Acoustics Harmonic Distortion Rs Cert Inhabited

The certification object

In mathematics, a certificate is a packaged collection of properties that, once checked, guarantees a result without further argument. The declaration cert_inhabited in the Recognition Science library proves that at least one such certificate exists for a certain abstract cost structure. That is the whole content of the theorem: the structure HarmonicDistCert is not empty, because the library explicitly constructs one instance of it.

The certificate itself bundles three properties of a cost function, a measure of discrepancy between two quantities. The cost is defined as Jcost (m / e), where Jcost is the framework's forced cost function. The three properties are: the cost is zero when the two inputs are equal; the cost is nonnegative for positive inputs; and a certain threshold constant, phi - 3/2, is positive. Each of these is proved by a separate theorem, and the certificate simply collects them into one object.

The proof that the certificate is inhabited is a direct construction: it takes the three theorems and packages them. It does not rely on any acoustic measurement or physical assumption. The name of the file mentions harmonic distortion, and a research note in the source suggests a connection to audibility thresholds, but the formal proof itself never defines what m and e mean for a real acoustic system. The definitions are left abstract.

What this means in practice is that the library has a reusable template: the same three-property certificate appears in many other modules, all sharing the same structure. The acoustic module is one instance of that template, but it is not a theorem about sound. To make it one, someone would need to define the physical quantities in acoustic terms and prove they satisfy the properties. That step is not done here.

The value of the declaration is therefore modest but real: it shows the framework's cost structure is consistent enough to instantiate in a new domain. It does not show that harmonic distortion behaves according to the cost function, nor that the golden-ratio threshold has any acoustic meaning. Those claims remain open.

theorem cert_inhabited : Nonempty HarmonicDistCert := ⟨cert⟩

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

What this page does not claim

cert_inhabited does not prove that harmonic distortion follows the cost function. The declaration does not give an audibility threshold in decibels or percent. The proof does not depend on any acoustic measurement or physical law.

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: