Approach of agents with restricted fuel tanks

The proof establishes a lower bound of $\Omega (D^2\sqrt{D})$ on the time of approach for two agents, first in the case of arbitrary start delays, and then extended to account for arbitrary orientations. The initial part uses algebraic arguments involving sets $Pi$ and $Qi$, defined based on walk sets related to geometric instances, and Lemma 4 to show that if the approach were guaranteed for all time-shifted instances, a contradiction arises regarding the required time extent of the walks. This is supported by an expectation argument involving Lemma 6 and Lemma 5 concerning the overlap ratio $ri(t)$. The second part addresses orientation-shifted instances, leveraging Lemma 7 to show that the limited set of geometric instance parameters $\mathcal{I}$ cannot be supported by a sufficiently small set of walk pairs. The final step uses a contradiction argument based on an assumed fast approach time and geometric inequalities derived from positional information to establish the lower bound.

The transition from the time-shifted case to the orientation-shifted case highlights a subtle but critical dependency: the information derived from geometric constraints (the $gi$ segments) is insufficient to guarantee approach unless significant time elapses, and this insufficiency is exacerbated when orientations are introduced. The structure of the proof suggests that the lower bound $\Omega(D^2\sqrt{D})$ is fundamentally constrained by the geometric necessity for agents to cover angular space relative to their bases ($2\pi/\sqrt{D}$ complexity) combined with the spatial separation $D$. The use of Lemma 6 and Lemma 5 in Theorem 6 implies that the interplay between timing (delay $t$) and walk configurations ($pi, qi$) is tightly coupled. When moving to orientation dependence, the analysis pivots from temporal overlap to angular constraints defined by $\phi(I{i,j})$ and $\gamma(I{i,j})$. The contradiction in Lemma 7 regarding disjointness of derived geometric segments reveals that the set of possible walk pairs cannot simultaneously encode all the required relational information specified by the orientation-shifted instances. This suggests that adding continuous rotational freedom does not fundamentally change the $\Omega(D^2\sqrt{D})$ barrier but requires a more complex, rotation-aware analysis to establish precise dependency on $D$. The implicit assumption in the final proof for Theorem 7 about fuel consumption and geometric positioning highlights how movement constraints interact with kinematic possibility.