Assuming that you’re referring to “pulling information out of your ass”, you could try: “Pulling facts out of thin air” or “making it up on the go”. Getting more creative with it: “unscripted intellect”, “on the spot expertise”.

A real gentleman would probably say "conjecture" or "unfounded information", but a less explicit word is "nonsense"

Somebody is trying to send a spicey work email lmao

Rectally sourced figure

Malarkey Nonsense Fabrication

An "ad hoc solution"?

Wingging it.

Pulled out of thin air

deus ex machina

Amateur hour

i'm just spitballing here.. i'm thinking perhaps.. maybe.. does this resonate.. have you considered.. from another perspective.. possibly.. informed opinion.. its likely.. in the ballpark of.. something like..

Cut from whole cloth

Hexbear argument.

Colonjecture.

Made up on the spot.

"A proctologist was needed to extract that information from its source."

“Pulled An Araki”

Alternative logic

"He's just pulling these figures out of his hat."
"This step in the argument is a bit of a handwave."
"I'm not sure we can afford to just guess here; let's gather some real data."
"Pay no attention to the man behind the curtain." (PNAMBIC for short.)
"I'm concerned that the Emperor's latest tailors may be in the pay of the Nudist Association."

Deus ex Machina?

Goatse