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Admittedly, this could be clearer, and it got lost in final edits.
The starting amount of Integrity is 10, the minimum amount of Tokens that the game's requirements section says you'll need. 

Thank you so much!

If the game is run as multiple sessions with the same characters/plot, should the integrity renew each new session or do the 'when it's out, it's out' rules apply to each campaign?

I'd say When It's Out, It's Out applies per campaign/investigation.