Learning Generated Controls under Fractured Geometry: Projective Residualization and Variation-Allocation Frontiers

Many two-stage estimators assess the first-stage learner by prediction error, even when the next stage uses its residual. In control-function instrumental variables, that residual must preserve the latent control direction without removing the treatment variation that identifies the structural response. A scalar prediction score does not reveal how the learner allocates this variation. Under piecewise-smooth graph geometry, interpolation can suppress the control, whereas isotropic smoothing can leak systematic variation across boundaries. We formulate this as a variation-allocation problem and introduce Adaptive Anisotropic Instrumental Heat Flow (A-IHF). The method uses pilot treatment contrasts to adapt edge conductance, takes the complement of a sparse graph resolvent as the generated control, and selects candidates without consulting outcomes. For a linear control-function regression, the generated control is identified only by its span. Working in that projective geometry, we derive an exact finite-sample fidelity--relevance frontier, spectral identities for remaining treatment variation and coefficient distortion, and a lower bound for monotone fixed-graph residual filters. A connected construction proves that adapting conductance can remove the corresponding fixed-graph obstruction. In a 54-cell benchmark, the A-IHF family wins 32 cells; its guarded observational variant lowers mean nonlinear response error by 8.3%, with the largest gains in fractured designs. Controlled rewiring explains when the graph should be used, replaced by a fallback, or rejected. The resulting lesson is task-specific: a first stage for generated controls should be judged by control fidelity, downstream relevance, and graph compatibility together.