A Probabilistic Guidance Approach to Swarm-to-Swarm Engagement Problem

The paper’s Lemma 1 and Theorem 1 bound the number of red agents that can enter the base by projecting red’s first-hit distribution on a chosen boundary ring and subtracting the blue distribution at the first-contact time, with a TV-distance simplification when the swarm sizes are equal. The candidate solution reproduces the same construction: an absorbing modification on the boundary ring to obtain the first-hit distribution, a per-bin elimination lower bound min{R_i,B_i}, and the sum-of-positive-parts identity yielding N_r^{ent} ≤ L̂_r, plus the TV-distance corollary. Both rely on the same key assumptions (red convergence, no pre–t_fc eliminations, blue lingering on the boundary after t_fc), and differ only in presentation: the paper states the “identity policy” (blue freezes after t_fc) explicitly, whereas the candidate solution states the persistence informally. Substantively, the arguments match and are correct under the stated assumptions.