Acoustics Harmonic Distortion Rs

Harmonic distortion in the ledger

Harmonic distortion is the unwanted change a sound system adds to a signal, usually measured as the ratio of new frequency components to the original. A common audibility threshold is around 0.5 to 1 percent total harmonic distortion (THD). In the Recognition Science framework, the module Acoustics.Harmonic_Distortion_RS attempts to connect this threshold to the framework's core cost function, but the connection is not established.

The module defines domainCost as the framework's cost function, a measure of recognition effort, applied to a ratio of two real numbers m and e. The machine-checked library proves three general facts about this definition: the cost is zero when the two numbers are equal, it is never negative for positive inputs, and a certain threshold constant involving the golden ratio is positive. These are all true statements about the cost function itself, not about sound.

The library's own documentation is explicit: the module proves nothing specific to acoustics, because domainCost is defined without any reference to a physical quantity like pressure or frequency. The research note in the module suggests a possible threshold of about 1.39 percent, derived from the cost function squared, but this is a recorded intention, not a proved result. The module is one of 2383 siblings sharing the same generic body, each awaiting a subject-specific definition of m and e.

What the module does establish, in plain language, is a small set of properties that any recognition cost must satisfy: zero cost for perfect agreement, nonnegativity, and a positive canonical threshold. These are the same properties that appear across the framework's domains. The acoustics application remains a target, not a theorem.

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 cert_inhabited : Nonempty HarmonicDistCert := ⟨cert⟩

What this page does not claim

The module does not prove that harmonic distortion audibility is 1.39 percent. The module does not derive the golden ratio from acoustics. The module does not establish any property specific to sound 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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: