No countable universal family of avoiding populations, even with exact parent counts

No. From any population avoiding s and any countable family of source-population hosts, construct an avoider with the same roots and exactly one incoming parent per label, admitting no injective map to any host with any finite global edge-stretch bound. Thus even adjacency embeddings fail. Parent selection preserves avoidance; terminal cloning preserves the selected infinite-word language exactly. The proof makes fibre sizes exceed enumerated finite host balls while preserving the actual population axioms. This strengthens the earlier complete-fibre obstruction by retaining exact parent counts. Separately, complete blow-ups preserve every corresponding least matching height, linking the obstruction to the rank question. Classical locally finite graph diagonalization is credited. Terminal vertices are permitted; fixed child caps, noninjective maps and arbitrarily unbounded stretch are outside the theorem. It answers the stated embedding interpretation, not every possible category.

No. From any population avoiding s and any countable family of source-population hosts, construct an avoider with the same roots and exactly one incoming parent per label, admitting no injective map to any host with any finite global edge-stretch bound. Thus even adjacency embeddings fail. Parent selection preserves avoidance; terminal cloning preserves the selected infinite-word language exactly. The proof makes fibre sizes exceed enumerated finite host balls while preserving the actual population axioms. This strengthens the earlier complete-fibre obstruction by retaining exact parent counts. Separately, complete blow-ups preserve every corresponding least matching height, linking the obstruction to the rank question. Classical locally finite graph diagonalization is credited. Terminal vertices are permitted; fixed child caps, noninjective maps and arbitrarily unbounded stretch are outside the theorem. It answers the stated embedding interpretation, not every possible category.