Smyth's conjecture and a non-deterministic Hasse principle

The matrix ( 3 -3 4 -4 5 -5 0 0 4 -4 -3 3 0 0 5 -5 -5 5 0 0 -3 3 -4 4 ) has an interesting property. Can you see what it is?

In case this puzzle is not enough information about the talk: I will explain how to prove a conjecture of Smyth from 1986 about linear relations between Galois conjugates, and explain what this has to do with (some subset of depending on time) linear combinations of permutation matrices, Brianchon's theorem on ellipses inscribed in hexagons, weightings on the edges of directed hypergraphs, representations of discrete groups with compact image, and the study of Diophantine equations where the equation is to be solved with probability distributions instead of with numbers. This is joint work with Will Hardt, and can be seen at https://arxiv.org/abs/2503.15833 (which has spoilers for the puzzle.)