Feasibility Ceilings Before Learning: Sizing and Timing Bounds for Drone-Mesh Interception

Abstract

A mesh of interceptors defending a fixed site must spread over the predicted arrival point of an approaching object. This is a recurring task in multi-agent robotics: N agents, each covering a disc of radius r, spread over a probability density so that a point drawn from it falls within some agent's disc. We give closed forms for the quantities that bound performance before any policy is chosen, and validate each by Monte Carlo. The uncertainty of a point predicted by extrapolating a noisy linear track follows from least-squares variance propagation plus a delta-method term for the uncertain arrival time, agreeing with simulation to within 1.8%. The coverage probability of a packed mesh admits a closed form accurate to 0.007 absolute, with a volumetric analogue. A naive area ratio misses a ceiling: while agents stay at least 2r apart, no mesh exceeds 90.69% coverage in the plane or 74.05% in a volume. This ceiling belongs to the separation constraint, not to the problem - permitting overlap substitutes Kershner's thinnest-covering density and removes it, the better regime above a crossover at p=0.80. Hexagonal packing is optimal among five arrangements; the centroidal-Voronoi configuration coverage control supplies reaches only half its coverage at small budgets. The Gaussian assumption is conservative under heavy tails and anisotropy but fragile under bias: a one-sigma aim-point offset costs 14 points of coverage. A repositioning team must also choose a commit time, and the admissible window can be empty. Against the applicable ceiling, analytic placement attains 85% of what the configuration permits while centralised PPO averages 15% over three trainer seeds, with a paired bootstrap against a null policy containing zero at every seed. We argue such ceilings should be computed before any learning experiment and quoted alongside every result.