The extra digits are not useful as they are below. The integration noise. The extra digits are simply there because of the floating point calc the machine does.

This claim is imo totally unwarranted. Who says so? On what basis? May we get some documentation please ...

I connected a 5 V ref (voltagestandard.com) to the 34401A in 10 V DC 6 Digits Mode.

It is quite steady at 05.000,45 - some flickering to ....,44 and ....,46. The meter reads effectively 6 digits and the last digit resolves 10 microVolt.

I made a data logger from a RS232 cable, a level shift transistor, an Arduino and a Processing sketch. The graph in the video below plots whatever value falls within the last 100 microVolt span (i.e. 1000 lines of 0.1 microVolt on the 100 microvolt peak-to-peak screen). Example: 1.2345001 Volt gives a plot of 1, 1.2345099 gives a plot of 99 and 1.2345678 gives a plot of 678. This is a simple operation that can be done with correct numerical math arithmetic or as a string operation on the original RS232 string reading.

I measured the 5 Volt reference with and without a 10 Ohm resistor in series. The video clearly shows that the DMM reading shifts a few microVolts (5? +- noise) when the resistor is connected (the "square" form is from swithing in and out the resistor). With an input impedance of 10 megaOhm for the DMM this setup shows that Agilent 34401A has a resolution at around 1 ppm in serial RS232 mode. Whether it is accurate or not is irrelevant - the video shows it is fairly stable for 10 seconds or whatever - that is all the time I need to take two measurements for comparison with another reference.

And since this is an experiment anyone with the 34401A, a 10 Ohm resistor and a Data Logger with 1 microVolt resolution can reproduce, it shows that the quote over is totally unsubstantiated.