Acoustics Room Acoustics Sabine From Jcost Room Acoustics Cert

The Sabine certificate

In room acoustics, the Sabine formula T_60 = 0.161 V / A predicts how long sound lingers in a room: T_60 is the reverberation time in seconds, V is the room volume in cubic meters, and A is the total absorption area in square meters. The formula, published by Wallace Sabine in the 1890s, is the field's founding law. A room with too little absorption sounds dead, an anechoic chamber; too much absorption for its volume makes speech muddy. The optimum for music sits between these extremes.

The Recognition Science framework models this balance with its cost function, a measure of recognition effort on a discrete record of events. The framework's library, a machine-checked collection of formal theorems, defines a dimensionless ratio r := observed_absorption / critical_damping. It then derives a golden-section threshold J(φ) that marks the transition from music-hall resonance to lecture-room intelligibility. The framework's definition optimalT60 sets the optimal reverberation time to φ, the golden ratio ≈ 1.618 seconds.

What the certificate RoomAcousticsCert establishes is narrow and precise. It is a structure containing two proved facts: the optimal time lies in the band 1.61 < T_60 < 1.62 seconds, and it exceeds 1 second. The proof of the band uses the framework's constants, not any measurement. The certificate's second field, over_damped, records that φ > 1, so the optimal time is above one second. These are the only claims the structure makes.

The framework then compares this derived value against empirical surveys. The docstring cites Beranek's measurements: Carnegie Hall at 1.89 seconds and Vienna's Musikverein at 2.05 seconds. Both fall in the interval (φ, φ²) = (1.618, 2.618). The match is suggestive, but it is not part of the certificate. The certificate itself proves nothing about real halls; it only states the two inequalities about the number φ. The empirical agreement is a separate observation, not a theorem.

What the certificate does not claim matters. It does not prove that any real concert hall has a reverberation time of 1.618 seconds. It does not derive the Sabine formula from first principles; that formula is taken as the classical starting point. It does not predict the measured values at Carnegie Hall or Vienna; those are cited as external data. The certificate is a formal object: it packages two arithmetic facts about the golden ratio under the name of room acoustics. Its value lies in showing how the framework's cost function connects to a classical engineering law, not in replacing measurements.

/-- Optimal reverberation time = φ (RS-native). -/
def optimalT60 : ℝ := phi

structure RoomAcousticsCert where
  optimal_band : 1.61 < optimalT60 ∧ optimalT60 < 1.62
  over_damped : optimalT60 > 1

/-- Optimal T60 is in the empirical Beranek band for concert halls. -/
theorem optimalT60_band : 1.61 < optimalT60 ∧ optimalT60 < 1.62 := by
  unfold optimalT60
  exact ⟨Constants.phi_gt_onePointSixOne, Constants.phi_lt_onePointSixTwo⟩

/-- Over-damped bound: T60 > 1 (φ > 1). -/
theorem over_damped_below_one : optimalT60 > 1 := by
  unfold optimalT60
  have := Constants.phi_gt_onePointFive
  linarith

/-- Optimal T60 is in the empirical Beranek band for concert halls. -/
theorem optimalT60_band : 1.61 < optimalT60 ∧ optimalT60 < 1.62 := by
  unfold optimalT60
  exact ⟨Constants.phi_gt_onePointSixOne, Constants.phi_lt_onePointSixTwo⟩

What this page does not claim

The certificate does not prove any real concert hall has a reverberation time of 1.618 seconds. The certificate does not derive the Sabine formula from first principles. The empirical agreement with Carnegie Hall and Vienna is cited, not proved by the certificate.

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/RoomAcousticsSabineFromJCost.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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