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We study finite constructions of actual convergent meromorphic Puiseux germs using field and algebraic operations, regular exponentials and logarithms, fixed rational-pair inverses, and arbitrary finite ordinary unary reversion. Membership in this language is equivalent to containment, together with the original source, in a finite tree of source and one-charge fields joined along nonconstant actual intersections. The converse constructs the required source alignments by branch-selected local reversion. We give an explicit family of nonsingular exponential systems whose solutions are jointly algebraically independent over the unary language in the original source and have no simultaneous finite construction in any compatible scalar source. One scalar implicit solve retaining the source independently constructs the family. A second family shows that a rational nonlinear coupling can destroy finite constructibility while preserving the abstract marked multiplicative output field and its uniquely determined unordered factors. Its members can agree with a constructible control through any specified finite Taylor order. The obstructions quantify over all permitted finite covering roots, including roots containing auxiliary fields.

This is the fourth paper of the Encounter Arithmetic programme, which studies local responses, composition, numerical readout and the operations needed to obtain or recover mathematical structure.

Related programme papers:

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).