Basin Geometry, Routing Transitions, and Finite-Rank Operator Resonance in a Transcendental Newton Family

We study the Newton dynamics of C_a(z) = z - exp(-a/z) on the punctured complex plane and its inversion representation G_a(u) = a - u exp(-u). The change u = a/z exactly conjugates their Newton maps and pushes planar area forward with density a^2 |u|^(-4). The distinguished Lambert roots u_0 = -W_0(-a) and u_(-1) = -W_(-1)(-a) collide at the fold a* = 1/e, where conjugation changes from fixing two real roots to exchanging a nonreal pair. We combine the resulting fold and stopped-transport theorems with numerical observers of basin geometry and routing. A six-panel convergence-time atlas visualizes the lifecycle, while three crossing observers and a conditional-kernel factorization quantify central transport. Finite-rank Ulam-Galerkin estimators exhibit a bounded period-three spectral epoch: a leading transient triplet with phases 0 and plus-minus 120 degrees opens, organizes, and collapses within explicit parameter brackets before the later traffic floor. The epoch brackets persist across the tested realizations and absorber rules, with resolution robustness on the tested grids; the birth edge and complete tower depth are realization- and geometry-dependent, the organization statement is tied to the matched 80x80 construction, and under the tested domain stretch the live collapse endpoint was examined. All operator statements carry exact finite-matrix certificates and no continuum transfer-operator claim is made. On exact-conjugate evaluations, crossing presence and first-crossing step obey conjugation symmetry for all tested observers; terminal-label statements retain their observer and catalogue scope. This record contains the article PDF, the LaTeX source and submission metadata (supplementary zip), and the integrated reproducibility deposit: all code, frozen numerical artifacts, figure sources, claim ledger, and SHA-256 manifests supporting every reported result.