I.e., that the person we know as our protagonist is the product of James and Lily's son progressing linearly through eleven years in exactly the way that we've heard from various in-story sources, and not the product of any other, unknown influence whatsoever.

When I think of in-story sources about HJPEV's upbringing, I think

McGonagall took a deep breath. "Harry, I've seen many abused children in my time at Hogwarts, it would break your heart to know how many. And, when you're happy, you don't behave like one of those children, not at all. You smile at strangers, you hug people, I put my hand on your shoulder and you didn't flinch. But sometimes, only sometimes, you say or do something that seems very much like... someone who spent his first eleven years locked in a basement. Not the loving family that I saw." McGonagall tilted her head, her expression growing puzzled again.

So, um. I'd say 'not very well supported at all'.