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Question on the gate of infinity
It says at the beginning of the librarian's movement.

Would this be allowed?
A unit moves into coherency with the librian joining him. The libby then teleports this unit across the field.

I would say no. It can be thought so, but the fact is that models do not have "phases", only players playing the game do.

So, I believe that the line "beginning of the librarian's movement" is basically saying in the beginning of the owner of the librarian's movement.

This is all IMO of course, and I can see and sympathize with how you read it.

I don't have my SM dex on me, so I could have some of the exact wording wrong. I do believe it says 'in the beginning of the librarian's movement' as you wrote.

Based on BGB, an Independent character joins a unit if he is within 2" of it at the *end* of his movement.

So no, he can't join it and then teleport because the teleportation occurs before moving, and joining occurs after.

He could, however, teleport to within 2" of a squad and then join it.

See: Independent Characters Joining & Leaving Units, p.48.

"In order to join a unit, an independent character simply has to move so that he is within the 2" coherency distance of a friendly unit at the end of their Movement phase."

And:

"Note that after a character joins a unit, that unit may move no further than Movement phase."

So unless the Librarian starts the Movement phase joined to the unit, then he cannot join it until the end of the Movement phase.

Seeing as the Gate of Infinity is used at the beginning of the Librarian's Movement phase, and under the reasonable assumption that what happens at the end of the relevant Movement phase happens after what happens at the beginning of the relevant Movement phase, I would say:

"No, a unit cannot join a Librarian, after which the Librarian uses the Gate of Infinity to move across the field"

It's something to note that the joining/leaving relationship is one-way, in other words joining and leaving units is something Independent Characters do with unit, not something that units can do with Independent Characters.

So, the first part of this hypothetical situation is already false in two ways, since units cannot join Independent Characters (although Independent Characters can join units), and any joining occurs at the end of the Movement phase.

That the second part is false follows trivially, since it's past the time when the Librarian can use the Gate of Infinity, and he hasn't joined the unit.