Acoustics Room Acoustics From Phi Ladder Rt60

The RT60 ladder

In room acoustics, RT60 is the standard measure of reverberation: the time, in seconds, for a sound to decay by 60 decibels after the source stops. A concert hall might have an RT60 near two seconds; a small recording booth near 0.2 seconds. The classical definition is purely empirical: measure the decay, read the time. It says nothing about why one room lands at one value and another at a different one.

The Recognition Science declaration rt60 (a discrete record of a quantity at integer steps) takes a different route. It defines a sequence of values, one for each natural number k, by the formula rt60(k) = φ^k, where φ is the golden ratio, about 1.618. The declaration proves three structural facts: there are exactly five room-acoustic regimes (anechoic, heavily damped, semi-reverberant, reverberant, echoic); the ratio of consecutive rt60 values is always φ; and every rt60 value is positive. These are machine-checked theorems in the framework's library of formal proofs.

What the declaration does not do is predict a specific room's RT60 in seconds. The sequence rt60(k) is a dimensionless ladder, not a calibrated time. The framework's own docstring connects the ladder to speech intelligibility, placing the threshold in a band between 0.11 and 0.13, but that band is a stated observation, not a proved theorem. Nothing in the declaration says that a real concert hall must have RT60 = φ^3 seconds; it says that if you arrange the five regimes in order, the ratio between adjacent ones is φ.

The practical consequence is a scaling rule for thinking about rooms: moving from one regime to the next multiplies the reverberation time by about 1.6. That is a compact way to remember the difference between a dead studio and a live hall. The framework's contribution is to derive that ratio from its cost function rather than from measurement, but the ratio itself is a classical observation about how rooms feel.

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

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

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

What this page does not claim

The declaration does not predict a specific room's RT60 in seconds. The speech intelligibility band is a stated observation, not a proved theorem. The declaration does not claim that the golden ratio is the only possible scaling for room acoustics.

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.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: