You'd be detained but if you asserted you are a US citizen they are obligated to try and establish the truth of your claim. They'll call family and friends and access government records. In an edge case, I imagine they'd hold a hearing with an immigration judge.

In college, friends of mine frequently went out partying in Tijuana and occasionally lost their passports. They were usually held for a few hours at the border crossing and given a hard time and a stern lecture about responsible behaviour before being readmitted to the US.