For one wild moment Harry considered lying to the world and telling everyone he was Slytherin’s heir, just to deny Riddle the clout. If it came down to him and Riddle he’d be at an advantage, with the Potter name and family connections backing him up, while Riddle had no one and nothing to support his claim. For just one moment he waffled, but then he let the fantasy go: he couldn’t in good conscious take advantage of the leverage his blood status gave him over Riddle. It wouldn’t be right. Not to mention that Riddle would probably kill him to erase the competition, and he could really do without having Voldemort after him in this timeline.