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I cannot answer why but i can explain difference :

bicubic interpolation is a function f with 16 parameters. To solve a such system with 16 unknown you need 16 data =4x4 neigborhood. It is smallest compact neighborhood.

Using this neigborhood you are sure f(j,i)=I(j,i) I image function (j,i ) pixel of coordinate column and row

Now you can take larger neighborhood but f(j,i)<>I(j,i)

An example with y=a x^2+ b x + c , 5 datas x=[-2 -1 0 1 2]

y=[-5 -1 3 2 3.5]

Do you prefer red or green curve ?