Acoustics Harmonic Distortion Rs Cert

The certification's scope

Harmonic distortion is the unwanted change a device adds to a sound, measured as a percentage of extra frequencies in the output. Engineers commonly treat roughly 0.5 to 1 percent total harmonic distortion as the threshold where a listener notices the change. The Recognition Science framework contains a declaration named cert, a machine-checked certificate, that concerns a different quantity: a cost function J(x) = (x + 1/x)/2 - 1, applied to the ratio of two positive real numbers m and e.

The certificate proves three general facts about this cost function. First, when the two numbers are equal, the cost is zero: J(1) = 0. Second, for any positive inputs, the cost is never negative. Third, the number phi - 3/2, where phi is the golden ratio (1.618...), is positive. These are the entire contents of the certificate, stated as a formal structure with a proof of each part. The machine-checked library of formal theorems confirms each of these three statements holds.

In Recognition Science, this certificate does not establish anything specific to acoustics. The module defines domainCost as J(m/e) without connecting m or e to any physical quantity such as a sound pressure or a voltage. The docstring records a research note suggesting that J(phi)^2, about 1.39 percent, might match the audibility threshold, but that note is explicitly labeled as an idea for future work, not a result. The certificate itself proves only the three abstract facts about the cost function and the golden ratio constant.

What the certificate does offer is a reusable template. The same three facts, vanishing at equality, nonnegativity, and a positive threshold constant, appear verbatim in 2383 sibling modules across the framework. The universal version lives in a single shared location, and this acoustics module instantiates it. A future theorem about harmonic distortion would require defining m and e in acoustic terms, such as fundamental and harmonic amplitudes, and then proving the cost behaves as a distortion measure. That step remains open.

theorem cert_inhabited : Nonempty HarmonicDistCert := ⟨cert⟩

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]

structure HarmonicDistCert where
  cost_at_eq : ∀ r : ℝ, r ≠ 0 → domainCost r r = 0
  cost_nonneg : ∀ m e : ℝ, 0 < m → 0 < e → 0 ≤ domainCost m e
  threshold_pos : 0 < canonicalThreshold

What this page does not claim

The certificate does not prove that J(phi)^2 equals any audibility threshold. The certificate does not define what m and e mean for a physical audio system. The certificate does not establish that harmonic distortion in real devices follows the cost function J.

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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