Certificate-Carrying Distributed Model Predictive Control on Product Manifolds with $\mathrm{S}\mathrm{O}(3)$

This paper studies constraint certification in synchronous distributed model predictive control (DMPC) when neighboring predictions change between sampling instants. Before the parallel local solves, each agent communicates a shifted prediction and an announced update budget. A hard trajectory trust region makes that budget enforceable, while an edge-wise feasibility cap computed from the shifted packets keeps the fallback feasible without using any current optimizer output. Distance and relative-attitude constraints are tightened with explicit Lipschitz constants and two budget layers: one accounts for the simultaneous neighbor update and the other retains a checkable shift reserve. We prove hard pairwise constraint satisfaction and recursive feasibility under stated nominal-execution and terminal assumptions, give the additional residual caused by execution error, and derive a local practical value-decrease bound. A spacecraft formation example uses hard terminal and pairwise constraints, a geodesic relative- attitude constraint on $\text{SO}$, and reproducible terminal-set checks. Comparisons with fixed, trajectory-only, and windowed online margins show that the proposed budget reduces conservatism while preserving a positive shifted-feasibility margin.