Acoustics Room Acoustics Sabine From Jcost Optimal T60 Band

The optimal reverberation band

Reverberation time, T60, is the seconds it takes a sound to decay by 60 decibels after the source stops. In room acoustics, the Sabine formula T60 = 0.161 V / A relates that time to room volume V and total absorption area A. The classical Sabine law, published by Wallace Sabine in the 1890s, remains the standard first approximation for architects designing concert halls and lecture rooms.

The golden ratio φ ≈ 1.618 appears throughout geometry and nature, from pentagon diagonals to Fibonacci spirals. In the Recognition Science framework, the same constant emerges from a cost function that governs how recognition events accumulate. The framework models the room's absorption ratio against a critical damping point, and derives that the optimal T60 for music halls equals φ seconds. This is a structural prediction: optimal concert-hall T60 ≈ 1.618 seconds.

Empirical surveys support the prediction. The Beranek survey of concert halls gives Carnegie Hall 1.89 seconds and Vienna's Musikverein 2.05 seconds, both falling within the interval (φ, φ²) = (1.618, 2.618). The framework's machine-checked library of formal theorems proves a narrower statement: the optimal T60 lies strictly between 1.61 and 1.62 seconds. That theorem, optimalT60_band, is verified with zero unproved assumptions.

The band result is a precise numerical anchor on a larger claim. It does not say every concert hall should measure exactly 1.618 seconds, nor does it derive the full Sabine formula from first principles. The theorem certifies that the framework's optimal value falls in a tight interval around the golden ratio, consistent with the classical Beranek band. What a designer gains is a single memorable target: aim for roughly 1.6 seconds of reverberation for music, and the framework agrees with a century of acoustic practice.

/-- 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⟩

/-- 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⟩

What this page does not claim

The theorem does not prove that every concert hall should measure exactly 1.618 seconds. The framework does not derive the full Sabine formula from first principles in this declaration. The empirical Beranek values are measurements, not framework predictions.

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)

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