Dynamics of the Lambert W Newton Iteration: Basin Geometry, Routing Transitions, and Finite-Rank Period-Three Resonance

We study the global dynamics of Newton's method for the Lambert equation. In the coordinate u = −w, Newton's method for w exp(w) = −α is Nα(u) = (u² − α exp(u))/(u − 1), and inversion u = α/z conjugates it exactly to the Newton map of Cα(z) = z − exp(−α/z). The closed form gives a fixed pole and critical skeleton, a critical-to-pole relation at 3/e², and a fold at 1/e. Square-root rescaling at the fold yields quadratic Newton models whose two period-three cycles persist with multipliers tending to 8. A certified cycle census and frozen Ulam–Galerkin matrices separate exact dynamics from finite observation. At the pre-onset anchors 0.400 and 0.405, the exact cycle is fully resident and its principal cell transitions are closed, but length-five walks keep its shadow inside a period-one component; finite-rank onset is an isolation event at those anchors. Branch A exits the base window at 0.461085, while finite cells remain cyclic briefly. Moving the window shifts collapse to [0.476, 0.478] with the same cycle shadow. Although the preregistered scalar lag envelope is missed, an exact last-column return calculation accounts for both collapse brackets. We also certify non-root attracting period-three rows that capture the free critical orbit and derive a pole-ejection return law for million-step transients. The later traffic floor remains at 0.481. All operator claims concern specified finite matrices; no continuum resonance is asserted. This record contains three files: the final article PDF; a supplementary ZIP containing the XeLaTeX source, all article figures, final metadata, and the five-page Supplementary Numerical Tables and Controls document; and an integrated reproducibility tar containing article-supporting code, selected raw finite-rank matrices, exact-cycle and operator-certification artifacts, parameter-sweep summaries, figure sources, audit records, and SHA-256 manifests. The finite-rank raw layer is a selected compact evidence layer rather than a complete copy of all 189 campaign run archives; the included master manifest content-addresses the larger frozen campaign. Programme consolidation and supersession scope (22 September 2026). This article is Paper A12 in the publication plan. It consolidates the Newton-dynamics, routing, conjugation-parity, and finite-rank period-three research developed in Papers 20–29, 31–33 and RN1–RN6. For the results stated in this article, it supersedes the corresponding earlier programme presentation and is the current consolidated reference. Paper 33 contributes conjugation-parity material; its other uses remain separate. Papers 34–35 are linked as boundary references only and are not superseded as whole works. The separate direct-iteration/escape-time project A3 is outside this consolidation. The source records and files remain available for provenance. All operator claims retain the finite-matrix scope stated in the abstract.