Identity Manifolds and Representation Spaces for Divergent Series - A Geometric Analytical Framework for Recursive Reconstruction of Infinite divergent Series

Identity Manifolds and Representation Spaces for Divergent Series - A Geometric Analytical Framework for Recursive Reconstruction of Infinite divergent Series Infinite series are traditionally classified by convergence behaviour: convergent series are assigned well-defined scalar values, while divergent series are treated as ill-defined or handled via external regularisation techniques. In this work, we develop a structural representation framework in which infinite series are not reduced primarily through summation, but instead interpreted as geometric objects generated by recursive analytic reconstruction. Within this viewpoint, a convergent series corresponds to a collapsed identity structure that reduces to a single point in representation space. Divergent series, in contrast, generate extended geometric structures composed of recursively generated identity layers. These layers arise from independent reconstruction steps and become comparable only through projection into a shared reference frame. The resulting projection mismatch produces structured deviations, which are interpreted as intrinsic geometric features rather than analytic artefacts. Divergence is therefore not treated as a failure of summation, but as the manifestation of persistent non-collapsing identity geometry. We show that both convergent and divergent behaviour admit a unified description in terms of reconstruction depth: convergence corresponds to finite closure of identity layers, while divergence corresponds to infinite continuation of non-collapsing generative structure. This leads to a representation-theoretic reinterpretation of infinite series in which scalar limits are secondary to the geometry induced by recursive identity generation. In this framework, infinite series are analysed through their induced identity manifolds, projection structure, and reciprocal invariant relations. This work refines and extends previous CEGS- and TSO-based constructions by making explicit the separation between reconstruction levels, projection-induced spinor emergence, and reciprocal identity structure. Furthermore, this work illustrates how the conceptual structure of CEGS4QM extends naturally to domains in which a deeper analytic and geometric representation framework is required.