Acoustics Speech Intelligibility From Jcost Hearing Loss Penalty Zero

The zero-rung threshold

In acoustics, speech intelligibility is the fraction of words a listener can recognize. It depends on the signal-to-noise ratio (SNR), the power of the speech signal divided by the power of the background noise. A common clinical measure is the speech-reception threshold (SRT), the SNR at which a listener recognizes 50% of words. Healthy adults typically reach that threshold at about -7 dB, which corresponds to a signal-to-noise ratio of roughly 0.2.

The Recognition Science framework models this situation with a cost function on the SNR. The framework's central cost function, J(x) = (x + 1/x)/2 - 1, is forced by five plain conditions and is proved in a machine-checked library of formal theorems. Applied to acoustics, the framework defines a speech-recognition cost that is zero when the signal-to-noise ratio equals 1, meaning the signal power matches the noise power. At that point, recognition is free. Below that ratio, the cost rises and intelligibility drops. The framework also defines a hearing-loss penalty at discrete steps, or rungs, of SNR degradation. One rung corresponds to a factor of the golden ratio phi, about 1.618, in SNR reduction.

What the declaration hearingLossPenalty_zero establishes is precise and narrow: at zero rungs of degradation, the hearing-loss penalty is zero. This is not a claim about real ears. It is a theorem about the framework's own definitions, proved from the fact that the speech-recognition cost at an SNR of 1 is zero. The theorem is one of five properties bundled into a certificate that the framework's acoustics model satisfies its basic sanity conditions: zero cost at the threshold, symmetry between a ratio and its reciprocal, nonnegative cost everywhere, and nonnegative penalty at every rung.

The clinical picture in the framework's documentation goes further, but that further content is not part of the theorem. The documentation notes that healthy adults perform at an SRT near -7 dB, with a signal-to-noise ratio around 0.2, and that hearing-impaired listeners show SRT shifts of +5 to +15 dB, described as one to three rungs of penalty. Those are empirical observations and a modeling choice, not consequences of the formal theorem. The theorem itself only says that the penalty function starts at zero.

In plain terms, the declaration is a boundary condition. It guarantees that the framework's model of hearing loss has the right starting point: no degradation, no penalty. The interesting content, how the penalty grows with each rung and how it maps onto measured SRT shifts in real listeners, is a separate empirical question that the framework's library does not settle.

/-- Speech-recognition J-cost on the SNR ratio. -/
def srCost (r : ℝ) : ℝ := Cost.Jcost r

/-- The hearing-loss penalty at zero rungs is zero. -/
theorem hearingLossPenalty_zero : hearingLossPenalty 0 = 0 := by
  unfold hearingLossPenalty
  simp
  exact Cost.Jcost_unit0

theorem srCost_zero_at_threshold : srCost 1 = 0 := Cost.Jcost_unit0

/-- The penalty is nonnegative at every rung. -/
theorem hearingLossPenalty_nonneg (k : ℕ) : 0 ≤ hearingLossPenalty k := by
  unfold hearingLossPenalty
  apply Cost.Jcost_nonneg
  exact zpow_pos Constants.phi_pos _

What this page does not claim

The theorem does not claim that real human hearing has zero penalty at any SNR. The framework's documentation on healthy SRT values and hearing-impaired shifts is empirical observation, not a formal theorem. The declaration does not establish how the penalty grows with each rung of degradation.

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/SpeechIntelligibilityFromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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