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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.

Programme origins and related work. This investigation grew out of the cover-function studies in The Geometry of the Critical Line programme. The scaled constraint Cα(z) = z − e−α/z = 0 reduces under w = −α/z to wew = −α, as developed in Paper 9, “The Analytic Vortex and the Lambert Spectrum” (in the Papers 9–10 record). Its Newton dynamics are studied separately in “Dynamics of the Lambert W Newton Iteration”. The present article develops the local algebraic questions suggested by this setting, studying fields generated by exponentials and normalized Lambert-type inverses. The cited programme papers document the motivation and related developments; the algebraic results are established independently here.

Version 2.0. adds an expanded introduction to the Encounter Arithmetic programme, updated citation presentation, clearer appendix headings and revised typography. The abstract, theorem statements and proofs are unchanged from version 1.0.

Continuation paper. A continuation paper, Escaping endpoints and controlled extensions of encounter fields, is in preparation. It studies finite admission at regular endpoints of continued expressions and controlled extensions of the fields developed here. Reserved DOI: https://doi.org/10.5281/zenodo.23453384. Manuscript in preparation; DOI reserved.