Encounter fields: exponentials, Lambert-type inverses, and an exact algebraic boundary

This article establishes local algebraic foundations for Encounter Arithmetic, a research programme on states, composition, and continuation. For each nonzero polynomial g with g(0) = 0, let J_g be the normalized local inverse at the origin of P_g(t) = t exp(g(t)). The paper studies the smallest field A of meromorphic germs containing C(t) and closed under exponentiation of holomorphic members and under J_g on holomorphic members vanishing at the origin. Every finite subset of A lies in a differential rational-function subfield inside A with algebraically independent exponential coordinates. Consequently, C(t) is relatively algebraically closed in A, equivalently algebraic degrees are preserved. For a rational map R regular at zero with R(0) = 0 and R′(0) ≠ 0, its local inverse h has degree [A(h):A] = deg R. Thus h belongs to A exactly when R is Möbius. Let K_F be the field of rational expressions in polynomial-encounter histories with labels over F. The paper classifies algebraicity over K_F for zero-based Lambert outputs W_0(s) of finite F-linear combinations and rational expressions in histories, with s holomorphic and vanishing at zero; algebraicity is equivalent to membership. In particular, W_0(t cosh t) belongs to A but is transcendental over the full complex-label history field. The inverse-only tower of repeated W_0 adjunction has purely transcendental layers, is strict at every depth, and is not closed under composition. The iterates of H(t) = W_0(t/(1-t)) witness exact successive depths and are algebraically independent over the history field, although every derivative readout is one. Polynomial translates give independent families sharing any prescribed zero-constant finite Taylor jet. A separate theorem proves transcendence over A for regular Lambert solutions at nonconstant rational inputs with nonzero constant term. The arguments use Ax's power-series theorem, not Schanuel's conjecture. The paper includes complete proofs and states the limits of its effective procedures.