During one of my runs, I have thought that it might be very useful to do a screenshot of a map while you are Cartographer, and to use it later when exploring location with other character. Also, sometimes it is useful to take written notes or use open materials (for example, boss damages or relics effects). Are these things available during speedruns?

I can't speak for the other people here or the moderators, but it isn't explicitly stated that you can't so I wouldn't see why not as long as you are following all of the other protocols.

I think the idea of speedruns is you can have only the game and video recording programs be used during your speedruns. In case of taking a screenshot it's like you have another active program that directly helps you during speedruns. Castle generation information is present only during your current run, and is supposed to be unique. But if we speak about information that is always the same (boss damages, relics effects), having written notes or using open materials (I suppose you can do that via phone, but won't it slower your run as you need to search for information?) about them, in my opinion, is not a thing that can directly affect your CURRENT run. Thus, writting notes are legal to use. I think there is no difference if you have knowledges about the game in mind or written somewhere.

@Vilks, along those lines, what if someone were to write a program that watches their speedrun, and analyzes and aggregates castle information into a HUD on their second monitor which is positioned such that the runner can actually see it in their peripheral vision for the remainder of the run. Some questions:

1. Is this strategy allowed in the current rules?
2. Should this strategy be allowed in the current rules?
3. Is this strategy even detectable if the runner doesn't explicitly come clean about their structure?
4. What is the natural extension of this strategy, and how do we deal with further improvements? (It is trivial to come up with a sequence of stronger strategies in the same direction)

I also had thoughts about creating a program that analyzes a world gen and writes what relics spawned using that seed. Is this allowed in speedrunning? No. Can we detect this? Well, we can't. If the runner doesn't do some suspicious movements (heading directly to heirlooms, relic rooms, boss rooms and finds no dead ends) then it's highly unlikely for that person to be found cheating.

It can't really be tracked, however, there have to be limits on what the program can do. Since we can pause to think (and theoretically take notes) a program that shows you basically a found hud of what you have already discovered during your run should not be disallowed as you can achieve the same result in more paused time during your run with handwritten notes.