Every Interaction Has a Price: An Exact Calculus for Tree Ensembles

A tree ensemble on binary inputs, gradient-boosted or otherwise, is exactly a finite sum of AND-rules whose coefficients are readable from the leaves alone, in closed form, with no data. We develop this identity into a calculus with three operations. Read the largest coefficients at supports too large to enumerate, under certified monotone bounds. Price the removal of any family of rules in closed form, before any edit. Cut exactly the chosen rules by appending a few small trees: nothing else changes, to floating point, and the result is a tree ensemble. Removal policies over any feature set form a lattice with exactly modular prices: one pass prices the policy space, the efficient frontier is a minimum cut, infeasibility is certifiable. A single coefficient decides whether to drop a feature or only cut its interactions, a verdict no summary of its Shapley attribution's distribution can supply. There is a wall: mediation, confounding, distribution shift, and learner bias leave no trace in (fitted model, data distribution); it appears five times, each paired with the sharpest survivor. Every theorem ships with a verified implementation, every number to script and log. Every interaction has a price: computed, paid, or proven unpayable.