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 = 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 (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 (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 now certifies the assembled endpoint: the concrete fourteen-manifest Atlas is constructed and the theorem tlCompact : TLCompact is accepted by the pinned Lean 4.32.2 kernel with axiom audit exactly [propext, Classical.choice, Quot.sound]. The successful, dated v4.30.0 adversarial replay by a second project instance on the same machine reproduced the internal audits; it has not been rerun under v4.32.2, and no independent 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 half of the problem is the subject of a separate paper in preparation by the author. 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. The present paper discharges the contraction hypothesis left open in Paper II and proves the per-field periodicity theorem for orders. Version 3 of the manuscript, dated 4 September 2026: typographical revision, two errata and wording corrections. Every change is listed in the paper's Version history section, each wording correction with its v2 text and its v3 text; the wording corrections change no statement, and no theorem, constant, figure content, table content or abstract sentence is modified. Erratum (one token): the marker emitted by the archived script problem4/regfree_state_count_certificate.py is REGFREE_FINITE_REGRESSION_OK; v2 printed UNIFQUANT2_OK, a name absent from the pinned archive. Erratum (one sentence): the Methods statement now excepts the analytic import listed in the owner table (namely the u_3 owner) and the companion-paper interfaces imported by the assembly, which are proved in the companion papers and are not replayed from this archive; v2 stated the replayability claim without this exception. Appendix B and the archive it pins (v5, SHA-256 3bd11bdf9a4858f7d63b16fb7649b2c11186faa661452ccfb89ae515ed19f175) are unchanged. The LaTeX sources v7 are the published sources v5 plus exactly these changes; sources v6 was the working copy shipped with archive v6. Companion records: reproducibility archive https://doi.org/10.5281/zenodo.21447482 (this PDF pins version 5 by its printed SHA-256; version 6 is the current release), Lean kit https://doi.org/10.5281/zenodo.21447484 (the paper cites version 3 by its version DOI https://doi.org/10.5281/zenodo.21447485 ). 75 pages. Preprint SHA256: 07df3a1fd6e4497cc71622cb13d967f9e4b2595a2a60a991b058aae41cd93386 Sources SHA256: 44a8f2547c5a6527c66a1e52062199144374fe05523d05f6b8581f449cbf7edd