Projective Graph Residualization: Variation-Allocation Frontiers for Control-Function IV
Control-function instrumental-variable estimators pass an estimated first-stage residual to an outcome model. The residual must retain the latent control direction while leaving enough treatment variation to identify the structural effect. These demands conflict when the systematic signal is locally smooth but discontinuous across unknown feature-graph boundaries: interpolation can erase the residual, whereas isotropic smoothing can move systematic variation across a boundary. We propose Adaptive Anisotropic Instrumental Heat Flow (A-IHF), which adapts edge conductance from pilot treatment contrasts, extracts a generated control with a complementary sparse resolvent, and tunes without outcomes. For a linear control-function regression, only the generated control's span matters. This projective view yields an exact finite-sample fidelity--relevance frontier, spectral identities for residualized treatment variation and coefficient distortion, and a lower bound for monotone residual filters. A connected construction shows that conductance adaptation can escape the corresponding fixed-graph obstruction. Across 54 benchmark cells, the A-IHF family wins 32 and guarded observational A-IHF reduces mean nonlinear response error by 8.3 percent, with gains concentrated in fractured designs. Exact distortion correlates 0.998 with realized linear coefficient error. Controlled graph rewiring further shows that connectivity alone is inadequate: an outcome-free compatibility screen triggers foldwise fallback and, under severe corruption, abstention.
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