Best start with the shelves; easiest to check edition dates there. She looks for books that are boat- or cat-themed, first. Does Old Man and the Sea have an appropriate edition? Lovecraft is probably also a good bet, given the subject matter--anything about Innsmouth? The Cats of Ulthar?

If she finds something with an appropriate edition, she'll skim the listed pages for anything pertinent.

Old Man and the Sea doesn't, but Lovecraft has a book with a boat on the cover - At the Mountains of Madness and Other Novels - the edition date of which is 1985. Page 306 is the start of The Dream-Quest of Unknown Kadath, which mentions cats at least once.

A boat on the cover, and the indicated page significant. Not to mention "Dream-Quest". Puzzle solved, looks like. Is page 407 the end of the story?

She sets to reading, and spends some time hacking through adverbs like "loathsomely" and "affrightedly". It's unclear how she's supposed read into it--are dreams supposed to correspond to the literal "dreamland" in the story, or "those cryptical realms known only to cats"? The student presumably meant to analogize herself to the cats in the story; is she implying that she finds it difficult to form English because she speaks some metaphorical "speech of cats"? Is she implying that she's a protector, of some sort--but then from what, and why?

(Jane spends a little time searching around for inexplicable deaths during sleep, and various other keywords from the story.)

And perhaps more importantly--if there are people going around visiting people's dreams, how on earth could she not have heard of it before?

Do searches for "cats of ulthar", "dreamland", "cat speech", "toad-things", or "cryptical places dark side moon" show up anything unexpected?

Page 407 is indeed the end of the story.

Past infancy, people don't tend to die in their sleep entirely inexplicably - people do sometimes have heart attacks, or blood sugar crashes, or similar deadly problems, but usually there's signs of an undiagnosed underlying condition there. Some people have claimed they died in a dream, then woke up with assorted neurological symptoms that faded after a few days to a few weeks. (Mostly 'hallucinations, executive dysfunction issues, psychosis, delusions.' That some patients will experience temporary mental illness, and usually report being attacked in their dreams, is documented, but the current overall scientific consensus is that the mental illness causes the dreams, not the other way around.)

There's a few obscure myths of significant meetings in dreams, and people claim to have met someone in their dreams every now and then (at much lower rates than they claim to have seen ghosts or aliens or used ESP), but like ghosts and aliens and ESP no one's managed to produce replicable results in a study of any quality.

Dreamland gets a thing talking about how Lovecraft was probably referring to contemporary theories (since fallen out of favor) that the recurrent locations in dreams are actually part of a shared consciousness, or else that humans are able to project their consciousness to another realm. There's never been any proof of this; someone claimed to have recordings of Lovecraft's Dreamland in the 1990s, but the video was pure static. A few lucid dreaming groups also seem to refer to the recurrent locations as 'personal Dreamlands', with discussion about how best to modify it.

Cats of Ulthar mostly gets Lovecraft and the descended cosmic horror genre and discussion about Lovecraft. Cat-speech gets a lot of fiction, cat body language, and cat whisperer things; in conjunction with stuff about dreams, it also gets someone claiming to have been visited by talking cats in his dream. (The response to this is mostly people asking what he was on at the time.)

'Cryptical places' gets the text of the same story she just read, and a password locked site that looks like it hasn't been updated since the days of early Geocities-style designs being popular.

Oooo, conspiracies. She's done this song-and-dance before, admittedly with a lot less personal anecdata.

She leaves the book to be re-shelved and finds somewhere to sit down, because hovering around with a phone is getting tiresome.

(She succeeds at not forming any hypotheses, and is very proud of herself.)

Can she find anyone dying of environmental phenomena in dreams and experiencing the same effects? How about being killed and not experiencing any ill.

And: if she can find a password-locked site by searching, someone clearly hasn't been very diligent with their security. She's not exactly a hacker, but she can poke around the address and check whether it's getting an unusual amount of traffic. Maybe there's a registry where she can check who owns the domain?

