I have Day 1 data and Year 1 Data, and I used R to separately calculate their geometric mean as Gmean(low$day1) and Gmean(low$year1)
and their change as $year\, 1 - day\, 1$: Gmean(low$diff)
but why Gmean(low$year1) - Gmean(low$day1) does not match with their changes in geometric mean Gmean(low$diff)?
Even if you only had two values of each, I would not think this might be true except in very special cases.
There is no reason to think $\sqrt{y_1 y_2} - \sqrt{d_1 d_2}$ should equal $\sqrt{(y_1-d_1) (y_2-d_2)}$.
It would be different with arithmetic means: $\frac{y_1+y_2}2 - \frac{d_1+d_2}2 = \frac{(y_1-d_1)+(y_2-d_2)}2$ because everything is nicely linear