Full-Support Solutions and Rigidity for Three-Dimensional Evolution Algebras

We disprove the conjecture of Camacho, Khudoyberdiyev and Omirov that \(x_i^2 = \sum_j b_{ij}x_j\) has a solution with every coordinate nonzero for every nonsingular complex matrix \(B\). Dimension three is the smallest in which this conjecture fails. In that dimension, all counterexamples are equivalent under simultaneous coordinate permutations and nonzero diagonal rescalings. The exceptional locus in \(\mathrm{GL}_3(\mathbb{C})\) is the disjoint union of six smooth, closed, three-dimensional algebraic tori, each a complete intersection of codimension six. Every exceptional matrix has exactly seven nonzero entries, the minimum possible for a counterexample. A regular three-dimensional evolution algebra with at most six nonzero structure constants has a full-support idempotent, and this bound is sharp. We also give polynomial equations characterizing the exceptional locus. The proof uses elimination, traces in an Artinian complete intersection, and local multiplicities.