Jellyfish: the Holonomist of the Sea - What a bell taught an instrument about memory, hovering, and the loop that encloses nothing

A jellyfish swims with a stroke that, on paper, cannot work: rim and lip contract and relax as one motion run forward and back, a closed loop in the space of shapes that encloses no area, and in a fluid without memory such a loop carries a body nowhere. This paper builds an instrument for that stroke — an axisymmetric viscous solver with a moving bell, a leash, a valve and a wake reader — and reports what the loop does when the water remembers. Seven findings. The stroke lands on its own wake: the next cycle's displacement falls by a third of a radius for every period of rest inserted before the stroke, on a straight line, at both grid resolutions. With a one-way apex cap that holds its breath, the same stroke goes almost nowhere — station-keeping within the instrument's floor at both grid resolutions tested, with the sign of the residual drift not yet resolved. The energy the bell parks is in the stop vortex under its rim, not in the ring it launches, and at water viscosity on a slow stroke that vortex is shed rather than kept. The ring's memory grows with Reynolds number while the body's does not, held or free. The free bell runs at a power factor of −0.08: ninety-nine percent of what it does to the water comes straight back. A valved bell on a long return becomes a through-flow duct, a jet the animal does not have. And the instrument has a floor, stated first: a rigid reciprocal squeeze at Reynolds number one moves a twelfth of a radius per cycle where the scallop theorem demands zero. Every figure regenerates from a named script and run file; every limit is on the page.

Preprint; not peer reviewed; limited literature pass, stated on the declaration page. The record deposits the paper, the axisymmetric viscous solver that produced every number (with its fidelity checker and one-click batch files), every run file cited in the provenance, the author's frame sheets, and the nine figures.

Keywords-

jellyfish propulsion; scallop theorem; swimming holonomy; vortex ring; stopping vortex; passive energy recapture; axisymmetric Navier-Stokes; station-keeping; power factor; speculative engineering; memory in fluids; Reynolds number