Analytic constraints in m_c–a_c space for three initial semimajor axes a_b = 0.3, 1, and 5 au. Light gray shading with no overlaid, colored hatch pattern indicates the observationally (2σ) and dynamically forbidden region. The solid black line delineates the observationally forbidden region. Each color corresponds to a different outer companion eccentricity: e_c = 0.1 (dark purple), e_c = 0.5 (violet), e_c = 0.7 (pink), and e_c = 0.8 (orange). The colored solid lines delineate where ﹩{t}_{\mathrm{GR}}/{t}_{\mathrm{quad}}=1﹩, above which ZLK precession dominates over GR precession. The dashed curves indicate Hill stability boundaries with a separation criterion Δ = 5, below which the system remains dynamically stable. Colored regions represent the parameter space where the outer perturber satisfies all three criteria: (1) is dynamically stable (below the dashed curve), (2) ZLK precession dominates over GR analytically (above the solid line), and (3) is observationally allowed.