Observational and Dynamical Constraints on an Unseen Outer Perturber in the GJ 436 Hot Neptune System

Hot Neptunes in the sub-Jovian desert offer unique insights into planetary system evolution, retaining signatures of dynamical processes that shaped their present-day architectures. Many of these planets exhibit polar orbits, yet the mechanisms responsible for these misalignments between the stellar spin axis and planet orbit normal remain under debate. GJ 436 b stands among the very few hot Neptunes with both a polar and an eccentric orbit, thereby preserving dynamical signatures that may have otherwise been erased by tidal circularization. We investigate the unusual orbital architecture of GJ 436, exploring von Zeipel-Lidov-Kozai migration induced by a distant companion as a mechanism to explain the present-day orbit of GJ 436 b. Using $\sim$20 years of archival radial velocity measurements and astrometric data from the Hipparcos-Gaia Catalog of Accelerations, we constrain a potential companion to $a_{c}<5.4$ AU for $m_{c}>0.05$ $M_{Jup}$ and $a_{c}<64$ AU for $m_{c}>24$ $M_{Jup}$ in the GJ 436 system at the $2\sigma$ confidence level, providing the most stringent constraints to date. We further perform three-body hierarchical secular simulations to determine which companion configurations could reproduce GJ 436 b's present-day orbit within the observationally constrained parameter space. Our dynamical modeling favors sub-Jovian masses on orbits with $a_\mathrm{c} \gtrsim 6.8$ AU, suggesting a substellar perturber. These observational and dynamical constraints can guide future companion searches and illuminate formation mechanisms for hot Neptune desert planets on polar orbits.