Let L = C(x)(u1, …, ur), where r ≥ 2 and ui = W0(cix) are the ordinary holomorphic zero germs at distinct nonzero complex scales. We study the rational differential field generated by h = b exp(b), where b belongs to L, is centered and holomorphic, and generates all of L under differentiation. The logarithmic observation R = xh′/h generates a subfield B over which L is finite Galois. Its group consists exactly of the root permutations fixing R. We identify the retained field L ∩ C(x)⟨h⟩ with B and compute the residual degree. We also characterize algebraic exponential adjunctions and determine their exact degrees. For the fixed pair W0(x), W0(2x), proper loss occurs on an explicit signed-integer family with unique parameters and has degree two. For every permitted triple of scales, every full input preserves the original field. Recovery throughout means rational operations and differentiation of whole germs. The unique analytic inverse b = W0 ∘ h exists even when this differential field omits part of the input field.
This is the third paper of the Encounter Arithmetic programme. It continues Escaping endpoints and controlled extensions of encounter fields, version 1.0, 10.5281/zenodo.23453384. The programme's founding paper is Encounter fields: exponentials, Lambert-type inverses, and an exact algebraic boundary, version 2.0, 10.5281/zenodo.23453449.
The arguments use the ordinary differential theorem of Ax. The paper supplies the common germ and differential-field foundations, the finite-loss mechanism, the complete classification for the fixed pair above, and preservation for full inputs at arbitrary distinct nonzero triples of scales. Its derivative bounds concern whole local functions, not finitely many numerical measurements at a point.