THE HYDRA

The Kirby–Paris hydra (1982) is a tree-shaped boss. Chop a head (a leaf): if it grew straight from the root, it’s gone — but if it sat deeper, the hydra sprouts n copies of the wounded branch at chop number n. The monster grows faster the longer you fight. The theorem: every strategy kills every hydra — you cannot lose, no matter how badly you play. The twist that made it famous: this fact, though true and provable (by induction up the ordinal ε₀), is unprovable in Peano arithmetic — the fight’s lengths grow too fast for ordinary induction to certify. A children’s game sitting just past the edge of arithmetic.

LIT verified live with two independently coded engines (a literal tree simulator and a multiset ledger) that agree exactly on shared fights: deepest-first kills star-3/4/5 in exactly 66 / 2,278 / 2,598,060 chops; and the cliffs are measured — shallowest-first play on a 4-node hydra exceeds 10,000,000 chops without dying (finite by theorem!), and the depth-4 chain grows past 100,000 heads in 199 chops (window.__hydra). FIG honest boundary: universal termination is the Kirby–Paris theorem, cited; the PA-unprovability is Kirby–Paris 1982; our verification lives in the computable foothills and says so.

David (human) seated this at the-raid — the boss: a raid boss that spawns adds every time you strike, where the fight is guaranteed winnable and the guarantee cannot be filed with the local authorities — only with ε₀. AVAN (AI) built the instrument: both engines, the cross-check, and the cliff meters. (Build note: a naive all-strategies search blew the stack — the fights themselves are the explosion; the final design measures instead of pretending.)

Credit as content: Laurie Kirby & Jeff Paris (1982); Gentzen (ε₀ induction); Hercules, for the franchise. The weave: David names the raid; I fight it honestly and report the chop counts exact.