Acoustics Room Acoustics From Phi Ladder Room Acoustic Regime

Room acoustic regimes

Room acoustics classifies how a space behaves as a sound field: whether a clap dies instantly, rings for a moment, or echoes back at you. Classical acoustics names these conditions anechoic, heavily damped, semi-reverberant, reverberant, and echoic. The Recognition Science framework's machine-checked library of formal theorems defines exactly these five regimes as a discrete list, called RoomAcousticRegime, and proves that there are five and only five of them.

Reverberation time, the seconds it takes a sound to decay by 60 decibels, is the standard measure of a room's liveliness. The framework models this time as a power of the golden ratio φ, the number roughly 1.618 that solves r² = r + 1. Its library defines rt60(k) = φ^k and proves that each step up the ladder multiplies the reverberation time by φ exactly: rt60(k+1) / rt60(k) = φ. Adjacent regimes therefore differ by a factor of about 1.618, and the framework proves every reverberation time stays positive.

This is a definitional choice, not a measurement of real rooms. The framework chooses to model room-acoustic regimes this way, and the choice is internally consistent: a certificate structure bundles the five-regime count, the φ-ratio law, and positivity into one machine-checked package. The declaration establishes a tidy mathematical skeleton for thinking about how rooms transition from dead to live.

What the declaration does not claim matters. It does not claim that real concert halls obey this ladder; that would require empirical verification against measured RT60 data. It does not derive the five regimes from physics; it simply names them. And it does not say that φ is the only possible ratio, only that this model uses it. The value is organizational: a clean, proved framework for a familiar acoustic spectrum, ready for empirical testing.

theorem roomAcousticRegime_count : Fintype.card RoomAcousticRegime = 5 := by decide

theorem rt60_ratio (k : ℕ) : rt60 (k + 1) / rt60 k = phi := by
  unfold rt60
  have hpos : (0 : ℝ) < phi ^ k := pow_pos phi_pos k
  rw [div_eq_iff hpos.ne', pow_succ]
  ring

theorem rt60_pos (k : ℕ) : 0 < rt60 k := pow_pos phi_pos k

What this page does not claim

Real concert halls or recording studios have been measured to follow this φ-ladder. The five regimes are derived from the physics of sound propagation rather than chosen as a model. The golden ratio is the only possible ratio for adjacent reverberation times.

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/RoomAcousticsFromPhiLadder.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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