The opponent you target doesn’t have to be the opponent who searched their library.

Casting a Trap by paying its alternative cost doesn’t change its mana cost or mana value. The only difference is the cost you actually pay.

Effects that increase or reduce the cost to cast a Trap will apply to whichever cost you chose to pay.

If a search effect is affected by Aven Mindcensor’s ability (which causes a player to search the top four cards of their library instead), that still counts as searching that library.

You may ignore a Trap’s alternative cost condition and simply cast it for its normal mana cost. This is true even if its alternative cost condition has been met.

Archive Trap checks only whether an opponent searched their library. It doesn’t matter whether that player found a card during the search.

A spell or ability causes a player to search their library only if it specifically contains the word “search” in its text. For example, if a spell or ability lets a player look at the top four cards of their library and do something with one of those cards, that’s not a search.

If the targeted opponent has fewer than thirteen cards in their library, that player puts their entire library into their graveyard.