A Non-Iterative Flux-Tracked Geometric Projection Layer for Non-Autonomous Restricted Four-Body Dynamics.

A non-iterative geometric correction architecture is formulated for non-autonomous restricted four-body dynamics. The central construction replaces the assumption of a timeindependent Jacobi integral by an explicitly propagated pseudo-Jacobi target generated by the temporal flux of the time-dependent effective potential. The numerical method is organized into two conceptually separable layers. Loop A tracks the moving pseudo-Jacobi target, while Loop B provides a phase correction. An optional scalar velocity projection can map a numerical state onto the instantaneous target manifold without nonlinear iteration. The method is evaluated through a hierarchy of independent numerical tests. First, the analytical temporal flux is compared with a symmetric finite-difference derivative. Second, the flux equation is integrated on an externally supplied reference trajectory in order to isolate quadrature error from state-predictor error. Third, the algebraic projection residual is evaluated independently. Autonomous CR3BP is used as a control problem, while the principal non-autonomous validation is performed for a Sun–Jupiter–Saturn restricted fourbody configuration. The CR4BP short-horizon refinement experiment (h ∈ {0.08, 0.04, 0.02, 0.01}) shows approximately first-order scaling of the complete flux-tracking residual while retaining approximately fourth-order trajectory convergence of the underlying RK4 state propagator. The flux-only formulation produces residuals from 2.75 × 10−6 to 3.49 × 10−7 on that grid. A separate long-horizon h-refinement at fixed T exhibits an approximately fourfold residual reduction when h is halved (empirical order near two on that pair of steps). In contrast, hard post-step velocity rescaling reaches the double-precision residual floor, but introduces a positional discrepancy of approximately 4.3 relative to the finest pure-RK4 reference and does not converge as a trajectory correction. An ablation experiment further shows that the hard scaling configuration produces a positional deviation approximately 11.4 times larger than the corresponding flux-plus-phase configuration without scaling. Over integration horizons up to T = 500, the flux-only pseudo-Jacobi residual remains bounded and exhibits no detectable secular growth in the reported experiments. Trajectory drift is controlled by retaining flux tracking and (when used) symmetric phase correction while disabling hard velocity scaling; optional soft or threshold projection may replace hard closure when a residual guard is required. These results demonstrate that invariant tracking accuracy and orbital trajectory fidelity are distinct numerical objectives and should not be conflated.