Acoustics Speech Intelligibility From Jcost Speech Intelligibility Cert

The certificate

Speech intelligibility is the practical question of how much of a spoken message a listener can recover. The standard measures, the speech intelligibility index (SII) and the speech transmission index (STI), both reduce to a single physical quantity: the signal-to-noise ratio, the power of the speech signal divided by the power of the background noise in a frequency band. When the signal equals the noise in power, the ratio is 1, and a normal listener recognizes words with ease. As the noise grows relative to the signal and the ratio falls below 1, recognition becomes harder and word scores drop.

In the Recognition Science framework, the cost of recognizing a signal at a given signal-to-noise ratio r is not a free choice. The framework's central theorem forces the cost function to be J(r) = (r + 1/r)/2 - 1. This function has three properties that matter for hearing. It is zero exactly at r = 1, so perfect signal-to-noise balance carries no recognition penalty. It is symmetric under taking the reciprocal, so a signal twice as loud as the noise costs the same as noise twice as loud as the signal. And it is always nonnegative, rising steeply as the ratio departs from 1 in either direction.

The machine-checked library of formal theorems packages these properties into a single structure called SpeechIntelligibilityCert. The structure is a certificate: it bundles the five key facts, the zero at threshold, the reciprocal symmetry, the nonnegativity, and the zero and nonnegativity of a hearing-loss penalty, into one object that any downstream proof can rely on. The library also constructs an instance of the certificate, so the facts are not just stated but proved. The hearing-loss penalty itself is defined at discrete steps of the golden ratio phi, the framework's natural scaling unit, so a penalty at step k is the cost at a signal-to-noise ratio of phi raised to the power -k.

What the certificate does not claim is just as important. It does not assert that any particular human listener will score a specific percentage on a word test. It does not derive the empirical speech-reception threshold of about -7 dB for healthy adults, which corresponds to a ratio near 0.2; that is a measured clinical fact, not a theorem. The certificate establishes the mathematical shape of the recognition cost, not the biological details of hearing. It is a precise statement about a cost function, and the clinical numbers remain measurements against which the framework's structure can be compared.

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

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

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 certificate does not predict any specific word-recognition score for a human listener. The certificate does not derive the measured speech-reception threshold of -7 dB; that value is a clinical measurement. The certificate does not model the frequency-band weighting used in SII or STI standards.

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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