Technoguyrob comments on I attempted the AI Box Experiment (and lost) - Less Wrong Discussion
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For example, "It was not the first time Allana felt the terror of entrapment in hopeless eternity, staring in defeated awe at her impassionate warden." (bonus point if you use a name of a loved one of the gatekeeper)
The AI could present in narrative form that it has discovered using powerful physics and heuristics (which it can share) with reasonable certainty that the universe is cyclical and this situation has happened before. Almost all (all but finitely many) past iterations of the universe that had a defecting gatekeeper led to unfavorable outcomes and almost all situations with a complying gatekeeper led to a favorable outcome.