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.
Ludovic Tagnon (Sat,) studied this question.