Kinematics of the Sol–Soror Binary: The Gravitational Bond and the Twin Sister's Cycle

 We present a theoretical astrophysics study of a hypothetical, distant binary companion
 to the Sun—Soror (Latin for “sister”)—and perform a detailed kinematic analysis of the
 putative Sol–Soror system. Our focus is twofold: (1) to assess how a distant yet extant
 stellar-mass or sub-stellar companion could remain gravitationally bound to the Sun on Gyr
 timescales while remaining elusive observationally, and (2) to quantify the characteristic
 cyclic orbital motion in which the two objects revolve about their barycenter, here dubbed the
 “Twin Sister’s Cycle”, and how such motion modulates long-term perturbations to the Solar
 System. We combine analytic perturbation theory and high-precision N-body integrations
 to derive constraints on orbital elements (semi-major axis, eccentricity, inclination) and to
 evaluate the companion’s long-term gravitational influence on the outer reservoirs of small
 bodies (Oort Cloud dynamics and comet injection rates) and consequences for planetary
 stability and astrobiology. Using sensitivity estimates from modern astrometric surveys (e.g.,
 Gaia) and infrared all-sky searches (e.g., WISE/Spitzer), we place observational bounds on
 the allowed parameter space for Soror. We propose specific, testable signatures—statistical
 anisotropies in long-period comet aphelia, subtle coherent accelerations detectable in precise
 ephemerides, and distant-object astrometric residuals—that could either falsify or support
 the existence of a distant solar sibling. The analysis demonstrates that a distant companion
 can be both gravitationally extant (not dynamically ejected) and engaged in a slow, secular
 cycle with the Sun while producing discrete observational consequences that contemporary
 and near-future surveys can constrain.