Acoustics Room Impulse Response From Jcost

Room impulse response and its cost

A room impulse response is the acoustic fingerprint of a space: the sound you hear after a sharp clap, including the direct sound and all the echoes that follow. Engineers measure it to design concert halls, tune loudspeakers, and build virtual reality audio. A key summary number is the reverberation time, often called RT60, the seconds it takes for sound to decay by 60 decibels. The standard formula is T_60 = 0.161 V / (A × α), where V is room volume, A is total absorption area, and α is the average absorption coefficient of the surfaces. The Sabine formula, named after Wallace Sabine who published it in 1898, remains the workhorse of architectural acoustics.

The absorption coefficient α measures how much sound a surface soaks up, from 0 for a perfect mirror to 1 for an open window. In Recognition Science, the framework models this coefficient with its central cost function J(x) = (x + 1/x)/2 - 1, evaluated at the golden ratio φ ≈ 1.618. The framework's library proves three general facts about J: it equals zero when its input equals 1, it never goes negative for positive inputs, and φ - 3/2 is positive. These three facts, applied to the ratio of measured to expected absorption, give a recognition cost, a discrete record of how far a measurement strays from expectation.

In Recognition Science, the acoustics module defines the cost of a room's absorption mismatch as J(measured / expected). The module proves this cost vanishes exactly when measured equals expected, and stays nonnegative whenever both quantities are positive. The threshold φ - 3/2 ≈ 0.118 appears as a canonical boundary, though the module proves only that this number is positive, not what it means acoustically. The framework's library states these results once, universally, and the acoustics module inherits them by plugging in its own ratio.

The plain-language payoff is a consistency check, not a new measurement law. The module certifies that the cost function behaves sensibly: perfect agreement costs zero, disagreement costs something positive, and the golden-ratio threshold sits where the framework says it should. What the module does not do is derive the Sabine formula or predict RT60 from first principles. The definition of measured and expected absorption in acoustic terms remains a research note, not a proved result.

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]

def domainCost (measured expected : ℝ) : ℝ := Jcost (measured / expected)

theorem cert_inhabited : Nonempty RoomImpulseCert := ⟨cert⟩

What this page does not claim

The module does not derive the Sabine formula or predict RT60 from first principles. The module does not prove that the golden-ratio threshold has any acoustic meaning. The definition of measured and expected absorption in acoustic terms is a research note, not a proved result.

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.

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: