IndisputableMonolith.Acoustics.SpeechIntelligibilityFromJCost
The module defines the speech-recognition J-cost srCost applied to the signal-to-noise ratio together with its basic properties and a certification for intelligibility. Acoustics researchers working in the Recognition Science framework would cite it when modeling perceptual thresholds from multiplicative costs. The module consists of definitions and direct algebraic lemmas that inherit properties from the upstream J function.
claimDefine the speech-recognition cost by $srCost(SNR) := J(SNR)$ where $J(x) = (x + x^{-1})/2 - 1$. The module proves $srCost(1) = 0$, $srCost(x) = srCost(1/x)$, $srCost(x) = 0$ exactly at threshold, $srCost(x) > 0$ off threshold, and non-negativity, together with a hearing-loss penalty term and the certification $SpeechIntelligibilityCert$.
background
Recognition Science obtains all costs from the J function introduced in the Cost module, where J satisfies the Recognition Composition Law and arises from the T5 uniqueness step of the forcing chain. The Constants module supplies the native time unit τ₀ = 1 tick. This acoustics module specializes J to the SNR ratio, introducing srCost as the direct J-cost on that ratio, the auxiliary hearingLossPenalty, and the SpeechIntelligibilityCert object.
proof idea
This is a definition module, no proofs. The listed lemmas are immediate consequences of the algebraic identities already established for J in the Cost module.
why it matters in Recognition Science
The module supplies the J-cost foundation for speech intelligibility within the acoustics domain of Recognition Science. It links the core J function and constants to perceptual modeling and prepares the ground for any later theorems that quantify auditory performance on the phi-ladder. No downstream declarations are recorded, so the module functions as a self-contained building block.
scope and limits
- Does not derive the J function or its composition law.
- Does not incorporate frequency-dependent or multi-band effects.
- Does not treat multi-speaker interference or reverberation.
- Does not provide numerical evaluation or simulation of intelligibility scores.