Acoustics Speech Intelligibility From Jcost

The cost of hearing

Speech intelligibility is the measure of how much of a spoken message a listener can understand, often scored as the percentage of words correctly recognized. The most common way to quantify the listening challenge is the signal-to-noise ratio (SNR), the power of the speech signal divided by the power of the background noise. A higher SNR means clearer speech; a lower one means the noise competes with the talker. In audiology, the speech-reception threshold (SRT) is the SNR at which a listener recognizes 50 percent of words; for healthy adults this sits near -7 dB, where the signal power is only about one fifth of the noise power.

Recognition Science models this everyday situation with a single function called the recognition cost, a number that rises as a task becomes harder. The framework's central result, proved in its machine-checked library of formal theorems, gives a unique cost function J(x) = (x + 1/x)/2 - 1. Applied to speech, the module defines the cost on the SNR ratio r. At r = 1, where signal and noise powers match, the cost is exactly zero, matching the idea that a barely audible signal is the threshold of intelligibility. Below that ratio, the cost climbs as the signal weakens relative to noise, and intelligibility drops accordingly. The cost also treats a ratio and its reciprocal as equally costly, a symmetry that reflects the listener's symmetric difficulty with too little signal or too much noise.

The module then connects this cost to hearing impairment. It defines a hearing-loss penalty at discrete steps k of SNR degradation, where each step is a factor of the golden ratio phi, about 1.618. The penalty at zero steps is zero, and it is never negative at any step. The clinical picture fits this ladder: hearing-impaired listeners show SRT shifts of +5 to +15 dB, which the module reads as one to three phi-rungs of extra recognition cost. A healthy listener at -7 dB sits at a cost near zero; a listener who needs +8 dB to reach 50 percent recognition has climbed roughly three rungs of penalty.

In plain terms, the module packages the framework's cost function as a tool for speech audiometry. It proves that the cost is zero at the 1:1 signal-to-noise threshold, that it is symmetric under swapping signal and noise, and that it is always nonnegative, with the hearing-loss penalty sharing those properties. These are not empirical claims about how ears work; they are formal consequences of the cost function's definition. The framework's contribution is to show that a single forced cost function, with no free parameters, reproduces the qualitative shape of intelligibility loss. The quantitative match to real SRT values remains a comparison against measured data, not a theorem about the ear itself.

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

theorem srCost_reciprocal_symm {r : ℝ} (hr : 0 < r) :
    srCost r = srCost r⁻¹ := Cost.Jcost_symm hr

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

This module does not prove that real human hearing follows the cost function; it only proves formal properties of the defined cost. The clinical SRT values cited are external measurements, not theorems derived within the framework. The module does not establish a mechanism for how the ear computes recognition cost.

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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