Hercules and the Hydra
Eduardo Piza · Zenodo (CERN European Organization for Nuclear Research) · 2026
Hydra battles combine finite combinatorial rules with transfinite termination arguments. After a self-contained introduction to Kirby–Paris hydras and Goodstein sequences, we study battle lengths under fixed, adaptive, and freely chosen reproduction laws. We prove a criterion for uniform boundedness under adaptive reproduction and identify the free transition rank of every finite hydra with its structural ordinal below ε_0. A six-vertex example shows that pointwise larger reproduction laws need not yield longer maximal battles; a simulation argument restores monotonicity when the larger law is nondecreasing. For explicitly tabulated reproduction schedules, with an explicit initial tree and binary copy counts, we prove membership in NP. A persistent compressed representation gives a deterministic algorithm with running time (v +2)O(r^2) poly(L), where v is the initial number of vertices, r the table length, and L the input length. Thus every fixed reproductive horizon is polynomial-time solvable. We distinguish these results from arithmetical independence and relate their conventions to formalized hydra developments. The complexity classification for variable horizons and the detailed bibliographic positioning of the formulations remain to be settled.