Exact Restoration and Historical Memory in Delayed Priority Networks with Three-Dimensional Motion

Abstract

We study restoration of probability-valued priorities after finite changes in network membership. For fixed uniform or unequal directional communication delays, the endpoint depends on the complete initialized history. We derive the exact returned endpoint after one admission, finite mixing and proportional departure. Priority restoration is an affine equation in the admitted conditional preference; restoration of a three-dimensional objective centre is its projection through the anchor matrix. A causal packet protocol preserves origin-dependent historical transport, uses reception ordinals to order simultaneous readiness, and yields bounded-age stochastic dynamics after finite membership changes. Under bounded joint connectivity and positive current self-weights, those dynamics converge with an explicit sufficient contraction bound. Restoration over unknown admissible futures is characterized by finitely many endpoint functionals; a finite polynomial characterization additionally requires positive-polynomial initial and bootstrap data. We prove historical survivor-query memory bounds in distinct smooth-state and deterministic finite-bit models. A reference service retaining unmixed rows provides eventual restoration under a modified update law. Finally, a double-integrator controller converts priority convergence into point rendezvous or separated formation convergence. A staged finite-pool construction parks retired bodies in reserved finite-depth berths, while preserving all-body separation and speed and acceleration bounds. All conclusions retain their stated information, timing, storage and geometric assumptions.