traumatic arithmetic

Solution 1:

The phrase traumatic arithmetic is a metaphor unique to the author. It means that the traumas had (metaphorically) "added" (or melded) those things together. This resultant "sum" could serve as a proof.

Sums don't normally serve as proofs, but the metaphor's pretty mixed.

Solution 2:

Quique had discovered that {the traumatic arithmetic}[1] {that melded}[2] {a past}[3] and a moustache {could function as proof}[4] of a set of privileged experiences,

[1] the traumatic circumstances

[2] that mixed together

[3] some part of someone's history/past experiences

[4] was also useful as proof