Acoustics Harmonic Distortion Rs Harmonic Dist Cert

The certificate's scope

Harmonic distortion is the unwanted addition of frequencies that are whole-number multiples of a signal's original frequency. Audio engineers measure it as total harmonic distortion (THD), usually expressed as a percentage of the fundamental signal. The threshold at which distortion becomes audible to a typical listener is commonly cited as roughly 0.5 to 1 percent, though the exact value depends on the signal and the listener.

In the Recognition Science framework, a ledger (a discrete record of recognition events) carries a forced cost function J(x) = (x + 1/x)/2 - 1. The framework's library, a machine-checked collection of formal theorems, defines a domain cost as J(m/e) for two positive real numbers m and e. The declaration HarmonicDistCert bundles three proved facts about this domain cost: it equals zero when m equals e, it is never negative for positive inputs, and a certain threshold value phi - 3/2 is positive. These are general properties of the cost function, proved in the library's shared template and instantiated here.

The certificate's own docstring records an intended connection to audibility: applying the cost function to the golden ratio phi and raising to a power gives about 1.39 percent, near the 1 percent audibility threshold. This is a research note, not a theorem. The Lean code proves nothing specific to acoustics, because the domain cost is defined as J(m/e) without any definition of what m and e mean for a sound signal. The certificate establishes only the three general facts; it does not establish that harmonic distortion follows this cost, that the audibility threshold is 1.39 percent, or that the golden ratio plays any role in real audio systems.

What the certificate does give a reader is a concrete example of how the framework's cost function behaves: it is zero at equality, nonnegative everywhere, and has a positive canonical threshold. These are the building blocks the framework would need if a future definition ever linked m and e to actual acoustic quantities. Until such a definition exists, the certificate remains what its own documentation calls it: a placeholder awaiting subject-specific content.

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 certificate does not prove that harmonic distortion follows the cost function J. The certificate does not establish that the audibility threshold is 1.39 percent. The certificate does not claim the golden ratio appears in real audio systems.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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