It is of course the Collective Bargaining Agreement...
From Article 10. Section 1.
The player so designated may be one who would otherwise be a Restricted
Free Agent. Except as set forth in Section 2(a)(i) below, any Club that designates a Franchise Player shall be the only Club with which such Franchise Player may negotiate or sign a Player Contract during the period the player is so designated, notwithstanding the number of his Accrued Seasons. The period for Clubs to designate Franchise Players will begin on the twenty-second day preceding the first day of the new League Year and will end at 4:00pm New York time on the eighth day preceding the first day of the new League Year.
The Franchise Tag is essentially a one-year contract and is very strict and well defined within the CBA. If no other deal can be made prior to the deadline the amount and term of the tag takes effect. Since you can't extend a Franchise Tag there would not be any ability to alter the deal once signed.
From Article 10. Section 9.
A Club may designate a Franchise Player only during the periods and in the numbers specified in Section 1 above; otherwise, the Club’s right to such designation expires. However, a Club may designate a player to whom the Club has rights as a Franchise Player with respect to any first future League Year during the term of this Agreement for which such player is anticipated to be an Unrestricted Free Agent. For any such players, the Club shall be deemed on the first day of the first future League Year in which the designation takes effect to have automatically tendered the player a one year NFL Player Contract...
The NFL's website puts this in a less "legalese" way:
July 16: At 4:00 p.m., New York time, deadline for any club that designated a Franchise Player to sign such player to a multiyear contract or extension. After this date, the player may sign only a one-year contract with his prior club for the 2018 season, and such contract cannot be extended until after the club’s last regular season game.