Acoustics Room Acoustics Sabine From Jcost Optimal T60

The golden reverberation time

The Sabine reverberation time, T60, is the number of seconds it takes for a sound to decay by 60 decibels after the source stops. The classic formula T60 = 0.161 V / A ties it to the room's volume V and its total absorption area A. For a concert hall, the ideal T60 sits in a band that balances warmth against clarity; too long and speech turns muddy, too short and music sounds dead. The empirical surveys of Leo Beranek, the standard reference for concert hall acoustics, placed the optimum for large halls around 1.6 to 2.1 seconds, with Carnegie Hall at 1.89 and Vienna's Musikverein at 2.05.

In Recognition Science, the framework's cost function J(x) = (x + 1/x)/2 - 1, which the library proves is forced by five plain conditions, applies to the ratio of observed absorption to critical damping in a room. The framework models the optimal reverberation time as the golden ratio phi, approximately 1.618 seconds, and its library defines this as the constant optimalT60. The machine-checked library of formal theorems proves that this value lies strictly between 1.61 and 1.62 seconds, and that it exceeds 1 second, meaning a room tuned to it is not over-damped to the point of anechoic deadness.

The structural prediction is that the optimal concert-hall T60 equals phi seconds, which lands inside the empirical Beranek band. Carnegie Hall at 1.89 s and Vienna at 2.05 s both fall within the interval (phi, phi squared) = (1.618, 2.618). This is a comparison against measured values, not a theorem about real halls; the framework's proof establishes the number phi as the optimum within its own model, and the match to Beranek's survey is an empirical check that the model's chosen optimum lands in the observed range.

What the declaration does not claim is broader than what it does. It does not prove that every real concert hall should be built to 1.618 seconds; the Sabine formula itself is an approximation that assumes diffuse sound fields and moderate absorption. It does not derive the Sabine formula from first principles, nor does it account for the frequency-dependent absorption that real materials have. The framework's contribution is a single clean number, phi, derived from its cost function and shown to sit in the empirically optimal band, not a full theory of room acoustics.

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

/-- 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 framework does not derive the Sabine formula T60 = 0.161 V/A from its cost function. The framework does not prove that every real concert hall should be built to exactly 1.618 seconds. The framework does not account for frequency-dependent absorption or non-diffuse sound fields.

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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