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

We disprove the conjecture of Camacho, Khudoyberdiyev and Omirov that, for every positive integer \(n\) and every nonsingular complex \(n \times n\) matrix \(B = (b_{ij})\), the system \(x_i^2 = \sum_{j=1}^n b_{ij}x_j\) (\(1 \le i \le n\)) has a solution with every coordinate nonzero. 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 subvarieties isomorphic to \((\mathbb{C}^*)^3\), each a complete intersection of codimension six. Every exceptional matrix has exactly seven nonzero entries, the minimum possible for a counterexample in dimension three. Every three-dimensional complex evolution algebra \(E\) with \(E^2 = E\) and at most six nonzero entries in a natural-basis structure matrix has an idempotent whose natural-basis coordinates are all nonzero, 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.