CAUSAL REVERSAL AND IDENTIFICATION UNDER COEFFICIENT MODULATION

A coefficient that changes with the volatility state can reverse the direction preferred by causal discovery methods built for a fixed mechanism. We certify this reversal in a matched two-variable model: changing only the coefficient's state dependence makes both residual mutual information and the deployed DirectLiNGAM score prefer the wrong arrow. On a ten-variable benchmark, DirectLiNGAM, ICA-LiNGAM, VARLiNGAM and a constant-coefficient heteroskedastic-SEM score all fall below chance as positive modulation grows. Conditioning on the state does not settle the direction by itself: without a restriction on how the coefficient moves, a reverse model reproduces every state-conditional covariance. We then show how the same variation can identify a graph. When coefficients and relative noise variances are log-affine in an observed state, sinks have the smallest signed curvature of the log precision diagonal. This criterion recovers the DAG from three state-conditional covariances, provided coefficient growth does not exactly offset relative noise growth. We give a recursive estimator, HetCD-$\Omega$, with an exact-state recovery guarantee and a matching fourth-order lower bound near this balance condition. On the same benchmark, a fixed-window implementation reaches $0.90$ order accuracy under strong modulation of either sign, compared with $0.32$ and $0.54$ for DirectLiNGAM. These experiments also expose the limits of the repair: fixed windows lie outside the guarantee, one-signed modulation creates a variance-sorting shortcut, and an outcome-derived state proxy substantially reduces accuracy. A daily financial panel provides a consistency check only. The results specify when coefficient variation reverses a pooled criterion, when it identifies structure, and what state information that identification requires.