Ah ha... :) I wouldn't see division as exact opposite (or symmetrical inverse) of multiplication. Multiplication is both left-distributive and right-distributive, and thus distributive. Unlike multiplication and addition, division is not commutative, neither associative in general.
It seems as my logic holds in the sense and case that it is “an inversion of meaning in some way” - whether that can be argued as an opposite, inversion, chiral disposition towards 0 or away from 0, who knows!
But most intriguingly it applies that subtraction and division don’t have the properties you mention which is a symmetric “anti” feature of these operators and seemingly can be reduced to the primitive function of addition or subtraction as multiplication and division are just meta operators acting as a loop.
At least this is how I’m viewing it.
There’s an intuitive sense that symmetry breaks when you take away (reduce) compared to add which is in parts of a whole.
And then to realize you can use addition to implement subtraction is where my brain starts to melt even though it’s relatively straight forward. Everything is just addition, the holy monad, the primordial dimensionless unit.
I similarly went down that route and found the answer to be at least plausibly more definitive than undefined.
It also still reaches an intuition on what dividing by zero means, and it is not that it equals zero, but that the amount of cycles to do are 0. As in index is still 0 in the array.
