Generic dichotomy for homomorphisms for E₀N

We prove the following dichotomy. Given an analytic equivalence relation E, either E₀N≤BE or else any Borel homomorphism from E₀N to E is "very far from a reduction", specifically, it factors, on a comeager set, through the projection map (2N)N→ (2N)ᵏ for some k. As a corollary, we prove that E₀N is a prime equivalence relation, answering a question on Clemens.