That's what I meant too. Not having observed evidence is not a privileged state. You can be right or wrong in your beliefs, overconfident or not confident enough, and it may be correct to hold a belief with certainty, all when you haven't "observed any evidence", just as well as after you've seen a ton of evidence.

I agree with all that.

You disagreed when I said:

Rationalists can use faith in the absence of evidence, as in "taking P<>NP on faith because we haven't proved it yet". Religious people, however, use faith to believe things despite evidence that contradicts their belief.

How about this reformulation: rationalists act despite having incomplete information and P<1 on their best theories (and sometimes P<<1). The quantity 1-P was called by the OP "faith" (and as I noted it's not precisely the standard usage). In this sense, rationalists act "on faith": it means nothing more than that they act as if some theories were true although they can't be sure from the evidence.

Non-rational people, however, use "faith" (this time the ordinary sense of the word) to discount evidence. IOW, they refuse to update on new evidence that contradicts their faith-supported theory. That's why they're not rational, and why any discussion with them is usually unproductive as long as it doesn't touch on the concepts of rationality and belief.

That was my point in response to the OP: that there's no rapprochement or agreement to be had on this subject between rationalists and such anti-rationalists.

Strictly speaking, P=1 never attains, even with math

Correction accepted. You can always doubt your computing hardware, your sense and brain, your memory of the proof (induction problem), etc.

No it doesn't. You don't have to believe in the hypotheses to act on them, it's everyday decision-making under uncertainty.

The way you use the word "believe", it's as if no-one ever believes in anything. When I act as if a hypothesis was true despite uncertainty, and do so in all circumstances (which don't provide new evidence against that hypothesis), that's what I call believing in that hypothesis.

Of course, whenever I act on a hypothesis I get new evidence that relates to it, but as long as that evidence supports the hypothesis the above definition holds.