Translation-Equivariant Disturbance Recovery in Delayed Open Priority Networks

Abstract

We study recovery of relative three-dimensional geometry without an absolute restoring force on the formation translation. A least-squares translation removes the mean target error. For equal-gain double integrators, the resulting common mode is a damped integrator and the relative modes are stable second-order systems. A causal realization uses complete same-time origin snapshots, a shared commit clock and a current local residual innovation. Its estimation errors sum to zero exactly, although individual estimates are stale. A sufficient small-age condition proves boundedness and recovery for the coupled sampled feedback under actual communication contracts, including directional and state-dependent delays. An actual delayed priority process supplies convergent inputs under bounded use ages and joint connectivity of consumed influence. Finite cohort changes retain surviving physical and estimator states and preserve departed bodies under a separate law. Shared anchor tables cancel from terminal relative shape; fixed local profile tables yield an explicit priority-dependent shape and a nonconstancy criterion. Finite-variation velocity impulses, conditional active-body tubes and an exact nominal event map are treated separately. Finite checks illustrate the formulas. No rigid-body implementation, continuous trajectory safety or novelty claim is inferred from them.