Refining the Gaia DR3 Parallax Zero-point: A Hybrid Approach Combining Global Parametric Correction with Local Refinement

The Gaia Data Release 3 (GDR3) parallaxes are affected by a complex bias that depends on stellar magnitude, color, and celestial position, with amplitudes reaching tens of microarcseconds ($\mu$as). Standard global parametric models (e.g., Lindegren et al. 2021, hereafter L21) effectively remove large-scale trends but struggle to resolve small-scale spatial systematics due to functional rigidity. We aim to construct a flexible, data-driven calibration map that eliminates these residual local systematics without imposing rigid functional forms. We propose a "Global Pre-correction + Local Refinement" hybrid strategy. First, we utilize the L21 model as a baseline to remove the dominant magnitude and color-dependent biases. Second, we model the residual zero-point using a Local Non-parametric method based on a Sliding Window technique. This approach fits local trends using k-nearest neighbors from quasars (for faint stars, G>18) and wide binaries combined with Large Magellanic Cloud (LMC) (for bright stars, G < 18). Our hybrid model demonstrates significant improvements over the standard L21 solution. Validation against different samples reveals a remarkably flat residual map with near-zero bias across the full sky. Our mathematical attempt at calibrating the parallax zero-point is expected to provide a useful reference for the zero-point correction in future Gaia DR4, and to help move towards a physical resolution of this issue.