Acoustics Speech Intelligibility From Jcost Sr Cost Nonneg

The nonnegativity theorem

Speech intelligibility, the fraction of words a listener correctly identifies, depends on the signal-to-noise ratio (SNR): the power of the speech signal divided by the power of the background noise. In the Recognition Science framework, this dependence is modeled by a recognition cost, a number that measures how hard it is for the system to recognize a signal. The declaration srCost_nonneg is a machine-checked theorem stating that this cost is never negative for any positive SNR. In plain terms: recognizing speech never has a negative price; it costs something, or at worst nothing, but it never produces a credit.

The theorem is one of several that together form a certificate of basic sanity for the model. Its companion results state that the cost is exactly zero when the signal power equals the noise power (SNR = 1), that the cost is symmetric under swapping signal and noise (the cost at SNR r equals the cost at 1/r), and that the cost is strictly positive whenever the SNR differs from 1. The nonnegativity theorem is the one that rules out the absurd possibility of a negative cost, which would mean that noise improves recognition. The proof is carried out in a machine-checked library of formal theorems, meaning the argument has been verified step by step by a computer; the declaration reports zero unproven assumptions and zero axioms.

The theorem does not claim that the model is a complete description of human hearing. It does not establish that any particular listener will understand any particular sentence, nor does it specify the exact shape of the intelligibility curve beyond the properties listed above. The clinical numbers in the framework's documentation, such as a speech-reception threshold near -7 dB for healthy adults, are empirical observations, not consequences of this theorem. What the theorem does establish is a consistency condition: within the model, the cost function behaves as a cost should, never rewarding the system for failing to recognize.

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

/-- Speech-intelligibility-from-J-cost certificate. -/
def speechIntelligibilityCert : SpeechIntelligibilityCert where
  threshold_zero := srCost_zero_at_threshold
  reciprocal_symm := srCost_reciprocal_symm
  cost_nonneg := srCost_nonneg
  penalty_zero := hearingLossPenalty_zero
  penalty_nonneg := hearingLossPenalty_nonneg

What this page does not claim

The theorem does not establish that any particular listener will understand any particular sentence. The theorem does not specify the exact shape of the intelligibility curve beyond the listed properties. The clinical numbers in the documentation are empirical observations, not consequences of this theorem.

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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