A wizard in my party recently took Spell Sniper and chose Chill Touch.  The spell appears on his character sheet, but not in his spell book.  Is there a specific reason why the chosen cantrip would be displayed separately from every other spell in his spell book?

Quote from Cistern >>

A wizard in my party recently took Spell Sniper and chose Chill Touch.  The spell appears on his character sheet, but not in his spell book.  Is there a specific reason why the chosen cantrip would be displayed separately from every other spell in his spell book?

 Because that feat does not add a spell to a Wizard’s spell list, just their brain.

Thanks for the response.  I see you're in SE PA.  Me too!  Lol.  

Also, it makes sense now that I think about it, since he was a wizard choosing a wizard cantrip.  If he would have chosen a cantrip from a different class, then of course it wouldn't be added to his spell book.  

Technically speaking, even the “regular” known cantrips should not appear listed under “Spellbook” for Wizards. They appear there as a function of the software and interface, but RAW, Wizards do not need to write down their Cantrips, and Cantrips are not subject to the rules for copying spells into the Spellbook.

Spellbook

At 1st level, you have a spellbook containing six 1st-level wizard spells of your choice. Your spellbook is the repository of the wizard spells you know, except your cantrips, which are fixed in your mind.

https://www.dndbeyond.com/sources/basic-rules/classes#Wizard