Bell state is not a join

In plain terms

Here is the problem. "Entanglement means the whole is more than its parts" is repeated so often it has stopped meaning anything. Can it be made into a theorem with actual content -- a precise inventory of what the whole contains that the parts do not?

On the arena picture, yes. Combine two systems and there are two candidate constructions: the mere pairing of the two arenas -- every joint configuration is just a left point plus a right point -- and the genuine composite arena the theory demands. The theorem: the Bell state lives in the composite and is provably NOT any pairing. No left point and right point jointly produce it.

That is entanglement as an address: the strict gap between product and composite, with the most famous quantum state exhibited inside the gap.

In CSD

The composite arena is mode concatenation, with the join of two component points given by the Segre construction -- the arena-level tensor product. bell_not_join is the arena-side signature of tensor-versus-Cartesian: the composite strictly exceeds the image of the join map, and the Bell ray witnesses it.

The follow-up question -- what the excess DOES -- has its own answer: the reduced state of the Bell ray is the maximally mixed ensemble on the correlated patterns, so entanglement is exactly what turns a local context's pure-state weights into mixed-state weights, while the remote pattern labels drop out entirely.

Mathematically

bell_not_join: the Bell ray over distinct correlated configuration pairs is not in the image of the arena join. Companions: the join factorises densities as Kronecker products with exact marginal and local-tomography laws; the composite algebras are generated by the mode-local subalgebras with the reconstruction theorem consumed arena-natively; and the reduced density of the Bell ray is the equal mixture with local weights exactly one-half.