Acoustics Room Impulse Response From Jcost Room Impulse Cert

The certificate's scope

In acoustics, a room impulse response describes how a room transforms a sound into its echoes and reverberation. A key summary is the reverberation time, often written RT60, the seconds for a sound to decay by 60 decibels. The Sabine formula estimates it as T_60 = 0.161 V / (A × α), where V is room volume, A is total absorption area, and α is an average absorption coefficient. The Recognition Science framework models α with its cost function J(x) = (x + 1/x)/2 − 1, evaluated at the ratio of measured to expected absorption.

The declaration RoomImpulseCert, a certificate in the framework's machine-checked library of formal theorems, establishes exactly three facts about that cost function. First, when the measured value equals the expected value, the cost is zero: domainCost r r = 0 for any nonzero r. Second, for positive measured and expected values, the cost is never negative: 0 ≤ domainCost m e. Third, a threshold defined as φ − 3/2 is positive, where φ is the golden ratio. The certificate packages these three properties into one structure and proves the structure is inhabited, meaning the properties are consistent and hold together.

The certificate does not claim anything specific to room acoustics. The cost function is defined as J(measured / expected) with no definition of what measured or expected mean for a real room. The docstring states this plainly: the module proves general facts about J and nothing specific to its subject. The RT60 formula and the identification of α with J(φ) appear in the research note, not as a proved theorem. What would make this a theorem about acoustics is a definition of measured and expected in acoustic terms, such as a measured absorption coefficient and a theoretical one.

The three proved properties are the foundation such a definition would need. A future theorem could use cost_at_eq to show zero cost at perfect prediction, cost_nonneg to guarantee a meaningful minimum, and threshold_pos to separate the diffuse-field optimum from other regimes. Until then, the certificate is a general-purpose tool, available for any domain that can supply its own definitions of measured and expected.

theorem domainCost_at_equilibrium (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]

theorem cert_inhabited : Nonempty RoomImpulseCert := ⟨cert⟩

What this page does not claim

No specific room impulse response or reverberation time is derived or predicted. The Sabine formula and the identification of α with J(φ) are a research note, not a proved result. No definition of measured or expected is given in acoustic terms.

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/RoomImpulseResponseFromJCost.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.

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