Hidden Assumptions in Bell's Theorem?

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Nullstein said:
In the no-BSM case, QM predicts that there is no entanglement between 1&4 in the full ensemble
Yes.

Nullstein said:
and that there is an entangled subensemble.
No, QM does not predict this. Cherry picking "predicts" this, but I don't agree that's a valid method.
 
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Nullstein said:
This fact follows from the axioms of QM
We've already been over this multiple times. The math is one thing. Claims about what the math means and what math is relevant is another.