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

Abstract

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.