Since SB doesn't remember Monday, she cannot feel the difference but the structure of the experiment KNOWS the difference.So if she is asked twice, Monday and Tuesday, that only happens with tails outcome. Even without memory, her credence may shift, but because the setup itself is informative. — Kizzy

It's also part of the protocol that although SB knows that she is being awakened a second time on Tuesday if and only if the coin landed tails, on each occasion where she is being awakened, she isn't informed of the day of the week. As specified in the OP (and in the Wikipedia article), she doesn't know if her awakening is happening on Monday or Tuesday (though she does know, or rather can infer, that it's twice as likely to be occurring on Monday). Hence, the relevant information available to her for establishing her credence is the same on each occasion of awakening.

I think I understand and see my mistake, I was confused from the start as I misunderstood the problem because of the way its laid out and my brain,[thing to note208am] and since I was and am assuming SB is using prior knowledge plus new information Tuesday, without amnesia Sunday to Monday. I just find it to be seemingly difficult to not use some sort of self locating knowledge to assign degree of belief in something, that something being a coin heads up with or without amnesia Sunday and the chance for belief to update come Tuesday.

Since I didn't account for the amnesia from Sunday to Monday, only Monday to Tuesday. My pov, though in error for mistaking PROTO, understood the OG problem (see wiki for Sleeping Beauty Paradox*) to mean after she goes to sleep Sunday, the experiment does not start until Monday, where she goes into the experiment asleep and awakened [inevitable], and asked her credence, put to sleep and either asked again Tuesday with amnesia or sleeps through it. awakening Wednesday and experiment is officially over. .

I do not put it passed me, that even though I just very well may have completely misinterpreted the problem before my own eyes, to be standing corrected, as I have to in order to carry on...

However, now knowing Sunday to Monday she is given amnesia, something about knowing the experiment duration and rules tells me self locating knowledge must come from somewhere, from something in time and our relation to it is relevant to the possible knowing, yes being awake is evidence of some thing but what else gives?

Thanks for your contributions to this thread. And also to your AI exploration experience and work that you've shared here on the forum as well. Of interest...

BUT even though, I was and still am off, in this way:

What else besides knowing what she signed up for, prior know ledge and new intel [ i am awake now] does not alter her credence Monday, only verifies indirect info about SB maybe? If she was given amnesia Sunday and put to bed, awakened Monday then what awakening is she forgetting? *see quote from wiki below, note bold words that are throwing me.....

Anyways, either way -

OH never mind, OF course if she knew it was Monday she wouldn't say 1/3, but what if she was off...and Tuesday comes around and it changes to 0? the chance to change or update belief still exists if tails and asked twice. On Monday she does not know for certain if heads or tails only gives her degree of belief in heads, knowing nothing Wednesday when experiment ends, tomorrow she will be awakened or sleep through the day, she can still guess reasonably participating, I think? I don't know, perhaps I am in over my head here...again!

* per Wikipedia, "On Sunday she will be put to sleep. Once or twice, during the experiment, Sleeping Beauty will be awakened, interviewed, and put back to sleep with an amnesia-inducing drug that makes her forget that awakening." https://en.wikipedia.org/wiki/Sleeping_Beauty_problem

OH never mind, OF course if she knew it was Monday she wouldn't say 1/3, but what if she was off...and Tuesday comes around and it changes to 0? the chance to change or update belief still exists if tails and asked twice. On Monday she does not know for certain if heads or tails only gives her degree of belief in heads, knowing nothing Wednesday when experiment ends, tomorrow she will be awakened or sleep through the day, she can still guess reasonably participating, I think? I don't know, perhaps I am in over my head here...again! — Kizzy

You'll more easily wrap your head around the problem if you don't overcomplicate things (even though it will remain a tough problem). The purpose of the drug merely is to make it impossible for Sleeping Beauty on any occasion of awakening to know if this occasion was a first or a second one in the experiment (which she could otherwise deduce if she had a memory of the previous one or the lack thereof). This makes all three possibilities—Monday&Heads, Monday&Tails and Tuesday&Tails—indistinguishable from her subjective perspective although she knows at all times that over the course of the experiment all three of those situations could be experienced by her (without knowing which one it is whenever she's experiencing one of them). You can now place yourself in her shoes and start pondering what the chances are that the coin landed tails.