A Unified Analytic Framework for Microlensing Caustics: Geode Solutions and Hyper-Catalan Signatures

Abstract We give a preparation-invariant analytic description of image formation near microlensing caustics. After a local Weierstrass preparation at any multiple image (order d ≥ 2), the lens mapping reduces to a single geode variable m satisfying m = U φ ( m ), where U is a prepared source coordinate and φ is an image-side kernel. The coefficients of m ( U ) obey closed hyper-Catalan (HC) recurrences, allowing termwise derivatives and truncation control from the characteristic system. We also use the same form for a short HC predictor–corrector: evaluate the series within its certified radius and apply a brief Newton polish near the boundary. We define an HC signature (first nonzero kernel coefficients) and an HC spectrum (branch points and analyticity radius ρ U ), which quantify sparsity, stiffness, and safe evaluation domains. The construction covers folds and cusps of any global degree. For an artificial fold, a physical binary-lens cusp, an artificial decic with a resonant unit, and two physical triple-lens cusps, HC seeds followed by a few Newton steps recover the exact images to machine precision within the certified domains and maintain continuity under continuation. The resulting single-series templates (with Sig R , and ρ U metadata) are ready for photometric and astrometric modeling.