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

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 ρU), 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 SigR and ρU metadata) are ready for photometric and astrometric modeling.