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

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. Version 2 (2 August 2026): 80-page corrected edition. The Lean layer is migrated to the corrected Lean 4.32.2 kernel and fully rebuilt; the concrete instantiation is complete (tlCompact : TLCompact accepted with the three standard axioms); the companion software record 10.5281/zenodo.21447485 is published alongside. All self-references now use concept DOIs. Preprint SHA256: 8196f501025bf5e7b7a3b83a732a5fcbdb83f2d7f56c0776216529189dbe0709Sources SHA256: a5b1a0091017bf16ff2ceae1159613e66675de29bf27ea27906e6d94b4fbe181