We introduce a geometric theory of payment channel networks that centers the polytope $W_G$ of feasible wealth distributions; liquidity states $L_G$ project onto $W_G$ via strict circulations. A payment is feasible iff the post-transfer wealth stays in $W_G$. This yields a simple throughput law: if $ζ$ is on-chain settlement bandwidth and $ρ$ the expected fraction of infeasible payments, the sustainable off-chain bandwidth satisfies $S = ζ/ ρ$. Feasibility admits a cut-interval view: for any node set S, the wealth of S must lie in an interval whose width equals the cut capacity $C(δ(S))$. Using this, we show how multi-party channels (coinpools / channel factories) expand $W_G$. Modeling a k-party channel as a k-uniform hyperedge widens every cut in expectation, so $W_G$ grows monotonically with k; for single nodes the expected accessible wealth scales linearly with $k/n$. We also analyze depletion. Under linear, asymmetric fees, cost-minimizing flow within a wealth fiber pushes cycles to the boundary, generically depleting channels except for a residual spanning forest. Three mitigation levers follow: (i) symmetric fees per direction, (ii) convex/tiered fees (effective flow control but at odds with source routing without liquidity disclosure), and (iii) coordinated replenishment (choose an optimal circulation within a fiber). Together, these results explain why two-party meshes struggle to scale and why multi-party primitives are more capital-efficient, yielding higher expected payment bandwidth. They also show how fee design and coordination keep operation inside the feasible region, improving reliability.

翻译：我们提出了一种以可行财富分布多面体 $W_G$ 为核心的支付通道网络几何理论；流动性状态 $L_G$ 通过严格循环映射到 $W_G$ 上。当且仅当转移后的财富仍位于 $W_G$ 内时，支付才是可行的。这导出了一个简单的吞吐量定律：若 $ζ$ 为链上结算带宽，$ρ$ 为不可行支付的预期比例，则可持续的链下带宽满足 $S = ζ/ ρ$。可行性具有割区间视角：对于任意节点集 S，其财富必须位于一个区间内，该区间的宽度等于割容量 $C(δ(S))$。基于此，我们展示了多方通道（资金池/通道工厂）如何扩展 $W_G$。将 k 方通道建模为 k-一致超边可在期望上拓宽每个割，因此 $W_G$ 随 k 单调增长；对于单个节点，其预期可访问财富与 $k/n$ 成线性比例。我们还分析了耗尽问题。在线性非对称费用下，财富纤维内成本最小化的流动会将循环推至边界，通常导致除残余生成森林外的通道耗尽。由此得出三种缓解机制：（i）每方向对称费用，（ii）凸/分层费用（有效的流量控制，但在不披露流动性的源路由中难以实现），以及（iii）协调补充（在纤维内选择最优循环）。这些结果共同解释了为何双方网状网络难以扩展，以及为何多方原语具有更高的资本效率，从而产生更高的预期支付带宽。它们还展示了费用设计与协调如何使操作保持在可行区域内，从而提高可靠性。