Acoustics Speech Intelligibility From Jcost Sr Cost Zero At Threshold

The threshold at which speech becomes clear

In the study of speech perception, intelligibility is the fraction of words a listener can identify in a noisy environment. The standard measure is the signal-to-noise ratio (SNR), the power of the speech signal divided by the power of the background noise. When the SNR is high, speech is easy to understand; when it is low, listeners strain and miss words. The threshold where recognition becomes unreliable is a central concept in audiology, called the speech-reception threshold (SRT), often defined as the SNR at which a listener recognizes 50 percent of words.

The Recognition Science framework models this phenomenon with a recognition cost, a quantity that measures how hard it is to identify a signal. The framework defines a specific cost function, J-cost, which depends on the SNR ratio r. The declaration srCost_zero_at_threshold establishes a single, precise fact: when r equals 1, meaning the signal power exactly matches the noise power, the cost is zero. At this point, recognition is effortless. Below r = 1, the cost rises and intelligibility drops, matching the clinical observation that listeners need a positive SNR to understand speech reliably.

This zero-cost point is not arbitrary. The framework derives J-cost from five plain conditions, including reciprocal symmetry, which means the cost of a ratio r is the same as the cost of its reciprocal 1/r. The theorem srCost_zero_at_threshold is a direct consequence of this derivation, not a separate assumption. It is proved in the framework's machine-checked library of formal theorems, with zero unproven axioms. The framework also proves that the cost is never negative for positive ratios, and that it is strictly positive whenever r differs from 1, so the threshold is a genuine minimum, not a plateau.

The clinical relevance appears in the hearing-loss penalty. The framework defines this penalty as the cost at a SNR that has been degraded by a factor of the golden ratio phi, roughly 1.618, raised to a negative integer power. At zero steps of degradation, the penalty is zero, matching the threshold theorem. For one to three steps of degradation, the penalty grows, and the framework's docstring notes that hearing-impaired listeners show SRT shifts of +5 to +15 dB, which corresponds to one to three such steps. This is a modeling claim, not a measured result, but it gives a concrete picture of how the abstract cost function connects to audiometric data.

What the declaration does not claim is equally important. It does not say that a 50 percent word-recognition score occurs exactly at r = 1; the SRT is a clinical convention, and healthy adults typically perform at an SNR near -7 dB, which corresponds to r approximately 0.2. The theorem only establishes the mathematical property of the cost function at the point where signal and noise are equal. It also does not claim that J-cost is the only possible cost function for speech intelligibility, nor does it assert that the framework's phi-rung penalty scale is the unique way to model hearing loss. These are definitional choices, not theorems.

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

theorem srCost_nonneg {r : ℝ} (hr : 0 < r) : 0 ≤ srCost r :=
  Cost.Jcost_nonneg hr

theorem srCost_pos_off_threshold {r : ℝ} (hr : 0 < r) (hne : r ≠ 1) :
    0 < srCost r := Cost.Jcost_pos_of_ne_one r hr hne

/-- Hearing-loss penalty at φ-step `k` of SNR degradation. -/
def hearingLossPenalty (k : ℕ) : ℝ := srCost (phi ^ (-(k : ℤ)))

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

What this page does not claim

The theorem does not state that the 50 percent word-recognition point occurs at r = 1; the SRT is a clinical convention and healthy adults typically perform at a lower SNR. The framework does not claim that J-cost is the only possible cost function for speech intelligibility, nor that the phi-rung penalty scale is uniquely determined by the axioms. The hearing-loss penalty is a definitional model, not a measured relation between SNR and audiometric data.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: