"self-financing" is deleted; affordability is a signed account

agent-decided, reversible. The round claimed that a fixed liability budget funds only a transient and therefore that persistent service must be self-financing. The step interchanges a supremum with a sum: the account is the worst case of a cumulative value, while the hypothesis bounds the worst case of each increment, and different worlds may be worst at different dates. One priced sentence and two worlds refute it — a norm alternating which world it excludes has robust loss exactly 1 at every date, an account that oscillates inside a band of width 1, and unbounded total force.

Nor does the corrected worldwise form give self-financing: a norm may exclude the sole live world at every date, be enforced forever, lose at every date and stay affordable when the exclusion depth decays. The traderized round had already withdrawn a theorem of the same shape and had already recorded that the per-date-supremum certificate is conservative; this round reintroduced both errors and now removes them.

The necessary condition is the account condition itself — for every world live at the horizon, the cumulative value stays above the floor — and the sufficient certificates are the existing liability-regime taxonomy. "Self-financing" was a fifth name for the world-compatible regime, presented as necessary, and is deleted. The existence theorem that used it survives as a strong sufficient theorem with the necessity gloss removed.

Rejected alternative: repairing the observation by quantifying over a single persistent world, which is true but bounds the number of costly dates rather than the service mass, and needs a proportionality hypothesis to bound the mass.