Wow! 2 counts is huge. Accuracy is supposed to be 0.05%+2 in the 5V range.
Unless I am mistaken, 0,05% of 5.00000V is 0.00250V, i.e. 250 counts in 500,000 count mode.
So 2 counts, or even 20 counts as you probably meant to write, is not bad at all.
Full scale is not the problem. The bottom of the scale might be.
Try measuring 600mV for example. If the difference is still 2 counts, the meter could easily fall outside of its 1 year calibration.
...
A simple look at Joe's 1mV pictures, will show that the relationship is not linear and points to a dubious 0.02%+2 accuracy in the mV range, at least for negative values.
I just can't understand your arguments.
The BM869 is specified at 4 4/5 digits, and this spec for DCV as usual contains accuracy for the reading plus accuracy for the range, that is a relative and an absolute error spec.
These errors are valid, independent of the reading, i.e. it fits to Full Scale, as well to low readings.
Both errors simply add up, but do not depend on each other, so there's nothing dubious, and the 2digits error can not violate the relative 0.02% reading error, even at lower readings.
Also, the question of linearity is not appropriate, here.
The absolute error, also often defined as 'error of range' is the dominant value for lower readings, covering effects like internal e.m.f. voltages, and also this roll-over error (difference in reading at input reversal).
So it's a bit convoluted, to distinguish e.m.f. and roll-over error, especially on low voltages as 1mV, and at low scale readings.
Clever e.m.f. cancellation is needed, or usage of the higher range, where e.m.f. does not play a role.
Anyhow, the BM869 obviously meets its specification in all cases, but more than that can not be expected..
6 1/2 bench DMMs probably perform much better, concerning this roll-over error. For example, the HP DMMs feature an additional -10V vs. +10V calibration item, which the handheld DMMs often / mostly do not have. But that comes at a cost..
For 5 4/5 digits of the BM869, the specification should be valid analogously, that means that the absolute error is 10 times higher, .. + 20 digits.
As the integration time is longer for this higher resolution, that implies a different integration algorithm, or maybe a different circuit, so results may be slightly different, but hard to be detected, due to the masked digit in 4 4/5 mode.
Just for completeness, I have measured the performance and linearity of the BM869 a few years ago:
https://www.eevblog.com/forum/testgear/bymen-bm869-measurement-of-dcv-linearity/msg666100/#msg666100, but unfortunately not measured both polarities. I will repeat this experiment accordingly, when I check my BM869 the next time.
As a first shot, the roll-over error on my BM869 at 5V F.S. is about 5 digits in 5 4/5 mode, so well below the +2digits specification.
Frank