Symmetric one-dependent colorings of regular trees

For every integer $d\geq3$, we construct a proper coloring of the infinite $d$-regular tree with finitely many colors whose law is one-dependent and invariant under every tree automorphism and every permutation of the colors. The construction uses symmetric tensors in a real Hilbert space. Orthogonal projections force adjacent colors to differ, tensor summation gives consistent finite-dimensional laws, and a uniform estimate on rooted trees proves their positivity. An explicit, unoptimized bound is $q(d)=2^{2d+16}d^2(2^d-1)9^{d-1}$ colors.