>>81941284 (OP) Low IQ post
Correlations are not transitive.
Suppose you have three random variables: X, Y, and Z.
Let:
corr(X, Y) = 0.99 (very strong positive correlation)
corr(Y, Z) = 0.99 (also very strong)
But corr(X, Z) = 0
This is mathematically possible and has been shown in real examples.
Pearson correlation measures linear relationships between two variables, based on how much one tends to increase or decrease when the other does.
But correlation is strictly pairwise. That means:
The relationship between X and Y depends only on how X and Y vary together.
The relationship between Y and Z depends only on Y and Z.
There's no mathematical rule that connects corr(X, Y) and corr(Y, Z) in a way that determines corr(X, Z).
An example using 3D vectors.
Suppose:
X = (1, 0, 0)
Y = (1, 1, 0)
Z = (0, 1, 0)
Now compute cosine similarities, which are proportional to correlation (after standardization):
cos(X, Y) is about 0.707
cos(Y, Z) is about 0.707
cos(X, Z) = 0
This shows:
X and Y are strongly aligned
Y and Z are strongly aligned
But X and Z are completely orthogonal (no correlation at all)
This means: even with strong correlation between X and Y and Y and Z, X and Z can be unrelated.
There is no mathematical rule or inequality that forces corr(X, Z) to be large (or even nonzero) just because both corr(X, Y) and corr(Y, Z) are large. Therefore, correlation is not transitive.