A dual reformulation of the complex sin2-algorithm: exact identities, descent, and finiteness

Version map: paper v2, reproducibility archive v3 (current release under concept DOI https://doi.org/10.5281/zenodo.21224266 ). The version DOI printed in this paper resolves to an earlier revision of the archive; the archive SHA-256 printed in the paper identifies revision v2, which remains available in the version history. The records of this series are versioned independently, so their numbers are not aligned. Series note (August 2026). This is Paper II of a series (Paper I: https://doi.org/10.5281/zenodo.21222497 ). Hypothesis (C_κ), to which this paper reduces per-field periodicity, has since been discharged in Paper III ( https://doi.org/10.5281/zenodo.21447478 ), which closes the one-gap collar, the remaining u₂-band subcases, the aligned stretched regime, and the compact minimax (R). The per-field theorem is now unconditional for orders. The abstract below is that of this paper as published, and states the results under the hypotheses as they stood in it; the series note above records what has been established since. We develop the structure theory of the deterministic sin²-type algorithm for complex cubic fields introduced in the companion paper, addressing the complex-signature case of Karpenkov's Problem 4. The selection rule is shown to be, exactly, the minimization of a conformal module: the hyperbolic cosine of the distance between the transverse complex structure of the state and the round point. All governing quantities are exact elements of the real embedding of the field and satisfy closed dual-type identities; in particular no isotropic candidate ever arises, and the transverse deviation lattice has exactly pinned covolume. We prove an unconditional soft-rebound lemma (the module can grow by at most the factor φ²=2.618… in one step), a finiteness theorem for states of bounded module and height at fixed coordinate discriminant, with explicit static constants, and a per-field periodicity theorem under two named hypotheses: (C_κ), contraction of the module in the high phase — partially reduced here to a fixed finite minimax over a five-parameter compact with rational objective, supported by sampled adversarial sweeps in exact or 40-digit arithmetic and by a guarded floating-point Bernstein pass screening 28.9% of the domain volume (almost entirely its out-of-scope branch; the guard-accepted descent part is 0.007%) — and (B), recurrence of bounded height — which we then prove under (C_κ) alone: a height-descent theorem (componentwise key-ratio dichotomy, backward comparison, integral floors) shows the height can never exceed max(H(s₀),C_H) with an explicit constant. The remaining program for per-field periodicity is reduced to (R) on the compact and to the proved stretched subcases, modulo the aligned stretched subcase, the remaining u₂-road band and subcases, and an open collar-contraction statement. All proved statements and certificates are finite and exact. A machine-checked core of the paper is sealed in Lean 4, kernel-only, under the standard axioms: the two-case analytic core of the height-descent theorem, the finiteness pigeonhole, the dual and conformal identity layer in embedding coordinates, and an abstract assembly theorem composing them through named interface hypotheses. The full reproducibility archive (Bernstein-enclosure certification pipeline, minimax sweeps, hostile numerical checks of the height-descent theorem, witness campaigns; three verification levels documented in its README and PROTOCOL) is available under its concept DOI, which resolves to the current release: https://doi.org/10.5281/zenodo.21224266 . The version DOI printed in this paper resolves to an earlier revision: https://doi.org/10.5281/zenodo.21224267 . v2: two local statement corrections following adversarial review (the hyperbolic-reading fixed-point clause; the corner-lemma orientation, with explicit counterexamples documented in the text), statement-level repairs (finiteness at fixed coordinate discriminant; per-orbit assembly; converse half with its hypotheses; boundary conventions of the compact), the Bernstein passes requalified as guarded floating-point screening with the exact volume decomposition stated, campaign figures restated from the shipped packages, figures redrawn, and the Methods statement now declares exactly which measurements ship as replayable packages, as historical data packages, or as development-log records. Every change is listed in the paper's Version history. The structural theorems of v1 are unchanged.