Analysis of the Baseline's Effect on Policy Gradient Expectation

In a policy gradient algorithm, a common technique to stabilize learning is to subtract a calculated value from the total reward of each trajectory before computing the update. This is done to reduce the variability of the updates without altering their expected direction. Which of the following calculated values, if subtracted from the total reward, would introduce an incorrect bias and potentially lead the policy updates in the wrong direction on average?

In the mathematical proof demonstrating that a state-dependent baseline b(s_t) does not introduce bias to the policy gradient estimate, the expected value of the baseline-related term, E[ (∇θ log πθ(a_t|s_t)) * b(s_t) ], evaluates to zero. Which of the following is the fundamental reason for this outcome?