A Mathematical Theory of Payment Channel Networks

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 $\zeta$ is on-chain settlement bandwidth and $\rho$ the expected fraction of infeasible payments, the sustainable off-chain bandwidth satisfies $S = \zeta / \rho$. 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 $C(\delta(S))$. 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.