= Solution
A sufficient causal condition is <conditional exchangeability>
$$
\{Y(0),Y(1)\}\mathrel\perp A\mid X,
$$
together with <consistency of potential outcomes> and <positivity in causal inference>. For the ordinary-least-squares coefficient itself to equal one common causal effect, also require the correctly specified additive conditional-mean model
$$
\mathbb E[Y(a)\mid X]=\theta_0+\theta^TX+\beta a.
$$
Then $\beta$ is the <homogeneous treatment effect>, and the adjusted coefficient consistently estimates both conditional effects and the <average treatment effect> in the analyzed population.
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