[LyricalAura]

I guess I'll go ahead and post my current understanding of the Rules. Most of this is from the time of EP6, with some minor revisions to take into account EP8 knowledge.

Spoiler for Rule X:

Rule X is the explosion which established the boundary of the catbox.

Bern's letter said the rule is like the game that always starts on June 20th. Looking at the Hinamizawa game, this is two days before the Hinamizawa Disaster, meaning it's already too late to make any moves to prevent it. Lambda also told Bern that she absolutely cannot win because of this rule. So, there should be an element on the Rokkenjima board that is already unstoppable at the opening of the game.

At this point, we know that all episodes are fictions which became games (became subjects of meta-discussion) sometime after the Rokkenjima Explosion Incident. In short, from the very beginning it was totally impossible for Bern to stop the disappearance of the Ushiromiya family or the total erasure of evidence of what really happened. Because the total destruction of information is necessary for Beato's existence, Bern refers to this rule as "the core of the witch Beatrice", and Beato herself calls the catbox her heart in EP4.

Spoiler for Rule Y:

Rule Y is Yasu's message bottles, which created the environment compatible with a witch's existence inside the catbox.

Bern said the rule is a new magic system which constitutes the existence of the witch Beatrice, and which Lambdadelta already has a small amount of experience with. According to Featherine, the identity of the witch Beatrice is really "an environment in which it's possible that Beatrice exists." That environment existed on the island due to a series of rumors, lies, and pranks by various people, but under normal circumstances the evidence of it would have been blasted out of existence by Rule X.

The "witch's environment" was able to become an element of the game because, after the two days had become a black box, Yasu's two anti-mystery message bottle stories were discovered and publicized. In the absence of any other information, they painted over the two days and became the pattern governing all interpretations of what happened. People began to think Beato could have existed on the island, and as a result, fantasy fragments having that premise came into existence.

Lambda's experience with this rule comes from the Hinamizawa game. Her piece in that game, Takano, created a number of these "environments in which X could exist" with her Document 34 stories, where X included the curse of Oyashiro-sama and an alien invasion conspiracy. However, the Hinamizawa Disaster catbox that Takano painted over wasn't anywhere near the perfect void of information that the Rokkenjima Explosion was, so her "magic" didn't reach Beato's level.

Spoiler for Rule Z:

Rule Z is Yasu's ownership of the gold, which allows her construct an infinite variety of serial murder illusions within the catbox.

Bern calls this a maze-like existence which generates phenomena not explained by the first two rules. Treating the sequence of events in the game like a maze, Bernkastel's power normally allows her to completely explore it and determine what results each action will have, but on this board, something causes the maze to change shape flexibly and makes Bern's mapping useless.

This phenomena is Yasu's ability to bribe the other characters to behave however she wants them to. Her accomplices can be chosen freely between games, and she can make them construct any closed room illusion she wants simply using the offer of vast amounts of money.