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

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 = 10⁶, 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 (Corollary 6.12). 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 (equivalently, to all real cubic vectors) remains open and is stated precisely in the assembly section. The proof assembles nine named interfaces; the analytic inputs left open by the first two papers of the program — the collar decomposition, the stretched u₂-road, the aligned regime, and the compact minimax (R) — are discharged by closed-form arguments and exact rational certificates. For genuine aligned states, the one-step proof is Appendix D at m ≥ 10⁶, without an alignment guard; its accompanying data are identified in Appendix B. For (R), a structural three-parameter proof is primary. Its deposited certificate is the exact GLRED4 quotient corpus, with 11,678 leaves; the 1,282,733-leaf exhaustive paving remains an independent alternative. Lean 4 has checked the 48,636 targets of the quotient corpus, proves the parameterized assembly theorem Atlas → TLCompact, and certifies the assembled endpoint: the concrete fourteen-certificate Atlas is constructed and the theorem tlCompact : TLCompact type-checks in Lean 4.32.2 with axiom dependencies exactly [propext, Classical.choice, Quot.sound]. No external re-verification is claimed (see Section 7). Here TLCompact is the compact structural statement, one owner of the thirteen, and not the per-field periodicity theorem: the nine interfaces, the thirteen-owner proposition and the passage to genuine states are not formalized and remain paper-level. No theorem of the paper depends on formal verification. The converse is outside the scope of this paper. This is Paper III of a series. Paper I (the deterministic algorithm and certified campaigns): https://doi.org/10.5281/zenodo.21222497 , arXiv:2608.23281. Paper II (structure theory: exact identities, height descent, finiteness): https://doi.org/10.5281/zenodo.21224268 , arXiv:2608.24750. Version 4 of the manuscript, dated 8 October 2026; 90 pages. The proof of Proposition 6.9 was incomplete in version 3 on the aligned branch. The contraction statement (C₁) is weakened by raising its threshold from 10 to 10⁶; the conclusion of eventual periodicity is unchanged. Appendix D provides the aligned proof; the threshold is not optimized. The Version note on the first pages lists each corrected statement with its section (fourteen items). The procedure narrative has been removed; the mathematics is unchanged except as listed there. Companion records. Reproducibility archive (this PDF pins version 7 by its printed SHA-256 1418d8de0eb534b4b4fb21a2bce8734f46bf198bddad213a485eae49ba733593): https://doi.org/10.5281/zenodo.21447482 Lean kit: https://doi.org/10.5281/zenodo.21447484 Earlier versions remain accessible. This work was assisted by AI. No independent mathematician has reviewed it to date. Mechanical verification does not address whether the statements express the intended mathematics. The author alone is responsible for the content. Preprint SHA256: 563908e5a3fed616295558bda6f208483edc07a35b458548404881c1187b4461 Sources SHA256: 6343d4287174bbd3186572690a3f9189bc6060ff3595494718bc6730cf3a1ef4