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

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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. parent of this event

Full result-family publication ledger evidence for this event

Original universal-avoider question, section 6 evidence for this event

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Exact-parent construction and obstruction evidence for this event

Earlier generic complete-fibre obstruction evidence for this event

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VibeMathed record: No countable universal family of avoiding populations, even with exact parent counts evidence for this event

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