Wait, I’m confused, I thought in the books the very first wight Jon saw WAS past the wall at the time it resurrected and tried to kill Mormont? Jon also tried to see if he could get another wight while locking corpses in the cells during Dance. Wouldn’t that disprove that it’s the Wall stopping them from reanimating once they go south of it and that some other factor contributed to the hand Ser Alliser brought to King’s Landing rotting away?

It was past the Wall, so there’s clearly some overlap. But the animated hand sent with Alliser stopped moving and rotted, so whether it’s a field effect or a recency effect or what have you, at some point, the magic fades.

And we know it’s the Wall doing this because:

“Why didn’t he come with you?” Meera gestured toward Gilly and her babe. “They came with you, why not him? Why didn’t you bring him through this Black Gate too?”
“He … he can’t.”
“Why not?”
“The Wall. The Wall is more than just ice and stone, he said. There are spells woven into it … old ones, and strong. He cannot pass beyond the Wall.”

“You are wrong. I have dreamed of your Wall, Jon Snow. Great was the lore that raised it, and great the spells locked beneath its ice. We walk beneath one of the hinges of the world.” Melisandre gazed up at it, her breath a warm moist cloud in the air. “This is my place as it is yours, and soon enough you may have grave need of me. Do not refuse my friendship, Jon. I have seen you in the storm, hard-pressed, with enemies on every side. You have so many enemies. Shall I tell you their names?”