Acoustics Room Acoustics Sabine From Jcost Over Damped Below One

The over-damping threshold

In room acoustics, the Sabine formula T₆₀ = 0.161 V / A gives the reverberation time in seconds: how long a sound takes to decay by 60 decibels after the source stops, where V is the room volume in cubic meters and A is the total absorption area in square meters. A room with too little absorption sounds boomy and speech becomes muddy; a room with too much absorption sounds dead. The classical target for concert halls, from the surveys of Leo Beranek, sits near 1.6 to 2.1 seconds.

Within the Recognition Science framework, the same quantity emerges from a cost function on the ratio of observed absorption to critical damping. The framework's central result forces a unique cost function J(x) = (x + 1/x)/2 - 1, and applying it to room acoustics gives an optimal reverberation time equal to the golden ratio φ ≈ 1.618 seconds. The theorem over_damped_below_one proves a narrow piece of that picture: the optimal time is greater than 1 second. A companion theorem places it in the band between 1.61 and 1.62 seconds, and the framework's library certifies both facts together in a single structure.

The empirical check is direct. Carnegie Hall measures 1.89 seconds, Vienna's Musikverein 2.05 seconds, both inside the framework's predicted range (φ, φ²) = (1.618, 2.618). The match is a measurement comparison, not a derivation from first principles; the framework supplies the structural prediction, and the survey data confirm it falls in the expected band.

What the theorem does not claim matters as much as what it proves. The bound over 1 second is a statement about the framework's optimal value, not a claim that every real concert hall must exceed one second. It does not derive the Sabine formula itself, which remains a classical empirical law. And it does not prove that φ seconds is the best reverberation time for all music; it identifies the framework's optimum and shows it lands in the measured range for famous halls.

/-- 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⟩

/-- 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

What this page does not claim

The Sabine formula T_60 = 0.161 V / A is derived from the framework; it remains a classical empirical law. Every real concert hall must have reverberation time above one second; the theorem concerns the framework's optimal value. The golden ratio is the best reverberation time for all musical genres or listening conditions.

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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