Escaping endpoints and controlled extensions of encounter fields

We study fields of convergent Puiseux germs generated by rational functions, holomorphic exponentials and selected finite polynomial-encounter inverses. Every field generated by a full finite primitive prefix has algebraically independent exponential coordinates with normalized charges in that same field. For a classical analytic Lambert remainder, we prove that its holomorphic endpoint is admitted at every nearby nonzero source but lies outside the old field at the endpoint. Guarded matched shifts admit this germ while preserving the exclusion of nonrational source-algebraic germs and logarithms of nonconstant rational units. Every monomial shift reduces constructively to linear shifts, with its exact endpoint branch retained, even under arbitrary finite nesting. For a mixed quadratic label and rational input functions, each selected output is either rational, on an explicit exceptional family, or transcendental over the entire linear shifted field. These results give a strict field chain from the old language through linear shifts to matched polynomial shifts. We classify the quadratic exceptional inputs and endpoint branches, and show why the rational-input hypothesis cannot be replaced by unrestricted generated inputs. This is the second paper of the Encounter Arithmetic programme. It continues Encounter fields: exponentials, Lambert-type inverses, and an exact algebraic boundary, version 2.0, 10.5281/zenodo.23453449. The arguments use the unconditional formal power-series theorem of Ax. The paper specifies its finite construction rules, branch data and local guards, and includes complete proofs and quantitative estimates.