Oof. Kinda sorta Stockholm's Syndrome here, by the looks of it. While the mechanic is interesting and in the case of survival, fitting, it makes me feel icky.
I'm guessing each day the temperature drops? There's a missing sentence telling us the rate of temperature drop. Rereading it, I see that the Dark Frost catches up every 4 hours. But we have a few days to get to Lazarus? I guess that needs to be played out with the narrator. Like, three hour sleep shifts, with an hour or less of flat out running? Or sleep marching (via The Long Walk)? Also, "few" days is an unnecessary vagary that could easily have been "four" or "five" in order to increase character and player stress. In general "few" is three or, at most, four.
And a trivial task is a 15? Even at 4d6 that's not trivial. It might have been better to say rolling a 3 or better on a die is a success, and you need 3 successes. That's at least a higher probability of success.
I apologize for sounding so harsh. I think the idea that someone who knows a hint of my dark secret can then sacrifice me in a daily event in order for an action to succeed (meaning we don't die, or whatever. But wow) is really off-putting. And then if they know my secret fully they can just abuse me at will. Ugh. The only reason I'm running with this other person is that I'm trying not to die and moving in groups is the best way to do that. Oh! This really grates my cheese!
I could swear this is the basis for some movies or reality shows. I'd have to play this with not-friends at a big party so I could avoid them until I left. This is a tough sell for me. It's good mechanics, even if the act of it is morally or ethically wrong. But then what other mechanic would there be? It is what it is, and it's decent. I really like the Dark Frost; would like to have seen that expanded. I don't love it overall, but I appreciate the work and thought behind this one.