Complete chains for fourth powers of complex cubics

Let $q$ be a monic complex cubic with all its roots in a disk centered at $o\in\mathbb C$ of radius $R\ge0$. There exists a complete chain $z_1,\ldots,z_{12}$ for $q^4$ satisfying

The first point is the centroid of the three roots of $q$, counted with multiplicity. All final 4 entries, from row 9 through row 12, coincide at one root of $q$. There are unit complex numbers $a,b$ with $\operatorname{Re}(a\overline b)=0$ such that both sequences $\operatorname{Re}(az_m)$ and $\operatorname{Re}(bz_m)$ are nondecreasing. They describe one common orthonormal coordinate frame, allowing reflection. Translation changes these coordinates only by constant offsets.

The statement includes repeated roots, zero radius, and degenerate centered coefficients. It assumes no supplied prefix, continuation theorem or genericity condition. The complete Lean proof has been accepted by Prove2me’s verifier. This restricted class theorem does not settle either unrestricted derivative-chain question.