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: a hyperbolic 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 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 arithmetic and by a Bernstein certification of 28. 9% of the domain volume — 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), CH) 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, and we outline a kernel-only formalization path. 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 at doi: 10. 5281/zenodo. 21224267. v2: two local statement corrections following external 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.
Ludovic Tagnon (Sat,) studied this question.