Dear Ryan,
I think your premise does not hold. The universe cannot be based on symmetry, otherwise physical manifestations would not exist. Also it cannot sum to zero, it can only sum to an infinite number times unity if you know what I mean.
There is another way to describe the universe assuming is consists of smaller elements (quoted from KVK Nehru):
In a closed group of operators, like [1 i j k], the result of the combination of any number of the basal elements is also a member of the same group. The result of any such combination can be known only if all the possible binary combinations of the elements are first defined in terms of the basal elements i, j and k themselves (besides, of course, the identity operator, 1). Let there be n basal elements (excluding the unit operator 1) in a group. Then the number of unique binary combinations of these elements, in which no element occurs twice, is n(n-1)/2. We can readily see that a group becomes self-sufficient (finite) only if the number of binary combinations of the basal elements is equal to the number of those basal elements themselves, that is
n(n-1)/2 = n.
The only definite solution for n is 3. (Zero and infinity are other solutions.) Therefore if we regard space (&time) as a group of orthogonal rotations, its dimensionality has to be three in order to make it self-sufficient dimensionally. Otherwise the number of dimensions either has to shrink to zero, or proliferate to infinity.
Good luck with the contest!
Steven Oostdijk