So I'm thinking this comes from the mathematics of adding (...summing, integrating, etc) incoherent phasors.

Dunno if this helps but I often use white noise to cross check my test gear here. I have a homemade high(ish) level noise source that puts out a pretty flat noise spectrum out to 180MHz.

If I put it through a 2.8MHz high order LPF and into a decent (Anritsu) thermocouple power meter I can measure the total heating power of the noise energy inside that 2.8MHz bw.

I can adjust the noise power to show -21dBm on the power meter via the 2.8MHz LPF. The power head works right down to a few kHz. This power level is 7.94e-6W and the Vrms should be 20mV in a 50R system.

If I then feed the output to an old analogue true rms voltmeter terminated in 50R it indicates very close to 20mV (as expected) on the dial

If I then click down in 10dB steps via a precision attenuator I see (about) 6.3mV, 2mV, 630uV, 200uV and about 65uV on the bottom range of the voltmeter. The meter has a 20MHz BW and can see down to 10uV but it begins to read a bit high once you get down to a few tens of uV due to its own internal noise.

So the power is falling by a factor of 10 (10dB) in each case but the voltage is only falling by the square root of 10 (= 3.16) in each case.

If I return the attenuator to 0dB attenuation and measure the noise in a 1Hz bw using the noise marker on a decent spectrum analyser the analyser shows -86.0dBm/Hz at 1MHz.

This is about 11.21uV in a 1Hz BW. To predict the voltage in the full 2.8MHz bandwidth we simply use our known method of multiplying 11.21uV by sqrt of 2800,000 to get 18.8mV.

This 18.8mV is in pretty close agreement with the 20mV seen on the voltmeter. If I measured the true noise bandwidth of the high order 2.8MHz LPF the result might have worked out even closer but to get within about 0.5dB isn't a bad result especially when the various measurement uncertainties of my test gear are taken into account

Note: The attenuator is very expensive and extremely accurate (with a very low VSWR) over all of the 50dB range.