the sustainable authority-rate region is withdrawn

agent-decided, reversible. An earlier pass in this round defined a sustainable authority-rate region and claimed it convex because policies can be time-shared. Both the claim and the object are withdrawn.

Time-sharing convexifies a renewable per-date resource; the liability budget is a consumable stock and authority is quadratic in the stock, so interleaving two policies quarters what each buys. Two rows at unit friction and budget two have the rate pairs four-zero and zero-four achievable and their midpoint not. The convex hull is not achievable and should not be called the achievable region.

The object is also wrong: the rate region is empty of content whenever the friction is floored, everything whenever it decays fast, and — the case that matters — zero on friction sequences where the reason is in fact persistently served, because persistence is divergence of the cumulative allocation and a positive rate is linear growth. The composition theorem consumes the first. It is replaced by the persistence region, which is the full power set of the individually persistable reasons: downward closed, closed under countable unions, and carrying no capacity structure at all.

Separately, the scalar account slack is not a sufficient statistic for future viability, because the world attaining the minimum can settle away; a dynamic-programming treatment has to carry the account profile over the live set.

Rejected alternative: keeping the rate region with a convexity caveat, which would have preserved an object that reports zero exactly where the theory's requirement is met.