A Unified Analytic Framework for Microlensing Caustics: Geode Solutions and Hyper--Catalan Signatures
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arXiv
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| Natura: | Preprint |
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2025
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| author | Berloff, Gleb Berloff, Natalia G. |
| author_facet | Berloff, Gleb Berloff, Natalia G. |
| contents | We give a preparation-invariant analytic description of image formation near microlensing caustics. After a local Weierstrass preparation at any multiple image (order $d\ge2$), 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. On a binary fold and cusp, an artificial decic with a resonant unit, and two triple-lens cusps of a challenging geometry, HC seeds plus a few Newton steps recover the exact images to machine precision within the certified domain and maintain continuity under continuation. The resulting single-series templates (with $(\mathrm{Sig}_R,ρ_U)$ metadata) are ready for photometric and astrometric modeling. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_15756 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Unified Analytic Framework for Microlensing Caustics: Geode Solutions and Hyper--Catalan Signatures Berloff, Gleb Berloff, Natalia G. Instrumentation and Methods for Astrophysics Astrophysics of Galaxies We give a preparation-invariant analytic description of image formation near microlensing caustics. After a local Weierstrass preparation at any multiple image (order $d\ge2$), 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. On a binary fold and cusp, an artificial decic with a resonant unit, and two triple-lens cusps of a challenging geometry, HC seeds plus a few Newton steps recover the exact images to machine precision within the certified domain and maintain continuity under continuation. The resulting single-series templates (with $(\mathrm{Sig}_R,ρ_U)$ metadata) are ready for photometric and astrometric modeling. |
| title | A Unified Analytic Framework for Microlensing Caustics: Geode Solutions and Hyper--Catalan Signatures |
| topic | Instrumentation and Methods for Astrophysics Astrophysics of Galaxies |
| url | https://arxiv.org/abs/2511.15756 |