What is the exact rule that says a showdown cannot end while there are items on a chain?

There isn't a single standalone sentence that says "a showdown cannot end while items are on a chain" — instead, the prohibition is structural across several rules. Here are the two clearest pathways:

Path 1 — State-based (Rules 310.4, 313.1, and 347.3):

• When a Chain exists during a Showdown, the turn is in a Showdown Closed state (Rule 310.4).
• A player can only Pass (the action that leads to ending the showdown) when they have Focus during a Showdown Open state (Rule 313.1: "Focus is the permission to take appropriately timed Discretionary Actions when the turn is in a Showdown Open State").
• The Showdown ends when all players Pass in sequence (Rule 347.3.a).
• So while items are on the Chain, the state is Closed — no one can Pass to end it. The Chain must empty first (Rule 340.2 / Rule 346), returning to Showdown Open, before Pass is even an option.

Path 2 — Cleanup lockout (Rules 321 and 347.3.a.1):

• Ending a Showdown performs a Cleanup (Rule 347.3.a.1).
• Rule 321 states: "while Chain Items are Resolving, a Cleanup cannot occur."
• If a Cleanup would be needed but Chain Items are resolving, the Cleanup becomes an Outstanding Task (Rule 321.1) — it's deferred until the Chain empties.

Bottom line: The mechanism is that the chain must fully resolve to empty before the state opens, Focus can be exercised to Pass, and the showdown-ending sequence can begin.