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Is crossed random-effect the only choice?

Hello Allo,

In my two-level data structure, I have a cluster-level variable (called
"X"; one that doesn't vary in any cluster). If I intend to generalize
beyond X's current possible levels, then, I should take X as a random
effect.

However, because "X" doesn't vary in any cluster, therefore, such a random
effect necessarily must be a crossed random effect (e.g., "~ 1 | X"),
correct?

If yes, then what is "X" crossed with?

Thank you,
Jack