I have recently got myself confused about CIA and wondered if somebody could perhaps help me disentangle my thoughts. Consider the regression
$Y_i = \alpha_0 + \Delta D_i + \beta X_i + \eta_i$
Where $X_i$ are controls/covariates and $D_i$ is the treatment dummy. The CIA says that $\eta_i \perp D_i |X_i$. If this holds the causal effect of the treatment can be estimated with $E[Y_{1i} - Y_{0i}|X_i] = \Delta$.
Is the CIA the same thing as saying $E[\eta_iD_i] = 0$? (considering $\eta_i$ should have mean zero because a constant is included in the regression?) Which then is the same thing as $Cov(\eta_i, D_i) = 0$? How is this related to $E[\eta_i|X_i] = 0$ Gauss-Markov assumption? Does this have anything to do with the exclusion restriction for IV which looks very similar?