The move-only converse and the ω-infinite island of the complex sin²-algorithm

We formulate the move-only converse for the complex sin²-algorithm — that an eventually periodic raw move word forces the input, up to positive scale, to be an embedded basis of an order in a cubic field — and we refute it: each branch of the family constructed here inhabits the ω-infinite island, has eventually periodic moves (period three) with an aperiodic sorting word, and no positive rescaling of its input lies in a cubic field. The refutation is computer-assisted: a written proof resting on finite exact computations, deposited and replayed in the archive; the construction has not been executed in the parameter domain for which it is stated. Conversely, we prove the augmented-digit converse: eventually periodic digits (moves together with their sorting permutations) make a nondegenerate Q-linearly independent input, up to positive scale, a Q-basis of a cubic field, with signature (1,1) exactly when the period module does not escape. Whether periodic digits force signature (1,1) remains open. The move-only converse is our own formulation; it is not part of Karpenkov's Problem 4. This is the converse-half paper of the closing pair of the complex sin² programme. Paper I (the deterministic algorithm and certified campaigns): https://doi.org/10.5281/zenodo.21222497 . Paper II (structure theory: exact identities, height descent, finiteness): https://doi.org/10.5281/zenodo.21224268 . The direct-half companion (per-field periodicity): https://doi.org/10.5281/zenodo.21447478 (concept DOI, as cited in the paper; version 3: https://doi.org/10.5281/zenodo.22285098 ). Version 1, dated 27 September 2026; 228 pages. The four-page statement document deposited with this record states, in closed notation, what the paper proves and with what status, and lists what is not claimed. Statuses in the paper have fixed meanings: "proved" means a complete written proof in the text, never a formal seal; "computer-assisted" means a written argument resting on finite computations whose deposit and replay status is stated alongside it; "replayed" means an execution recorded in the deposited archive, not a mechanized proof; no statement carries a Lean kernel seal. The death of the alternating witness gate (Theorem 3.13) is computer-assisted: its certificate programs are deposited and replayed from the archive, and maximal steering of the deep witness is exactly 36. The 40 finite calculations supporting the construction are indexed in replay/CONSTRUCTION_CALCULATIONS.tsv. Also included: a local theory of the native disc −31 cycle, including capture of eventually move-periodic greedy tails permanently confined to the radius-2⁻²⁶ chart (Propositions 3.125–3.128), quasiconformality, Birkhoff contraction and fibred obstruction tools, and 13-word A- and 31-word D-codes giving m_non > (251/250)m_site on periodic fully confined seven-return gates (Theorem 3.35). Companion record: reproducibility archive https://doi.org/10.5281/zenodo.23057746 ; this PDF pins it by its printed SHA-256 e46412e3f3a10bbb61cc39448f4d9f4164f4d19375ae587e32aab7a31d59e6d2. The archive holds the manuscript sources without the printed archive SHA-256; a PDF recompiled from those sources therefore lacks that printed fingerprint and has a different SHA-256 from the deposited PDF. The register records 135 executions: 76 with return code 0 and 59 with a nonzero return code (54 blocked, 5 path); 0 control failures. It covers 133 distinct program paths: 76 have a recorded return code 0 and 57 have none. The Python inventory contains 152 files: 137 in the main archive and 15 in the supplement; 19 are library modules without an autonomous entry point. The five recorded path outcomes are distinct from the two successful mapped executions. verify_extension and analyze_long_returns stop because the configured verify_uniform_seven_returns.py location does not resolve; neither has a successful mapped execution. external_records__bc_len20_exact_certificate reaches the output write after its mathematical checks, then stops because external_records/ does not exist; its separate declared-mapping execution returns 0. Here , denoted WORKSPACE_TOKEN in MISSING.md, is the unresolved workspace-location token in a configured path, not a supplied directory or an assembly field. external_records__extract_uniform_ratio_D stops because the configured external_records/verify_gloc2_D.py does not resolve; its separate declared-mapping execution returns 0. manuscript__figs__make_certification_summary stops because /etc/matplotlibrc is outside the declared isolated mounts; it has no successful mapped execution. replay/NUMBERS.tsv compares manuscript quantities with recorded outputs: 263 equal, 0 different and 90 not produced. “Not produced” means that a quantity is absent from the recorded output; it does not mean disagreement. The table identifies the extraction rule, manuscript location and comparison target. State-fingerprint comparisons are metadata comparisons, not mathematical results. The replay establishes neither the written proofs nor their adequacy to the stated claims, and supplies no formal seal. 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: 3e2ccc1b8b3ba3b6c94f0606cf3e2b5d136e59f26a9e0fb93d348ede67ab4003 Sources SHA256: cd16e8515a114efa004c72d3fe4f1ff2226ceb8baef0eb9055ef8e6a70e41d7b Statement document SHA256: c88a463865e50bfbec4d69f51e9506319000917c647e93f4e963a096687ef656