rt60
rt60(k) assigns reverberation time for the k-th acoustic regime the value phi^k on the Recognition phi-ladder. Acoustics researchers modeling room scaling within the framework cite it to enforce the constant adjacent-regime ratio phi. The declaration is a direct power definition imported from the phi constant.
claimLet $k$ be a natural number and let $phi$ denote the golden-ratio fixed point. The reverberation time for regime $k$ is defined by $rt60(k) := phi^k$.
background
The module establishes five canonical room-acoustic regimes with configDim equal to 5: anechoic, heavily-damped, semi-reverberant, reverberant, and echoic. Reverberation time RT60 scales on the phi-ladder so that the ratio between adjacent regimes equals phi. The speech intelligibility threshold is given by J(phi) lying in (0.11, 0.13). This definition supplies the explicit scaling form rt60(k) = phi^k that realizes the ladder. No upstream lemmas are required.
proof idea
The declaration is a direct definition that sets rt60(k) equal to phi^k. It serves as the base for the ratio theorem, which unfolds the definition, applies pow_succ, and uses ring to recover the adjacent ratio phi.
why it matters in Recognition Science
This definition supplies the explicit reverberation-time form required by RoomAcousticsCert to certify the five regimes and the phi-ratio property. It realizes the phi-ladder scaling from T6, where phi is the self-similar fixed point, and supports the Recognition Composition Law through power scaling. The construction connects to the eight-tick octave via successive phi powers. No open questions or scaffolding remain.
scope and limits
- Does not assign numerical RT60 values to specific room geometries.
- Does not incorporate frequency-dependent absorption or material coefficients.
- Does not derive the five-regime classification from the wave equation.
- Does not address non-ideal boundary conditions or diffraction.
formal statement (Lean)
30noncomputable def rt60 (k : ℕ) : ℝ := phi ^ k
proof body
Definition body.
31