Can two linear regression variables be perfectly correlated but not share a single causal chain ancestor? A causal chain lists event (or fact) $y$ with all its causal antecedents.
We make a model of the following form:
$$
y = \beta_0 + \beta_1x_1 + \beta_2x_2 + \epsilon
$$
$\hat\beta_1$ has a p-value <0.001 but the true $\beta_1 \ne 0$ because the true $x_1$ is perfectly correlated with the true $y$.
In a deterministic paradigm, is it possible for two true population variables to be perfectly correlated but share no causal chain or causal ancestor, i.e.,


*

*$x_1 \not\to ... y$ AND 

*$y \not\to ... x_1$ AND 

*$Z \not\to ... x_1, Z \not\to ... y$?

 A: I think perhaps you're mixing the concept of correlation with linear regression.  They are similar, but answer somewhat different questions.  Graphpad has a nice description here, for further reading.  @Juan offers a nice example, but here's something to build on it.
Correlation measures two variables' tendency to go up and down together (either in tandem for positive $r$, or in opposite for negative $r$).  This can imply some causal link, but the correlation coefficient alone cannot offer concrete evidence for that (or lack of evidence).  An amusing website I found while doing some quick Googling was this.  Beware thy cheese consumption.
The $P$-value you reference (presumably for $\beta_1$, not $x_1$) refers to the regression coefficient in the multiple regression.  Regression addresses the ability of $x_i$ to predict $y$.  The $P < 0.001$ implies it is very unlikely that the $\beta_1$ is equal to zero, and thus $x_1$ has some predictive power of $y$ given the specified model.  Even then though, this doesn't answer the question of causation, but rather the unit change in $y$ given a unit change in $x_1$.
Providing evidence for causation comes from experimental design, not from a statistical relationship.  I don't recall which statistician said it, but there is "no causation without manipulation."  If your experimental design has no basis to demonstrate causation nor solid theory to imply it, then the low $P$-value can't add evidence to the presumption of causation.  It can spur further experiments to find a causal chain, however!
EDIT: For a more technical read, see here.  See section 2 for the distinction between association and causation.  Regression is directly listed as an associational concept, not a causal concept.  In brief, the relevant concepts can be summed up like this from the paper, "An associational concept is any relationship that can be defined in terms of a joint distribution of observed variables, and a causal concept is any relationship that cannot be defined from the distribution alone."
A: Correlation is a tricky concept, since it denotes that two variables show dependence based merely on statistical concepts. Correlation does not care about the nature behind the variables, therefore you can create a variable X  = {1,2,3,4,5} and y = {2,4,6,8,10} (which I just made) , they don't share ancestors and have a correlation coefficient of 1 which is perfect. What I am trying to say, is that never trust just the correlation coefficient to judge relationship between variables.   
A: Yes, they can be $d$-connected by observed effects.  Since $x_2$ and $x_3$ are also observed, then for example:
$x_1 \rightarrow x_2 \leftarrow z \rightarrow x_3 \leftarrow y$
means that $x_1$ and $y$ can be correlated given $x_2, x_3$ observed despite neither being the cause of the other nor them sharing any common cause nor any common descendent.
For more information, read about $d$-separation.
