We introduce the Abu-Ghuwaleh Orbit-Singularity Series (AGOSS), an adaptive singular expansion obtained by repeatedly extracting a dominant singular atom from the asymptotic ratio jet of the Taylor coefficients of a holomorphic germ at the origin. In this first paper we work in a deliberately rigid regime: one complex variable, finitely many algebraic-logarithmic singular atoms, and strict radial separation of singular locations. On this class we prove five results. First, the dominant ratio jet canonically identifies the dominant atom. Second, the greedy AGOSS peeling procedure recovers the entire finite singular skeleton in finitely many steps. Third, each successful extraction lifts the analytic radius of the residual exactly to the next singular radius. Fourth, truncating Taylor only after skeleton extraction yields a strictly better exponential approximation rate than direct Taylor truncation. Fifth, the recovered skeleton is stable under asymptotically negligible perturbations of the ratio jet, which yields an inverse orbit-to-skeleton principle. We also explain how a simple confluent atom decomposes algebraically into two basic atoms, thereby motivating the next paper in the series.
Mohammad Abu-Ghuwaleh (Wed,) studied this question.