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

TL;DR

The paper develops a preparation-invariant analytic framework that reduces local microlensing image formation near caustics to a single HC series via a geode variable $m$ satisfying $m=U\varphi(m)$. It provides explicit steps (Weierstrass preparation, cyclotomic products, square-free reduction, Lagrange normalization) and yields HC coefficients, a certified radius $\rho_U$, and a majorant to bound truncation error, enabling stable seeds and Newton polishing to recover exact images within the certified domain. The framework is demonstrated on multiple examples (folds and cusps in artificial and physical multi-lens settings), including a decic cusp with a resonant unit and triple-lens cusps, showing machine-precision reconstruction and continuous continuation across catastrophes. By introducing HC signature and HC spectrum as preparation-invariant moduli, the work provides a degree-agnostic, analytic taxonomy that supports differentiable photometric and astrometric modeling and gradient-based inference, with clear paths toward real-time application in surveys like OGLE and Roman.

Abstract

We give a preparation-invariant analytic description of image formation near microlensing caustics. After a local Weierstrass preparation at any multiple image (order $d\ge2$), the lens mapping reduces to a single geode variable $m$ satisfying $m=U\,\varphi(m)$, where $U$ is a prepared source coordinate and $\varphi$ 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.

Figures (3)

• Figure 1: Binary‑lens geometry at $\epsilon=0.25$, $s=2$. (a) Caustics with a fold‑crossing track $\zeta(t)=0.30-0.05(t-1)i$ (red) and a cusp‑passage track $\zeta(t)=t$ (green). (b–c) Per‑image magnifications along each track. (d–e) Predictor and post‑Newton errors versus base step $\Delta t_0$ for HC truncations $K\in\{1,2,4\}$, showing the expected HC scaling and a quadratic reduction after one Newton step.
• Figure 2: Root trajectories for three polynomial systems as the parameter $\zeta$ (or $u=\zeta-\zeta_{*}$) varies. (a) Quintic fold given by Eq. \ref{['eq:P_fold_start']}. Black stars mark the roots at $\zeta\approx0$. Solid coloured curves (blue/red) show geode approximations valid up to $\zeta=0.6$. Black dashed lines indicate numerically exact roots extending to $\zeta=0.8$. Yellow–green circles mark the convergence radius $\rho_{\zeta}\approx0.271$. (b) Binary–lens cusp considered in Section \ref{['sec:binary-cusp']} emanating from $(z_{*},\zeta_{*})$. Three image branches (coloured solid: geode; black dashed: exact) are tracked from the cusp singularity as the parameter $u$ increases until $|U(u)|$ reaches the certified radius $\widehat{\rho}_{U}$. The black filled circle marks the cusp point. Inset: zoomed view of the local structure near the cusp with the third image trajectory too short to be visible on the main plot. (c) Decic polynomial (given by Eq. \ref{['eq:decic-P']}). Black stars correspond to all ten roots at $\zeta\approx0$. Coloured solid curves show the HC series combined with Newton polishing for the triple image. Black dashed lines show the exact roots trajectories emanating from the cusp. Cyan circles mark the safe radius ($0.95\,\rho_{U}$) where continuation begins, while yellow–green circles denote the convergence radius $\rho_{U}$. HC seeds from the unit–aware kernel $\varphi(m) = (1 + m^7)^{-1}$ are used only for $|U| < 0.95\,\rho_U$ with $\rho_U \simeq 0.651$; beyond this limit a predictor–corrector continuation uses the previous polished roots as seeds. All three panels demonstrate geode–based root tracking with high accuracy within their certified radii of convergence.
• Figure 3: Caustic structure and image trajectories for two triple–lens configurations.(a,b) Source–plane caustics for a triple lens with mass fractions $\varepsilon=(\tfrac{1}{2},\tfrac{3}{10},\tfrac{1}{5})$ at positions $s=(0,1,1+3i)$(a) and $s=(0,1,\tfrac{1+i}{2})$(b). The analysed cusp is marked by the red dot; the red dashed segment indicates the initial source direction (too short to be visible in (b)). Inset in (a): image trajectories (Im $z$ vs. Re $z$) for the short track $\zeta=\zeta_{*}-(2+3i)u$, $u\!\in\![0.001,0.1]$, reconstructed by the local geode construction: evaluate the HC series $m(U)$ from the preparation–invariant kernel $\varphi$, lift per branch $t=\omega\,m^{1/3}$ (with anisotropic seeds), and apply a short Newton polish on the full eliminated decic $P(z;\zeta, \bar{\zeta})=0$. Coloured solid curves: geode+Newton; black dashed curves: exact roots of $P(z;\zeta,\bar{\zeta})=0$. (c) Near–tangential cusp passage for the configuration in (b): image trajectories as the source moves along the real axis from $u=5\times10^{-4}$ to $0.033$. The same HC geode $\rightarrow$ lift $\rightarrow$ Newton construction recovers all three branches with machine–precision residuals; the apparent “jump’’ in one branch is a projection effect from crossing a nearby fold: continuation in the geode variable remains single-valued and smooth (Sec. \ref{['sec:triple-cusp']}).