The Disagreement Law: A Covariance Decomposition of Architectural Disagreement in Molecular Property Prediction and When It Fails

The absolute predictive disagreement D(m) = |pred_A - pred_B| between two graph neural networks with different inductive biases is a strong empirical estimator of prediction error in molecular property prediction, outperforming Monte Carlo Dropout and deep ensembles on scaffold-split data (Merlini & Merlini, 2026). This paper asks why. Working in a controlled system where the ground-truth function f* is known, we show that disagreement obeys an exact decomposition, E[D²] = MSE_A + MSE_B - 2E[e_A e_B], where e_X = pred_X - f* is the signed error of each model. The entire out-of-distribution (OOD) behaviour of the signal including the documented failure of single-architecture uncertainty methods is governed by the error-covariance term. We verify the identity to machine precision, recover it from data without supervision using symbolic regression (loss 6 × 10⁻¹⁴, a 13-order-of-magnitude drop over simpler forms), and map its failure region: when the two models share an inductive bias the signal collapses, exactly as the decomposition predicts. We then confirm the identity on real chemistry by reloading published GIN+EGNN checkpoints on four kinase targets (EGFR, VEGFR2, JAK2, BRAF). Finally, we show that a single scalar derived from the law—the inter-architecture error correlation ρ_AB—predicts the rank ordering of disagreement utility across the four targets (Spearman = -0.62, p = 0.013, across six protein families). This converts an empirical observation ("BRAF yields the strongest signal") into a quantity that can be estimated a priori, and yields a concrete design rule: the second model should be chosen for bias orthogonality, not accuracy.