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) is not established here 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 5 of the manuscript, dated 28 October 2026; 92 pages. Version 5 (this release): Target 2.1 now states the admissibility convention of its witness, gives the adaptive lever of its stretched tail as a witness rather than as the algorithm's selection and attributes the discharge of that tail to the exact collar cover of Remark 2.6, Target 3.1 now states that it is established for genuine states through its replacement and that its split and candidate bounds are not proved here and are not used, Target 4.1 that its finite-ledger certificates are reported (not in the reproducibility archive) and not used, and the last sentence of Theorem 5.3 now applies when the certified witness passes the isotropy filter, in particular for genuine states; one sentence of the proof of Proposition 6.9 now places the factor 2 of δ = 2|x|A⊥ in the embedding determinant; no other theorem, proposition, lemma, corollary or target, no other part of a proof in the chain from Proposition 6.7 to Corollary 6.12, and no certificate changes. In Remarks 5.10 and 2.5, the following are corrected: the admissibility of W on the corner band, the monotonicity in S of a fixed move, the ρ → 0 tail, the forcing argument, and the residual list of Remark 5.10, from which the ridge bound, certified there, is removed. Statuses are made explicit: the unimodality of μ* (numerical), the residual items of Remark 5.10 (items of its analytic presentation), the bound 27/100 (reported), the filters passed by the witnesses of Theorem 5.3 (genuine states) and the Lean 4.30.0 object of Remark 7.1 (historical certification); the definition of μ* is corrected. Remark 6.17 no longer states that the archived replay certifies the general count; that replay is a finite regression. Remark 5.14 now states that no uniform one-step margin holds on the wall face, by the face state of Remark 2.2, whose failure statement is now restricted to strictly admissible nonidentity moves. The expansion statements of Remarks 3.3, 5.10 and 5.11 are likewise restricted to nonidentity moves (of C₈ in Remark 5.10), and the face states of Remark 5.11 are no longer described as recorded. Remark 6.5 now attributes the cover of the in-window road to the subdivision of Remark 3.3 alone. Remark 4.3 now describes its cell comparisons as reported and not in the reproducibility archive, and no longer describes the cells as archived; its closed form now invokes the unimodular invariance of |x|A⊥, not of the projected area; and it now says that the cells-to-states bridge and the p-tail are not settled by the recorded cells, rather than genuinely open. Appendix B now lists the further computations that are reported, not in the reproducibility archive and not used by the proof of record, and the reported Lean checks of Section 7 that are not in the companion software record. Remark 6.16 now attributes the restriction of the states to bases of O to this paper. The abstract and the status ledger now say that the lattice extension is not established here, rather than open, and Remark 6.13 now says, as the second paper does, that the index of the stabilizer multiplies the regulator in the finiteness proof. Remark 3.2 now states its quadratic gain in the conditional, and Remark 6.17 says that the pigeonhole no longer needs regulator-dependent unit-class counting. Remark 3.3 now restricts its emptiness statement to the integer edge q* = x₂/x₃, Remark 6.18 attributes interfaces (H3), (H5) and (H9) as Section 1 does, and Appendix A marks the limiting wall crest as reported and not used by the main theorem. Removed: the last two sentences of Remark 2.2, the limit value of V(0,1,0) on the ridge in Remark 5.10, the attribution of the fringe p < 30 to the class 30 ≤ a ≤ 60 in Remark 5.13, the attribution of the corner of Remark 5.8 to the second paper, the index factor in Remark 6.17 and in the status ledger, and the comparison with the T* statement in Section 6. Marked numerical: the cone supremum ≈ 0.13 in Remark 5.10 and the two tight contacts of μ* in Remark 2.5 and Figure 4. The Version note (version 5) on the first pages lists these items; the Version note of version 4 is kept below it. The Lean kit and the reproducibility archive are unchanged.

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: 9f43e226ca1e18e50868ce509aee58826e98a62db1aba322b84a247e8bd7ff02

Sources SHA256: f83dde0210fd5ba119acc9abb0cc9a2edd4134907674df3435c8c696cdf75101