Uniform derivation of the color-0 congruence split modulo 12

Derive a uniform explanation for the color-0 split by m mod 12 in the Route E analysis of D3(m)=Cay((Z_m)^3,{e1,e2,e3}), showing that it arises as a cyclic reordering within a single residue-4 splice framework rather than as separate congruence cases.

Background

In the even-modulus Route E analysis, the color-0 first-return dynamics exhibit a case distinction depending on m mod 12 when describing the splice of arithmetic family-blocks.

The author asks whether this dependence can be derived uniformly by interpreting the observed cases as cyclic reorderings of a single residue-4 splice picture, eliminating the need for separate treatments.