Short, math‑heavy write‑up (PDF, 11 pp.) that re‑derives the classical initial‑algebra construction for ω‑continuous endofunctors and then isolates a “Node Deletion Theorem”: remove a single object from the building chain and the theorem tells you exactly which arrows vanish in the resulting colimit. In practical terms it gives a provably‑safe recipe for cutting nodes out of abstract‑syntax trees, graphs or other inductive data without reconstructing the whole structure—think incremental compilers, program transformers, or graph‑rewrite engines. The note also sketches a joint‑fixed‑point extension (Bekić‑style) and invites feedback, counter‑examples and real‑world applications. Comments and pointers to prior art welcome!