Exact random covers of metric trees: balanced rounding, duality, and sharp thresholds

What the math gives to ML

The paper gives an exact, constructive way to convert a fractional distribution of tree-metric balls into a random finite cover without exceeding a prescribed radius budget. The transferable asset is not graph burning itself, but the combination of budget-preserving trimming, balanced rounding, and a max-over-local-balls dual certificate. This suggests a tree-structured attention or graph-neural receptive-field mechanism whose stochastic neighborhoods cover all nodes while controlling expected locality cost. The dual inequality also provides a falsifiable coverage diagnostic and a surrogate regularizer for learning the radius distribution.