hearingLossPenalty
plain-language theorem explainer
Hearing-loss penalty at φ-step k of SNR degradation is defined by applying the speech-recognition J-cost to the ratio φ^{-k}. Auditory modeling researchers cite it when mapping observed SRT shifts of +5 to +15 dB to one-to-three rungs on the recognition ladder. The definition is a direct one-line wrapper around srCost.
Claim. The hearing-loss penalty at rung $k$ is $J_cost(φ^{-k})$, where $J_cost$ is the recognition cost on the signal-to-noise ratio $r$.
background
Speech intelligibility is governed by recognition cost on the ratio $r$ = signal_power / noise_power. The module sets srCost(r) := Cost.Jcost(r), which is zero at the intelligibility-1 threshold $r=1$ and rises for $r<1$, matching the clinical speech-reception threshold near $r≈0.2$ for healthy listeners. Hearing-impaired listeners show SRT shifts of one to three φ-rungs, each adding a positive penalty term.
proof idea
The definition is a one-line wrapper that applies srCost to the argument phi raised to the negative integer power of k.
why it matters
This definition supplies the penalty term required by SpeechIntelligibilityCert, which collects threshold-zero, reciprocal symmetry, nonnegativity, and penalty properties. It implements the module's mapping of clinical +5 to +15 dB shifts onto the φ-ladder (T6 self-similar fixed point) without new axioms. The construction closes the link between J-cost formalism and measurable hearing-loss data.
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