She wrote her phone number on a Kit Kat bar wrapper , which Lily accidentally gave away to a trick-or-treater later that night.
While the "long paper" isn't a singular named artifact in the episode like the "Hanging Chad" costume, it represents the specific, nerdy details Ted obsessed over for four years while waiting for her to return to the rooftop party. Key Context from the Episode
In the How I Met Your Mother episode , "long paper" likely refers to the Tootsie Roll
drink recipe or the that the Slutty Pumpkin (Naomi) and Ted bonded over during their first meeting in 2001.