A deterministic sin2-type algorithm for complex cubic irrationalities with exact periodicity certificates

Hermite asked in 1848 for a representation of real numbers whose eventual periodicity characterizes cubic irrationals. The totally real case was solved by Karpenkov's $\sin^2$-algorithm; the complex case, signature (1,1), is his Problem 4. We study a deterministic algorithm implementing his suggested analytic extension: the score expression is strictly negative on (1,1) data (closed form proved), the most negative score is selected, and exact score ties are resolved by a declared ordering. On a sample of 205 complex cubic polynomials, every run closes projectively with an exact unit certificate, each transition certified by exact comparisons in $\mathbb{Q}(α)$. An exhaustive campaign over the full box $[-3,3]^3$ closes 194/194. Across 457 deformed bases, the terminal cycle is an invariant of the marked lattice. Certified finite transition graphs are computed for four fields; the plastic case is machine-checked in Lean 4, kernel-only. All data ship in a public archive with a portable verifier.