We are told that if we assume $$E(u|x) = 0$$, then the population regression function can be interpreted as $E(y|x) = a + bx$, where "$a$" and "$b$" are the population parameters. However I can construct a counterexample where this doesn't hold: $$\begin{array}{c|c|c|} & \text{x} & \text{y} \\ \hline \text{} & 1 & 12 \\ \hline \text{} & 2 & 14 \\ \hline \text{} & 3 & 16\\ \hline \text{} & 4 & 20\\ \hline \text{} & 5 & 25\\ \hline \text{} & 6 & 29\\ \hline \text{} & 7 & 31\\ \hline \text{} & 8 & 40\\ \hline \text{} & 9 & 20\\ \hline \end{array}$$

Suppose the above is a population dataset (I'm assuming I know exactly the population, there is no sample). $E(y|x) = y$, for instance $E(y|x=1) = 12$ since $x=1$ has only 1 $y$ value which is equal to 12. If this is true whats the population regression line?

A linear regression yields gives us $Y = 10.87 + 2.41x$. However, this does not satisfy the property of $E(u|x)=0$, clearly for $x=1$, the predicted value is not equal to 12, there is an error. So how is it that linear regression satisfies $E(u|x)$?

Does this mean $E(u|x) = 0$ doesn't hold in the population and is just an assumption?

Even under the assumption that $E(u|x)$ is true, with the population dataset above I cannot find a linear equation that satisfies this. What are the implications of this in real world examples when it doesn't seem to hold?

Edit: Background information: I have two variables of interest, $y$ and $x$ that are linearly related via the following: $y = a + bx + u$, where "$a$" and "$b$" are fixed parameters to solve for, and "$u$" is the error term and captures the fact that there are other factors that also affect $y$, which aren't captured by "$a+bx$" alone. In this dataset, there would be 9 errors terms $u$ for each of the 9 $(x,y)$ pairs below.

We are told in textbooks that if we assume $$E(u|x) = 0$$ then the population regression function can be interpreted as $E(y|x) = a + bx$, where "$a$" and "$b$" are the population parameters. However I can construct a counterexample where this doesn't hold: $$\begin{array}{c|c|c|} & \text{x} & \text{y} \\ \hline \text{} & 1 & 12 \\ \hline \text{} & 2 & 14 \\ \hline \text{} & 3 & 16\\ \hline \text{} & 4 & 20\\ \hline \text{} & 5 & 25\\ \hline \text{} & 6 & 29\\ \hline \text{} & 7 & 31\\ \hline \text{} & 8 & 40\\ \hline \text{} & 9 & 20\\ \hline \end{array}$$

Suppose the above is a population dataset (I'm assuming I know exactly the population, there is no sample). $E(y|x) = y$, for instance $E(y|x=1) = 12$ since $x=1$ has only 1 $y$ value which is equal to 12. If this is true whats the population regression line that relates "$y$" to "$x$" in a linear form $y=a+bx+u$ where $E(u|x)=0$ actually holds?

If I solve for "$a$" and "$b$" using a regression calculator gives us $Y = 10.87 + 2.41x$. However, this does not satisfy the property of $E(u|x)=0$, clearly for $x=1$, the predicted value is not equal to 12, there is an error. So how is it that linear regression satisfies $E(u|x)$?

Does this mean $E(u|x) = 0$ doesn't hold in the population and is just an assumption?

Even under the assumption that $E(u|x)$ is true, with the population dataset above I cannot find a linear equation that satisfies this. What are the implications of this in real world examples when it doesn't seem to hold?