Chiral fixed points with high-order spectral cancellation in a three-dimensional cellular automaton

We construct pairs of finite chiral fixed points of the three-dimensional outer-totalistic cellularautomaton B5–7/S5–9 whose Fourier transforms have arbitrarily high-order contact at the origin.For every odd integer m ≥ 3, the two mirror-related configurations are distinct stationary states,all their Cartesian moments of degree less than m coincide, and the difference of their Fouriertransforms admits an exact product factorization with a zero of order m in the axial frequency.The construction is stable under up to two site errors per registered eight-site cube. We alsoderive an explicit discrimination bound for finitely many Fourier observations corrupted byGaussian noise. The proof combines a local cellular-automaton retraction with the classicalProuhet–Thue–Morse product identity