Durable Coupled Navigation: Exact Sets and Guarantees for Nodes Sharing a Sink

Paper 3 of this programme studied several nodes that export entropy into one shared sink, up to a finite horizon. This paper treats durable viability, and the worst-case guarantee of Paper 3, when the field moves according to a known map. For the centralised system, the exact set of the sequel carries over to n nodes through one absorbing state. On an integer lattice it is described by a critical sink reserve, the minimum initial credit of an energy game, attained by a policy that ignores the sink level. Without a cap on the export rate it is a cumulative budget along every admissible path, attained by least export. For two decentralised nodes under the cap min(c, E), the guarantee of Paper 3 is determined by the greatest fixed point of a monotone operator on knowledge positions. A fixed point on the product state is a larger object: on Paper 3's two-step grid it holds 67,607 worlds instead of 24,500. With two nodes and public rates the knowledge game collapses. A node is guaranteed if and only if it is guaranteed against the other node at the top of the other's possible stocks; when there is no top, any winning upper bound suffices, and one number of memory is enough. With private rates the collapse can fail. A threat bound shows that when the cap does not exceed the other node's reservoir range, and the other may be at its ceiling, a node can be guaranteed only where zero export keeps its own floor. On Paper 3's grids every guaranteed policy exports nothing within the horizon. All counts are on finite integer grids with equal weight per world; none is a property of a physical system. Code, data and review record. The 15 scripts and 13 result files deposited here are listed with their md5 checksums in Section 10 of the PDF, which also states which scripts reviewers have rerun. Section 7 describes the grids and what the counts do not show. The declaration of AI assistance and the version history, at the end of Section 10, record the review rounds and the corrections of this version, which have not been reviewed.