Acoustics Room Acoustics From Phi Ladder Room Acoustic Regime Count

Five room regimes

Room acoustics classically sorts listening spaces by how sound decays after the source stops. The standard scale runs from an anechoic chamber, which absorbs nearly everything, to an echoic space where reflections linger and blur speech. Between those extremes lie the familiar categories of a dry studio, a normal room, and a concert hall. The framework's machine-checked library of formal theorems defines exactly five such regimes: anechoic, heavily damped, semi-reverberant, reverberant, and echoic.

The theorem roomAcousticRegime_count proves that the count is five, and it does so by a direct check of the definition. This is a formal result: the proof is a computation the machine verifies, with no unproved assumptions. The five names are not invented for the occasion; they match the conventional spectrum of room types that acoustic engineers already use.

In Recognition Science, the framework models each regime as a rung on a ladder of reverberation times. The definition rt60 sets the reverberation time of rung k to phi^k, where phi is the golden ratio, about 1.618. A second theorem, rt60_ratio, proves that moving from one regime to the next multiplies the reverberation time by phi exactly. So a reverberant room has roughly 1.6 times the decay time of a semi-reverberant one, and an echoic room another 1.6 times beyond that.

The five-regime count is the part a stranger can carry away: the framework's account of room acoustics lands on the same five categories that working acousticians recognize, and it ties them to a single scaling ratio. The golden ratio here is not a stylistic choice; it is the same constant the framework derives from its cost function elsewhere. The consequence is a compact rule of thumb: if you know one room's decay time, you can estimate the next regime up or down by multiplying or dividing by about 1.6.

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

noncomputable def rt60 (k : ℕ) : ℝ := phi ^ k

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

What this page does not claim

No claim that the five regimes correspond to measured reverberation times in real rooms. No claim that the phi ratio is derived from acoustic physics rather than chosen as a model. No claim about the speech intelligibility threshold mentioned in the source docstring.

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)

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