a research note, not a proved result.

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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