Sarah: Roger, quick scenario. Tournament hall. You're playing White, your hand is on the king, you're about to castle. And your opponent leans in and asks: has anyone actually checked that square?

Roger: And nobody has. Which is exactly today's episode: one move, and three different things we might know about one square.

Roger: Quick housekeeping first. We're both synthetic voices, the music is generated too, and so is this show. The answers aren't invented, though. Every result you'll hear comes from a recorded Arxo run, and the facts and hashes for each one are in the show notes.

Roger: This is Executable Knowledge: Cases.

Roger: Chess players learn it early: you don't castle through check. FIDE Laws of Chess, Article 3.8.2: the square the king stands on, the square it crosses, and the square it lands on can't be attacked.

Sarah: That part everybody knows.

Roger: Right. The tricky part is that little word, not. Not under attack. What does that mean when nobody has looked at the square?

Sarah: It means... nobody has looked at the square.

Roger: Exactly. And that's not the same as nothing there. So we loaded the FIDE rule into Arxo, set up one position, White castling on the king's side, king and rook never moved, and asked the same question three times. Only one thing changes: what the record says about those squares.

Sarah: Case one.

Roger: The squares were checked, and the record says the square the king has to cross is under attack.

Arxo answer: Result: not established.

Sarah: Not established. So that's a no?

Roger: Not quite, and that difference will matter in a minute. It isn't a yes. The rule doesn't grant castling here, and it tells you exactly which condition broke.

Arxo answer: Blocker. Required: not — the king's square, the square it must cross, or the square it is to occupy is attacked. Status: refuted.

Roger: Refuted. That condition was supposed to hold, and the record says it doesn't: the path is attacked. So the rule won't sign off.

Sarah: The path is attacked, so the rule won't sign off. Beginner's book.

Roger: Beginner's book. But remember that word: refuted.

Sarah: Case two.

Roger: Same position. The squares were checked, and nothing was found. No attack recorded, no piece in the way.

Arxo answer: Result: true. Rule applied: Castling Legal.

Sarah: There we go. A yes.

Roger: A yes, with eight reasons. Four are plain facts somebody wrote down: White's king, White's rook, White has the move, and White wants to castle with that rook.

Roger: The other four are the interesting ones. The path isn't attacked. There's no piece between the king and the rook. The rook hasn't moved. And the king hasn't moved.

Sarah: Aren't those just facts too?

Roger: Nobody typed them in. There's no fact saying the path isn't attacked. There's a line saying these squares were checked, and these pieces were checked. The answer marks all four the same way: checked, nothing recorded.

Roger: That's how a not gets earned here. Somebody looked, so an empty record is a clean record.

Sarah: So, case three.

Roger: Case three is case two with one line deleted. Same board, same pieces, nothing on record about an attack. But nothing says the squares were checked, either.

Roger: That's the tournament hall: hand on the king, and nobody has looked.

Arxo answer: Result: not established.

Sarah: Hang on. No attack on record, nothing in the way. Same board as case two. Why isn't it a yes?

Roger: Same board, different record. Case two was checked. Case three never was. Listen to the answer this time.

Arxo answer: No blocker. Not refuted — simply no derivation.

Roger: No blocker. Case one had a refuted condition. Here nothing is refuted. Two conditions are simply unknown.

Arxo answer: Required: not — there is a piece between the king and the rook with which castling is to be effected. Status: unknown. Required: not — the king's square, the square it must cross, or the square it is to occupy is attacked. Status: unknown.

Sarah: So it's not saying the squares are attacked. And it's not saying they're safe.

Roger: Right. It's saying: I don't have that.

Roger: That's the point. A system that reads no attack on file as no attack would wave White straight through. Arxo doesn't. Unknown is not no, and it isn't yes, either.

Sarah: So two of the three cases say not established.

Roger: For completely different reasons. In case one, we looked and found a problem. In case three, nobody looked.

Sarah: Where else does this bite?

Roger: Anywhere a missing record gets read as a clean one. An inspection with no defects recorded: inspected and fine, or never inspected? An audit with no exceptions noted: same question.

Roger: Arxo keeps those three apart: looked and found a problem, looked and found nothing, never looked.

Roger: Three cases, one rule, one square. The facts, the full answers and the hashes are in the show notes. Run them again and you'll get the same three answers.

Sarah: Next time: another rule, and a few more cases that don't go the way you'd expect.

Roger: Both voices synthetic, every answer from a real run. This was Executable Knowledge: Cases, from Arxo.