Signed Tidal Evolution of Retrograde Earth Satellites: Spin Reversal, Darwin Stability, Eccentricity and Obliquity

A negative orbital-angular-momentum reservoir can satisfy a terrestrial spin budget without ensuring a stable tidal endpoint. This paper examines the subsequent evolution of imposed retrograde Earth satellites within a reproducible, finite-mass, Earth-tide-only model. A signed circular constant-time-lag calculation is compared with an independently integrated constant-magnitude sign-torque control. Complete-eccentricity equilibrium-tide functions are then expressed through regular axial and transverse spin components, retaining the terrestrial orbital-frame feedback. Satellite spin inertia and satellite tidal dissipation are consistently omitted in the primary model; the distinct historical Gate-3B spin ledger is preserved as an archival comparison. For a reversed present-lunar-mass circular reference, the inertial spin-zero semimajor axis is 243236.402398 km. A modern-recession-normalized CTL model reaches this crossing after 2.368554 Gyr, whereas the Q-like control reaches it after 1.476810 Gyr. These times are conditional model results. The closed Earth-only Darwin boundary occurs at initial eccentricity 0.693441427 for the specified anti-aligned family. At eccentricity 0.70, axial reversal occurs after 18.415193 Myr and the adopted Roche stop follows after 20.131547 Myr. Oblique cases retain transverse spin at axial crossing. A fixed-tilt projection diagnostic is separated from an aligned apparent-solar-motion threshold and from a general observer-horizon calculation. Fifteen eccentricity cases, five obliquity cases, independent circular quadrature, archived-source comparisons and fourteen automated CPU checks support the numerical results. No collision-generated remnant, solar-dynamical survival, or reversal of the actual Earth is demonstrated.