People mostly seem to die of falling off of or into things, or weird wildlife, or eating something in the dream that must have been poisonous, or starting a fire. No one's died of thirst or hunger in a dream that she can find. Someone claimed to have suffocated in a sandstorm, another in a flood. One person claimed that their dream-scape broke around them, though that person was documented as suffering apparently permanent delusions. Deaths of all causes that she can find seem to be correlated with the same general class of mental effects.

Traffic's very low to the site, and whoever owns the domain isn't registered somewhere she can find.

Alright, makes sense. And if that domain isn't getting much traffic, then it's unlikely to be important.

... Can she find anyone killing themselves in a lucid dream, and suffering the same effects? Also, do people typically die in their dreams again during the time that they're hallucinating or psychotic or whatever, or only at the start? It would make some sense for there to be something you could do to the brain to make it do a bunch of bad things and also to imagine dying -- it would be very strange if the imagining-dying always preceded the illness and wasn't ever concurrent with it.

Dream-suicide seems to have the same effects, but is more strongly correlated with existing depression and other mental illness; it's unclear what's causative there.

People who die in their dreams once seem to be more likely to die again than the general population (there's not a lot of people who only ever die once, at least not among 'people who wind up in medical studies'); sometimes this overlaps with mental illness episodes. Usually the people who die more often in their dreams describe those dreams as typically occurring somewhere hazardous - around wildlife, amidst steep cliffs, on a little island surrounded by a river...

Ugh, alright. Well, she'll do her due diligence and ask around for people who might know something, but it look's like she's going to have to learn how to lucid dream to determine anything with confidence. She knows that she isn't particularly mentally unhealthy--if she in a dream she kills herself--or better, if in a dream she does risky things that might result in death and eventually dies from them--and immediately her life goes to (temporary) shit, that would be definitive evidence that Something Pretty Wack Is Going On.

If she asks some med-school folks if they know anything about the etiology of dream-deaths, does she get anything useful?

Generally people seem to think dream-deaths probably are caused by underlying mental health problems or predispositions, and either one heralds a sudden onset mental illness or the trauma of dying, even in a dream, causes a break with reality? Patients very commonly report a sharp break in mental health from one day to when they wake up, and it's apparently happened twice (that this person knows of) that someone had a dream-death while under observation in a hospital, and the symptoms were noted then as dramatically suddenly onset, though one of them wasn't well documented and the other was a couple decades ago.

There's also a lot of variation in how badly people are affected. This mostly seems to track with resiliency? In that people with high resiliency generally do better after a dream-death. Weirdly, some preexisting mental illnesses - like bipolar disorder and schizophrenia - make you less likely to have a severe drop off after a dream-death. The dramatic effects are mostly in apparently mentally well people.

(There's apparently an ongoing medical debate if dream-deaths are caused by mental illness, themselves cause a very severe and possibly unique kind of PTSD, or both - opinions vary a lot.)

(Okay, uhm, maybe she shouldn't try it, then.) That's mildly scary. She has ever lost consciousness, though, and didn't experience it as so traumatic that she started hallucinating--how about people who have actually died, more-or-less? That is, someone who has a cardiac arrest, loses consciousness, and is resuscitated? Naively this should be a lot more traumatic than dying in a dream, but she's never heard of people becoming psychotic after near-death experiences. And--there shouldn't be any meaningful difference after the person has lost consciousness, right?

Having near death experiences is often traumatic but not to the same apparent extent.

Yeah, something very strange is going on here. It makes no sense that dying in a dream should be more traumatic than dying in reality. She thanks the med student--and, on second thought, reaffirms that she's investigating a bunch of weird dream stuff, so maybe if she has some Unfortunate Accident they'll remember and look into it.

Over the next few weeks, along with her normal activities, she starts looking into lucid dreaming. It takes a long time--much, much longer than the forum-goers suggested it would--to learn how to modify her dream-environment. There's a part of the exercise which is understanding the object--which she takes to well--and a part which is expecting the object, which requires a kind of mental gymnastics that she has great trouble with. Of course: once you have observed yourself to have some power, it is easy to expect yourself to continue to have it. It takes comparatively little time at all before she can produce arbitrary things to the limit of her understanding.

