Reverb Tail
One mode of a tonal reverberant tail. The kernel lives at eml-stdlib/gaming/audio/reverb.eml:tail(t) = exp(−α·t) · sin(ω·t + φ)Decaying envelope · sinusoidal carrier at the room's dominant resonance. Underdamped sibling of the critically-damped spring proven in MachLib.Applications.SpringCriticallyDamped — same chain-order-2 shape, different physics.
α (decay, 1/s)2.20
f (Hz)220
φ (phase, rad)0.00
verified obligation (from the .eml)
@verify(lean, theorem = "reverb_tail_envelope_bound")ensures
|result| ≤ 1.0 on the .eml-stated domain (α ∈ [0.05, 100], ω ∈ [6.28, 1.26e+5], |φ| ≤ 7, t ∈ [0, 60]).Proof status: not yet shipped in MachLib — the kernel mixes exp and sin, putting it outside the single-exp Khovanskii framework. Direct envelope-bound chain (|sin| ≤ 1, exp(−α·t) ≤ 1 for α,t ≥ 0) is a ~3-line proof obligation; queued as a follow-up Applications/ file.
EML cost (via 1op parser)
positive domain:
11n · general: 17n · naive baseline: 7nChain order 2 (one exp, one sin — not nested). drift_risk LOW per the .eml's `chain_order ≤ 2` constraint. The exp factor is EML-1; the sin factor is EML-3 (Euler bypass). The product stays at chain order 2 because they're independent (not composed). FPGA emit clock pin: 250 MHz (.eml).