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.