Ordinal certificates and exact pruning characterize sequence realization

Avoidance is equivalent to a strictly decreasing ordinal certificate on the reachable states; arbitrary vertex/label types are allowed. Under finite per-label branching, natural certificates suffice. It also proves that states surviving every finite pruning round form the greatest successor fixed point and are exactly those with infinite continuations. A start realizes the full target iff its phase-zero state survives. Every other state has an attained finite maximum h, is removed at round h+1, and has a pointwise least natural certificate, even in populations realizing the target elsewhere. Complete finite-fibre blow-ups preserve corresponding least heights. Reachability prevents confusing an unrelated suffix with the original word. These are classical well-founded and finite-branching methods made exact in the source setting. This is not a deciding algorithm for arbitrary infinite inputs or a rank hierarchy separating avoiders; the artificial-root history tree is a different object.

Avoidance is equivalent to a strictly decreasing ordinal certificate on the reachable states; arbitrary vertex/label types are allowed. Under finite per-label branching, natural certificates suffice. It also proves that states surviving every finite pruning round form the greatest successor fixed point and are exactly those with infinite continuations. A start realizes the full target iff its phase-zero state survives. Every other state has an attained finite maximum h, is removed at round h+1, and has a pointwise least natural certificate, even in populations realizing the target elsewhere. Complete finite-fibre blow-ups preserve corresponding least heights. Reachability prevents confusing an unrelated suffix with the original word. These are classical well-founded and finite-branching methods made exact in the source setting. This is not a deciding algorithm for arbitrary infinite inputs or a rank hierarchy separating avoiders; the artificial-root history tree is a different object.