Self-Stress Is the Supply of Transfer: Predictability of held measurements in root sheaves

Whether a training design determines a withheld measurement is an estimability question, and the number of held directions it leaves undetermined satisfies an exact identity: it equals the number of held measurements minus the number of states of self-stress that adjoining them creates. The second quantity is the Maxwell–Calladine count of the rigidity literature (Maxwell 1864; Calladine 1978). That count splits in two, into self-stresses carried by the held rows alone and those binding held rows to training rows, and the second part is the predictable held-direction dimension. Adjoining held rows one at a time converts the same accounting into the count of individually predictable measurements, so the Maxwell object is the supply of predictable held measurements. We give the number four coinciding descriptions, one in each of cohomology, rank, estimability and mechanics, and an arithmetic identity relating them. In the classical block design, where every incidence carries the same indicator, the identity reduces to the textbook criterion that connectedness makes all treatment contrasts estimable. Once incidences carry nontrivial restriction maps, connectedness stops being sufficient; the replacement is a path-composition criterion, sound on any training graph and complete on forests, and off a forest the witnesses branch. We then ask which structural choices a design has. Block structure splits the count but buys nothing across blocks; exchangeable reveal contributes only its rate; free stalks make every held measurement unpredictable uniformly in the design; and what does buy predictability is an equality constraint crossing levels. The paper closes with a construction whose folds take both extreme values of the support index under one model, which separates support from tie strength.