Acoustics Room Acoustics From Phi Ladder Rt60 Pos

The positivity guarantee

Reverberation time, usually written RT60, is the number of seconds it takes for a sound to decay by 60 decibels after the source stops. In the Recognition Science framework, this familiar quantity is modeled on a phi-ladder, a discrete ladder of values where each rung is the golden ratio φ ≈ 1.618 times the rung below it. The framework defines rt60(k) as φ raised to the k-th power, so the sequence runs φ⁰, φ¹, φ², and so on. The theorem rt60_pos states the obvious but essential fact: every value on this ladder is strictly greater than zero.

The proof is a one-liner in the machine-checked library of formal theorems. Since φ itself is positive, any positive number raised to a natural-number power stays positive. The declaration certifies this for every rung k, not just for a few checked examples. It is a universal statement, verified by the kernel of the proof assistant, with no axioms beyond the standard three and no unproved assumptions.

The theorem matters because it is one of three properties bundled into the RoomAcousticsCert structure, a certificate that packages the framework's claims about room acoustics. Alongside the fact that there are exactly five acoustic regimes (anechoic, heavily damped, semi-reverberant, reverberant, echoic) and the fact that adjacent regimes differ by a factor of φ, the positivity guarantee completes a minimal, machine-checked description of the model. It ensures the ladder never dips to zero or negative values, which would be physically meaningless for a time duration.

What rt60_pos does not claim is broader. It does not assert that real rooms actually have reverberation times that follow this ladder; that would be an empirical claim about physics, not a theorem about a definition. It does not say that φ is the only possible scaling for room acoustics, nor that the five-regime classification is exhaustive for all real spaces. The theorem is a statement about the internal consistency of a model, not a measurement of any actual room.

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

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

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

noncomputable def roomAcousticsCert : RoomAcousticsCert where
  five_regimes := roomAcousticRegime_count
  phi_ratio := rt60_ratio
  rt60_always_pos := rt60_pos

What this page does not claim

No claim that real rooms have reverberation times following this exact ladder. No assertion that φ is the only possible scaling factor for room acoustics. No empirical measurement of any actual room's reverberation time.

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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