Coupled Navigation: Multi-Node Viability in Shared Thermodynamic Sinks

Several nodes export entropy into one shared sink of finite capacity. We study when all their floors can be kept up to a finite horizon, with exogenous rates and no disturbance. Without a cap on the export rate, the answer is one inequality per step: the sink's cumulative supply must cover the sum of the nodes' individual overshoots. The pooled test that treats all nodes as one is necessary but not sufficient, because no node's slack can absorb another node's overshoot. Exporting as little as possible is then jointly optimal and needs no information about the other nodes; for one node the criterion coincides with the two-budget theorem of the sequel. With a shared cap this harmony fails. The smallest counterexample has two nodes and two steps: one node must export before its own floor requires it. Neither fully lazy nor eager (earliest-deadline-first) rules are optimal, whereas centralised feasibility remains a linear programme. For decentralised nodes, which see only their own data and the sink, we show that no largest set of situations can be served by one decentralised policy. We then measure, on exhaustive finite grids, how many situations decentralised policies can serve together, and how many are served by a guarantee that treats the other node as an adversary. On the two-step grid the centralised set has 147,067 situations. The best single decentralised policy serves 145,242 of them, and the worst-case guarantee serves 24,500. At three steps, observing the sink adds 102 of 93,492, through a signal carried by the timing of an export owed in any case. All numbers are counts on finite integer grids with equal weight per situation; none is a property of a physical system. Adversarial review. Before deposit, drafts of this paper went through three rounds of adversarial review by AI systems (DeepSeek and Meta AI). Each round used a written brief with numbered items and a standing instruction to find any false sentence; reviewers had to confirm the file's md5 and paste the output of any code they ran. Round C1 (2 November 2026) reviewed an outline; its cumulative form of the shared cap was adopted (Lemma 4.1), its instance pair was found broken although its conclusion held, and its proposed definitions of the decentralised sets were shown to be ill-posed or incoherent. Round C2 (3 November 2026) reviewed version 0.1; it found no false statement, contributed the counterexample now in Remark 3.3 (the per-step criterion needs non-negative gains), and corrected an overclaim in the comparison with Moser et al. Round C3 (4 November 2026) reviewed version 0.3 together with the scripts, their result files and the full text of Martin, Alvarez and Lavallée (2024); one reviewer confirmed the file's identity and regenerated, with pasted output, the two-step coverage (145,242), the centralised set (147,067), the guaranteed set (24,500), the incoherent-variant counts (146,683 / 30,011 / 34,690), the block optimum 876 and the one-node check. No false sentence was found. Not regenerated by any reviewer, and resting on the deposited scripts alone: the three-step counts (93,492; 88,882; 88,984; 102) and the signalling example; the minimality claim in Example 4.4 and the counts in Remarks 3.2 and 4.5; and the guaranteed set with known data (67,607 worlds) with the statements that it contains the guaranteed set and excludes every pure-contention world. All scripts are deterministic and their md5 checksums are listed in the paper.