Hyper–Catalan Single–Series Microlensing with Certified Radii and Kernels

We present a preparation-invariant and degree-agnostic framework for image formation in point-mass microlensing. By performing a local Weierstrass preparation around any multiple image of arbitrary multiplicity, we derive a single convergent power-series representation that unifies all classical singularities: folds, cusps, and higher catastrophes within the same analytic form. This "geode" expansion yields image positions, magnifications, and finite-source observables with certified convergence and explicit error bounds, eliminating the need to repeatedly solve the global lens polynomial. The method transforms the traditionally multivalued microlensing map into a single analytic branch whose coefficients obey closed Hyper-Catalan recursions, guaranteeing geometric convergence inside a rigorously defined domain. In this way, each local caustic behaviour, whether a fold pair or a cusp triplet, is captured by one ordinary power series whose derivatives with respect to physical parameters remain analytic, enabling stable, gradient-based inference. Applied to binary (quintic) and triple (decic) lenses, the method reproduces standard fold and cusp asymptotics while achieving substantial computational speed-ups and machine-precision accuracy across caustic crossings. Because only local invariants enter the construction, the approach extends directly to multi-point and multi-plane lens systems, providing certified, differentiable, and ready-to-fit analytic templates for both photometric and astrometric microlensing.