The Persistent Starter clause can be one of several clauses that are based on the implied importance of a player's starter Pokemon.
Version 1: Catch-limitingEdit
Also called a Monolocke, this version of the persistent Starter clause disallows any catches throughout the game - aside from making the occasional catch to have HM slaves or plot-necessary legendaries. Often, Nuzlockers will program in a starter Pokemon different from one of the three defaults to make application of this version more varied.
Using only one Pokemon throughout the entire game increases the difficulty and also the bond with one particular Pokemon, that being the starter, or the player's "partner Pokemon".
Version 2: FailureEdit
This version of the persistent Starter clause states that if a player's Starter faints/dies, the game is instantly over. (Note that, while this version can be applied to Nuzlockes by itself, it is also implicit in any application of version 1 of the clause.)
Because of the strength of the bond between a trainer and their starter, loss of the starter is thought to be so emotionally devastating that the trainer would quit trying.
Version 3: Team-inclusionEdit
The version of the persistent Starter clause forces the Starter to be on a player's team at all times during the game - it cannot be boxed.
The strength of the bond/friendship/partnership between the trainer and their starter is strong/respected enough that separating the two by boxing the starter would be betrayal.