Hearing-Aid Fitting from First Principles on a Phone: Gain Targets, Latency, and Feedback Bounds

Most hearing loss is never fitted because audiology is scarce and hearing aids are priced as luxuries. This paper derives a fitting a phone plus earbud chain can honestly perform. A loudness-restoration mapping from pure-tone thresholds gives gain T(1 - L/100), from which the half-gain rule follows at a 50 dB input (0.50T), with 0.35T at 65 dB and compression ratio 100/(100 - T). For a flat 50 dB loss, average-speech targets are 12.5, 15.5, 17.5, 20.5, 19.5 and 14.5 dB at 250 to 8000 Hz, agreeing with stated representative NAL-style reference values to 2.06 dB RMS (maximum 4.5 dB). End-to-end latency from stated component delays is 15.64 ms wired, 55.64 ms for LE Audio and 125.64 ms for Bluetooth Classic, with first comb-filter notches at 32.0, 9.0 and 4.0 Hz; only the wired chain stays below the 20 ms own-voice disturbance bound and the 45 ms lip-sync detection bound. Feedback caps maximum stable gain at 9.0 dB (open dome), 24.0 dB (closed dome) and 39.0 dB (foam or custom mould): an open fitting serves none of the modelled losses, a closed dome fully serves only a flat 30 dB loss, and a foam or custom seal is required for a flat 50 dB loss and a sloping loss reaching 70 dB at 8000 Hz. Profound loss, severe flat loss at soft-speech gain, severe recruitment, and single-sided deafness are outside the honest service window. All results are simulated, not measured; nothing was built, worn, or tested on people, and all computations are reproducible from the deposited compute.py.