We study cyclic codes over split two-branch finite local rings of the form Rℓ, m=Fpu, v/⟨uℓ, vm, uv⟩, ℓ, m≥2, whose radical filtration is governed by two independent nilpotent chains: u→u2→…→uℓ−1→0andv→v2→…→vm−1→0. For the structural part, we develop a residue–torsion framework in which a cyclic code is described by one residue layer together with ℓ−1u-torsion layers and m−1v-torsion layers over Fp. This yields divisibility constraints, a layered generator description, and an explicit cardinality formula in terms of the associated field cyclic codes. We then specialize to the split cube-zero ring R=R3, 3=Fpu, v/⟨u3, v3, uv⟩, a non-chain local ring of type (2, 2) with basis 1, u, v, u2, v2. For this ring, the general theory becomes a five-layer structure consisting of one residue layer, two first torsion layers, and two second torsion layers. Using an Fp-linear Gray map adapted to this split radical filtration, we show that, when gcd (n, p) =1, the Gray image is linearly equivalent to a direct sum of five cyclic codes over Fp, so the dimension is additive across the layers. The minimum distance, however, is not determined by this decomposition alone and requires separate analysis. When n=ps, we derive exact distance formulas by reducing the problem to the five associated repeated-root cyclic codes over Fp. For p=3 and n=9, we compute explicit examples whose Gray images are ternary codes of length 45, illustrating the theory and producing several optimal codes. These results give a structural and metric description of cyclic codes over the split local ring R3, 3 while placing its algebraic framework in the broader family Rℓ, m.
Saif et al. (Fri,) studied this question.