General periodicity of the complex cubic sin²-type algorithm

Prove that the deterministic sin²-type algorithm for complex cubic fields produces an eventually periodic digit sequence for every admissible trajectory, thereby establishing a general periodicity theorem for the non-totally-real case of Hermite’s problem.

Background

The paper studies a deterministic analytic extension of Karpenkov’s sin²-algorithm to cubic fields of signature (1,1), where the field has one real embedding and one pair of complex-conjugate embeddings. Extensive finite computations produce exact periodicity certificates for the tested polynomial samples and several finite transition graphs, but these computations do not establish periodicity for all complex cubic fields or all admissible starting states.

The unresolved issue is the general periodicity theorem originally posed as the complex case of Karpenkov’s Problem 4. The paper’s Conjecture T* formulates the remaining substantive claim as eventual periodicity of every admissible trajectory; the accompanying well-definedness clause is stated to have been settled separately.

References

The general periodicity problem remains open.

— A deterministic sin^2-type algorithm for complex cubic irrationalities with exact periodicity certificates  (2608.23281 - Tagnon, 24 Aug 2026) in Abstract; Section 1; Section 8.1, Section 8.2 (Conjecture T*)

We state the phenomenon as a conjecture. What the data supports is invariance for the algorithm as fixed here --- polynomial p, lattice \mathbb{Z}[\alpha], sorting and tie-breaking conventions included:

— A deterministic sin^2-type algorithm for complex cubic irrationalities with exact periodicity certificates  (2608.23281 - Tagnon, 24 Aug 2026) in Section 4.4, subsection “The conjecture”; Conjecture T*, Section 8.2

Unit-exponent spread is the best single structural discriminant of reachable versus out-window admissible states (AUC 0.95), though no exact invariant characterizing reachability was found; we state its identification as an open problem in \S8.

— A deterministic sin^2-type algorithm for complex cubic irrationalities with exact periodicity certificates  (2608.23281 - Tagnon, 24 Aug 2026) in Section 6.4; Section 8.2, OP3

OP8, sharpened: prove or refute that the terminal cycle depends only on the isomorphism class of (\mathbb{Z}[\alpha],\mathrm{conventions}).

— A deterministic sin^2-type algorithm for complex cubic irrationalities with exact periodicity certificates  (2608.23281 - Tagnon, 24 Aug 2026) in Section 4.4, Conjecture C*; Section 8.2, OP2 and OP8

Which pairs of unimodular moves are score-equal on which states, and why?

— A deterministic sin^2-type algorithm for complex cubic irrationalities with exact periodicity certificates  (2608.23281 - Tagnon, 24 Aug 2026) in Section 2.3; Section 8.2, OP5