Ecosystem Stability: Why Some Food Webs Persist and Others Collapse - Bounded-Denominator Cycles

One axiom. One operation. Zero free parameters. Some ecosystems persist for ages; others collapse. What makes the difference? This paper forces the distinction. Predator and prey populations are coupled fold orbits — the boom-and-bust of hares and lynx is two populations folding around each other — and an ecosystem is stable exactly when its coupled orbit has a bounded denominator, which makes it periodic: a closed cycle that repeats forever. A 3/5 predator-prey coupling is a periodic orbit, a stable cycle. Collapse is the loss of that bounded-denominator periodicity: pushed toward unbounded denominators, the orbit stops closing into a stable cycle and the system breaks down. Stability is bounded-denominator periodicity; collapse is its loss — the same eternal/transient structure that distinguishes a held memory from a fleeting one. Machine-checked; reproduces from one command. A standalone result within the Smithian Fold Theory of Everything (SFTOE). Full corpus, code, and the run-it-yourself VERIFY.md protocol: https://github.com/MettaMazza/Smithian-Fold-Theory