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Also, I thought it flakey, that Jack would care if Walt thought Jesse was his partner. He was going to kill him anyway.

Hubris. Kind of the same thing that let Walt deviate from his plan to net only $700K that became his empire-building $90M. Jack had too much ego to even let a man die thinking he was a lying dick. It was Jack's way of saying "say my name" except Walt ended up putting a bullet in his head instead. And Walt never thought twice about the hidden money Jack was bargaining for his life.