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 sin²-algorithm; the complex case (signature (1, 1) ) remains open — Problem 4 of the companion Monatshefte paper, with the suggested analytic extension of the sin² formula. We study a deterministic algorithm implementing that suggestion: on (1, 1) data the score expression is strictly negative (we prove a closed form), the algorithm selects the most negative score, and the frequent, provably exact score ties are resolved by a declared ordering. We report three groups of results with a strict separation of statuses. First, on a fixed sample of 205 complex cubic polynomials, every run of the sign-normalized campaign (195 seeds directly, 10 via the mirror generator −α) closes projectively with an exact unit certificate λ = ζ·εᵏ in OK^×, and every transition is certified by exact score comparisons in Q (α): the computed dynamics is the exact algorithm's. An exhaustive campaign over the full box -3, 3³ (194 polynomials, no sampling freedom) closes 194/194 under the same certification. Second, across 457 deformed starting bases the terminal cycle is always the same, by exact state-level comparison — an observed invariant of the marked lattice (p, Zα), demonstrably not of the field alone — and measured descent diagnostics quantify the reduction behaviour a proof would need. Third, for four fields we certify finite transition graphs; for the plastic field the full transition map is decided by exact comparisons in Q (α). The admissible set of bounded height is not confined to the certified window (checked at a +10 widening, plastic field) and shows no sign of saturating. The general periodicity problem remains open. All data and scripts ship in a public archive with a portable verifier. v2: editorial and statement corrections following two rounds of externaladversarial review — related work expanded, the strict-real-ordering lemmastated and proved, closure-instance exactness statuses made uniform, thepre-period indexing convention restated to match the recorded values, HAPDtie-halts restated exactly, statistical indicators labeled exploratory. Themachine-checked plastic transition graph (Lean 4, kernel-only) is nowreported and replayable from the companion archive (v9). Every change islisted in the paper's Version history section. No principal result orcomputed datum of v1 is modified.