What question did this study set out to answer?

This research addresses the per-field periodicity of the complex sin²-algorithm related to cubic fields and orders.

July 23, 2026Open Access

Closing the per-field reduction: collar contraction, the stretched road, and a certified minimax for the complex sin²-algorithm

Key Points

This research addresses the per-field periodicity of the complex sin²-algorithm related to cubic fields and orders.
Developed a periodicity theorem applicable to cubic fields of signature (1,1).
Utilized constants γ=51/50, m₀=10, and κ=1 to study the properties of the greedy orbit.
Included the use of rational certificates and machine-checked software for verification.
Demonstrated that the greedy orbit of admissible initial states is projectively periodic.
Closed the direct half of Karpenkov's periodicity problem specific to the complex case of orders.
Continued work is required to extend findings from orders to arbitrary invariant lattices.

Abstract

We prove a per-field periodicity theorem for the complex sin²-algorithm: for every cubic field K of signature (1,1) and every order of K, with constants γ=51/50, m₀=10, κ=1 independent of the field and state, the selected greedy orbit of every admissible initial state is eventually projectively periodic up to the stabilizer unit. This closes, per field and for orders, the direct half of the complex case of Karpenkov's periodicity problem; the extension from orders to arbitrary invariant lattices remains open (ongoing work by the author) and is stated precisely in the paper. Decision-bearing inequalities are carried by exact rational certificates (archived and replayed fail-closed in the companion dataset record) or by closed-form derivations recorded in the text and adversarially audited. The companion dataset record contains revision v4 of the reproducibility archive; a machine-checked Lean 4 layer will be released as a further companion software record of the same series. Preprint SHA256: 33a0651f07322b10d6c156665b0fa47fe90423be11c1389895611e3c9a563a56 Sources SHA256: 058cf82e1c4aede1919427c9e11e46a1228c32129f1ce9e540fd19d72a8b48a3

Ask AI

Helpful

Bookmark

View Full Paper