We study correction from finite observations when the returned analytic response must have a finite construction from polynomial encounters P_g(x) = x exp(g(x)) and their normalized inverse germs. For a Lipschitz exact-response map on a closed subset of R^n, whose values have histories of at most L letters with uniformly bounded polynomial coefficients, we construct an ambient Lipschitz correction with at most (2n+1)L letters. No uniform degree bound is required. Coefficient bounds and any supplied degree bound are retained, and explicit common disks support every forward primitive and the reversed inverse word. A positive two-linear model then separates recovery, correction and representation. Compatible observations determine whole responses Lipschitz-continuously, but correction confined to that family has sharp local exponent 1/3. A two-term convex sum restores Lipschitz correction. We classify all complex affine mixtures of the two orders of two linear inverse letters: except for endpoint weights or coincident histories, their analytic inverses are transcendental over the original encounter field. A separate three-letter correction retains both a Lipschitz response and an admitted inverse on prescribed disks. It has a continuous six-slot implementation whose eight coefficients are 1/3-Hölder. No exact selection in any fixed finite signed polynomial-word format has a better uniform pairwise coefficient exponent, even with cancelling auxiliary labels. Nevertheless, independent coefficient errors produce linear response and inverse errors. Finally, equality-respecting C^(2,alpha) parameter tests on the model admit transport exponent (2+alpha)/3, optimal uniformly over that class for 0 < alpha <= 1, including Lipschitz transport at alpha = 1. Native one-letter outputs and inverses realize sharpness. A separate isometric recalibration of primitive labels has attained sharp exponent 3/4 for the whole response and inverse. These results distinguish stable correction, executable representation and the stability of later operations.
This paper belongs to the Encounter Arithmetic programme, which studies mathematical structure accessible from analytic responses under specified operations. It distinguishes stable recovery from compatible finite observations, correction of incompatible observations, continuous finite representations, and stability under subsequent operations.
Founding programme paper cited in this manuscript: Encounter Fields: Exponentials, Lambert-Type Inverses, and an Exact Algebraic Boundary (version 2.0).