She spends a while messing around. She generates pretty shells and implausibly-large crystals of rare compounds, she makes two different houses in aggressively modernist styles, she makes nanomaterials that haven't been successfully synthesized yet, she tests reactions with reagents too rare to use in the real world--

She's making a bunch of francium to explode. In a vacuum, of course, because it'll immediately vapourize from the energy released by its decay; and if she has less shielding than she should, well, this body's not going to last past the night anyway. And so she closes her eyes, imagines a centimeter-sided cube of a grey metal with these properties and this structure, decides that it's going to appear now, and opens her eyes to--

A cube of grey metal. Glowing, and conspicuously solid. She's confused--tries to make more, tries to visualize better, ends up with a lot of glowing grey cubes and very little explanation for why this, a pure solid element in a visible quantity, is the first thing that she would fail to create. She tests one--and, yes, it reacts like should would expect francium to, which makes the whole situation just that much more baffling.

And it's then, a month after she met the singing girl in the desert and exactly twenty-one minutes and fourty-eight seconds after she created her first cube, that a half-centimeter by centimeter by centimeter box of glowing grey metal is deleted from existence, and Jane realizes the full magnitude of what she's gotten herself into.

She doesn't get anything more done that night.

Next morning, she starts hypothesizing. It certainly seems as though there's something happening without her brain as a substrate, given that she got very consistent understandable results up until she messed with things on too low a level. If her dreams ran entirely off of her expectations, she wouldn't have observed what she did; she was surprised at first they obeyed physical laws so consistently, and they did so for months. She guesses she should probably give this less credence than feels right, given that so far nobody who thinks that they've discovered something paranormal actually has.

She finds herself feeling again the long-forgotten childish want to have sleep now, although for the first time she is anticipating the night and not the morning. She lies awake for hours, shivering, unable to banish the adrenaline and the what if what if what if--

And somewhen she slips into that desert, less empty with two houses and all manner of shiny litter--

She finds and memorizes the boiling points of some alcohols and oils, writes them down when she wakes, checks. Anonymously emails the Independent Investigations Group:

Hello.

I believe that I have discovered a supernatural phenomena. I would like assistance in confirming that this is the case. For privacy reasons, I would prefer not to conduct correspondence through email. If you respond to this email with a time and a phone-number I will call you and explain.

Thank you.

Boiling points are in the approximate right area - close enough to feel the same to the touch - but are often rounded off to whole numbers, despite her equipment being fully capable of measuring fractions of degrees. They're, in fact, rounded off to whole numbers no matter which measuring system Jane uses, if she thinks to test both Celsius and Fahrenheit. 

The group gets back to her a few days later with a time and phone number, as requested.

... Spooky. (She does not think to check in two different units; weird dream-planes are one thing, but Fahrenheit being useful is too absurd to think of)

If there was ever a justification for skipping class, then an upcoming potential-Nobel-or-schizophrenia-diagnosis surely counts. She begs a phone from an acquaintance and locks herself in her office.

"Hi. Is this the IIG?"

"Yes. Is this - " And he rattles off her email.

"Yes, that's me. Would you mind if I asked who I was speaking to?"

He gives a name, title, and position easily enough.

"What was the phenomena you mentioned?"

She can quickly look him up while she's explaining.

"I have some preliminary evidence that suggests that dreams are run on a substrate other than the dreaming human's brain. In particular, it seems to be possible to produce information that the dreamer has no way of accessing; I have measured, while dreaming, physical properties that I did not otherwise know and found them to be approximately the same in reality. I would like help determining whether this is the case; is there any way in which your organization can provide such?"

"We can help with that! I'd want some more details, and time to get notes together - I can email you suggestions, or we could do another call? We might also be able to replicate some things if you describe what you were doing?"

"I can list for you the tests I have done and their results, although of course you'll have to take on faith that I followed any particular procedure. I would prefer not to correspond over email, if at all possible. It should be possible for any lucid dreamer to independently verify everything; otherwise, the only way I can think of testing would be to have someone generate information while sleeping under supervision, which might be difficult to arrange."

"We'll likely want to arrange that as a second step, yes," he says. "When would you be free for another call - ?"

She can give him a date and time. "Is that all you need from me for now?"

"Yeah. Thank you for contacting us. We'll be in touch."