But 0 is a value and therefore signifies that whatever the 0 is in relation to does exist even if you don't have it.

q = 0

q mod 0 = q by your explanation? This is the same as q mod (q + n) | n >= 1

The modulus defines what you have left once you deliver your pizzas to the parties equally.

12 pizzas/ 5 parties = 2 pizzas/party

12 pizza mod 5 parties = 2 pizzas

These 2 pizzas have no party association with them

2 pizzas mod 0 parties = 2 pizzas

these pizzas have no parties associated with them.

2 pizzas mod 3 parties = 2 pizzas

2 pizzas have no parties associated with them

Using the above 2 pieces you can determine this is a 1-to-many function and is not reversible.

You could use a predefined condition saying there were 0 parties but then you run into something similar to the Theseus's ship paradox.