Let k = C(x) with T = x d/dx, and for a nonzero polynomial g in X C[X] let P_g(z) = z exp(g(z)). For a nonconstant centered holomorphic germ b, the output h = P_g(b) generates a differential field K = k<h>, and the logarithmic observation R = Th/h generates B = k<R>. We determine the part L intersect K of the input field L = k<b> that the output retains. In general L = B(b) and K = B(h); the retained field equals B when exp(g(b)) is transcendental over L, and is an explicit cyclic intermediate field B(h^q), q = [L(exp(g(b))):L], when it is algebraic. The loss is finite exactly when b is algebraic over B, equivalently when there is no k-derivation partial: L to L commuting with T and satisfying partial b = b/(1+b g'(b)). A rigidity lemma for labelled encounters classifies the rational solutions of the equation (1/z+G'(z))z' = a(1/t+g'(t)) relating two labelled roots: they are z = A t^n with a = n and G(A t^n) = n g(t), and for equal labels they are the label rotations z maps to zeta z, g(zeta X) = g(X). On fields generated by normalized inverse encounters with arbitrary polynomial labels we compute the group of differential embeddings, a finite group of permutations with scalar multipliers within scaling classes of labels (after rescaling, a wreath product of a cyclic group of roots of unity by a symmetric group on each class), and prove that every full input in at least two independent roots has finite Galois loss with retained field the fixed field of the observation stabilizer. For full inputs in these families, infinite loss occurs exactly when the output is a linear germ. For reduced encounter histories, composition with P_g loses an ordinary transcendence degree exactly when the outermost letter of the history is J_g; for any label e and b = P_e(x) the loss is exactly the gcd of the exponents of g. Explicit examples mark each boundary, including a finite loss of degree three that is not Galois. For ordinary Lambert families we record the complete description of differential intermediate fields, the classification of full inputs with proper loss under z exp(z), and the rotation law for labels of degree at least two. The input, observation, output and retained fields consist of whole germs, and equality in those fields is equality of germs. Auxiliary algebraic extensions are used in the proofs.
This paper forms part of Encounter Arithmetic, a programme on analytic response laws, their numerical readouts, and the structure retained under composition and continuation. It studies retained differential fields and their finite or infinite loss under polynomial encounter composition.
The deposit contains the paper PDF, self-contained LaTeX source, and a supporting archive of exact calculation scripts and preserved outputs. Computational checks supplement the proofs; they are not human peer review. Substantial AI assistance is disclosed in the paper.
Programme papers cited: Encounter Fields: Exponentials, Lambert-Type Inverses, and an Exact Algebraic Boundary (version 2.0); Escaping Endpoints and Controlled Extensions of Encounter Fields (version 1.0); Recovery and Finite Loss under Exponential Composition in Lambert Fields (version 1.0).