A statistical learning framework for explainable forest disturbance-agent attribution with provable guarantees and theoretical foundations for spatial transfer

We develop the mathematical foundations of a framework that attributes a disturbance agent, namely wildfire, harvest, insect defoliation, windthrow, or drought, to each disturbance event detected in standardized satellite time-series change metrics. The attribution task is posed as a statistical decision problem on a feature space built from spectral-temporal descriptors, object geometry, terrain, and climate. Within this formulation we prove the Bayes optimality of the posterior-maximizing rule and of its cost-sensitive counterpart, a finite-sample generalization bound for the empirical-risk minimizer over a finite class, the variance-reduction identity that governs ensemble estimators together with a bias-variance decomposition, an ordering between the Gini and entropy impurities, the convexity and second-order optimality conditions of gradient-boosted trees, the gradient identities of the softmax and focal objectives, the efficiency, symmetry, and null-player properties of the Shapley attribution used for explanation, the order preservation of temperature scaling, and the distribution-free lower and upper coverage bounds of the marginal and class-conditional conformal predictors used for uncertainty. We further prove an unbiasedness and variance result for the design-based accuracy estimator and a decomposition of the cross-region transfer gap, together with an estimable form of that decomposition. Many of these results are specializations of established statistical-learning theory to the attribution setting; the contribution lies in their unified formulation on a standardized metric product, in the explicit statement of the assumptions under which each guarantee holds, and in a small number of new corollaries. The paper is theoretical: all results are established by proof and illustrated with synthetic numerical examples, no empirical forest-disturbance dataset is analysed, and the empirical transferability, calibration, and accuracy of an implemented system remain to be evaluated.