The log 0.04 value above paper white was chosen because it is close to the limit at which we can see tonal difference on typical papers at the bright end of the scale. The brighter the tone, the more sensitive we are to density differences. Darker tones require much greater density difference to be noticed.
I just did a quick test with a calibrated reflection densitometer. I have a contact print made from a Stouffer 0.1 step wedge using a a #00 Ilford filter on Ilford MG Warmtone paper. The base reflection density of white is log 0.07. In rather poor illumination (2.5m from a 40W frosted bare bulb), the first noticeable tone that I can clearly distiguish from white measures log 0.08 on the densitometer. The next band, which looks dramatically darker (relatively speaking), measures log 0.13. This implies that the log-0.04-above-white figure in the ANSI/ISO standard is reasonable in the conservative direction.
Doing the same quick test in the same illumination at the dark end of the scale requires a reflected density change of log 0.10 (from log 1.69 to 1.59) in order to see any distiguishable difference in tone.
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