A ring-native multipole method for the wakes of flapping wings: closed-form moments of vortex rings, the lumped wake they define, and where the force of a discrete wake comes from after the first period, draft v0.5 with the first runs and the sensiti

This deposit records the state of Paper 1 on 27 October 2026. It contains the draft manuscript (v0.5), the candidate Section 2.12 for the laboratory notes (the lumped wake rule for 3D vortex rings), the report of the first runs of the rule in the four-wing solver (hover and forward flight, 64 solver runs of 24 to 26 October 2026), the conceptual manual of those runs (each run: what was held fixed, the one change, the question, what was observed, what was learned, and the commands to repeat it), the run ledger, and the logs, figures and scripts. New in v0.5 against v0.4: the forward-flight reference run; the measurement that in hover the old rings grow as fast as they recede (so the old wake is never far); the checks of the spread of perturbed runs (a perturbation a thousand times smaller gives the same spread after three periods; the untouched run lies outside its own spread; a mirrored perturbation does not cure this); the spread of the lumped run itself (as wide as the untouched one); and the frozen-impulse result confirmed on a second family of perturbations (5 to 28 times narrower). The conclusion for the method: the hook that replaces old rings only as velocity sources cannot give a steady state; the equivalent ring must carry the impulse too (the removal run, not yet made).