Battler's sin

Axioms

• Battler says Six years ago, no person called Beatrice existed for me.
• The sin I am now demanding that you remember is not between Ushiromiya Battler and Beatrice.
• Ushiromiya Battler has a sin.
• Because of your sin, people die.
• Due to your sin, a great many humans on this island die. No one escapes, all die.

Corollaries

The story only talks about a sin, but we narrowed it down to a broken promise because of how often Beatrice remarks that witches, unlike humans, keep their promises.
The framing of the scene also makes it clear that the person who was hurt by this scene is not Sayo, whom the red truth identifies as Beatrice, but X, whom it does not.

So Battler made a promise to X, then forgot about it completely, leaving her desperate. People often make fun of him in a way that resembles teasing someone for treating a romantic partner badly.
Through George, the story places great weight on marriage proposals. And in Legend of the Golden Witch, the corpse who we believe to be X is found with a wedding ring on her finger.

Did Battler meet X under a different name, promise that he would marry her, only to forget and then return to Rokkenjima 6 years later? It's our best guess.