Actual rule reference for clock timing is 4-3-3:
ARTICLE 3. FAIR CATCH KICK DOWN. The game clock operator shall start the game clock for a fair-catch kick down when the ball is kicked.
And there's the other loophole that had time expired on the punt, the Pats would have been able to extend the game by an untimed down to fair-catch kick., 4-8-2(e):
ARTICLE 2. PERIOD EXTENDED. At the election of the opponent, a period may be extended for one untimed down, if any of the following occurs during a down during which time in the period expires:
[...]
(e) If a fair-catch is signaled and made, the receivers may choose to extend the period by a fair-catch kick down (10-2-4-a). If the first or third period is not so extended, the receivers may start the succeeding period by either a down from scrimmage or fair-catch kick (11-4-3).
Please note, that period extension is NFL only! Don't let your local high school team do that, as much as you would like to see the fair-catch kick.