Hmm, I think we can use this setup to make an even stronger claim about the non-linearity of a particular fractional voltage.

Consider a slightly simpler setup: a 5V reference which has 5 1k resistors, giving us taps of 1V, 2V, 3V, 4V and 5V.

Naturally, our resistors will have some error, so we cannot simply measure the 1V tap and declare that it should be 1.000000V.

However, if we measure every 1V span (0-1, 1-2, 2-3, 3-4, 4-5), since these spans must add up to the total, if we average the 5 measurements, we should be able to claim that the average is exactly 1/5 of Vref (no matter the error of each individual resistor, as long as they didn't drift during the measurements).

I threw together a spreadsheet to play with this idea. I randomly assigned some error to each resistor, then calculated what some of the average spans were. The result is very close -- possibly we are seeing cumulative rounding errors here?

(I did this twice, the second time with a more intentionally biased set of errors in the resistors)

(the bolded values at the bottom of each "span of" column are the average of all of the spans -- in theory these averages should be exactly 5.000000, 4.500000 and 2.000000)