Geometric Theory of Payment Channels

• The geometric framework rigorously characterizes off-chain payment feasibility using convex polytopes and combinatorial optimization.
• It quantifies liquidity and throughput by mapping network states to wealth polytopes and zonotopes derived from channel capacities.
• Geometric routing techniques employ virtual embeddings and Voronoi structures for scalable, privacy-focused, near-optimal path selection.

The geometric theory of payment channel networks (@@@@1@@@@) unifies the analysis of off-chain transaction feasibility, routing, scalability, and liquidity through the geometric properties of convex polytopes and combinatorial optimization. This framework rigorously characterizes which payments can be routed over a given topology and liquidity allocation, how multi-party extensions alter throughput, and how specific routing, fee, and rebalancing protocols interact with the fundamental geometric structure. Advances include polytope-based throughput laws, fast state sampling, and distributed virtual embeddings for scalable, privacy-preserving routing.

1. Geometric Models: Liquidity, Wealth, and Feasibility Polytopes

Let $G=(V,E,\mathrm{cap})$ be a PCN graph with $n$ nodes, $|E|=m$ undirected edges of capacities $c_e$, and total on-chain coins $C$. A liquidity state $\lambda$ assigns to each edge $e=(u,v)$ the pair $(\lambda(e,u), \lambda(e,v))$ subject to $\lambda(e,u)+\lambda(e,v)=c_e$ and $0\le \lambda(e,u)\le c_e$. The set of all such $\lambda$ forms the integer hyperbox $L_G \cong \prod_{e\in E}\{0,\dotsc,c_e\}$ (Pickhardt, 8 Jan 2026).

Each $\lambda$ induces a wealth vector $w\in\mathbb{Z}^n$ with

$$w_v = \sum_{e\ni v} \lambda(e,v), \qquad \sum_v w_v = C.$$

The set of feasible wealth distributions $W_G \subset \mathcal{W}(C,n)$ is precisely the projection of $L_G$, i.e., only those $w$ attainable by redistributing coins along channel restrictions. Consequently, $W_G$ is a polytope in $\mathbb{Z}^n$, typically a thin subset of the simplex $\mathcal{W}(C,n)$ imposed by topological bottlenecks.

A payment $i\to j$ of size $a$ is feasible off-chain if and only if the new $w'=w - a\,b_i + a\,b_j$ remains in $W_G$. Otherwise, on-chain settlement is necessary. The throughput of off-chain payments is thus limited by the volume and structure of $W_G$ (Pickhardt, 8 Jan 2026, Goel et al., 2020).

2. Cut-Intervals, Capacity, and Multi-party Channels

For any node set $S\subset V$:

• Internal edges: $E[S]$
• Cut edges: $\delta(S)$ (edges with one endpoint in $S$, one in $V\setminus S$)
• Total internal capacity: $\sum_{e\in E[S]} c_e$
• Cut capacity: $C(\delta(S)) = \sum_{e\in\delta(S)} c_e$

A fundamental result is the cut-interval lemma (Pickhardt, 8 Jan 2026):

$$\sum_{e\in E[S]} c_e \;\leq\; \sum_{v\in S} w_v \;\leq\; \sum_{e\in E[S]} c_e + C(\delta(S)),$$

so possible wealth in $S$ must stay within an interval whose width equals the cut capacity. Thus, bottlenecks in $C(\delta(S))$ directly restrict feasible transfers across $S$.

Multi-party channels (coinpools, factories) are modeled as $k$-uniform hyperedges. For $k$-party channels of capacity $c$, the expected number of hyperedges crossing $S$ (of size $s$) is $$q_k(s) = 1 - \frac{\binom{s}{k} + \binom{n-s}{k}}{\binom{n}{k}}$$. Expected accessible liquidity per node then scales like $k/n$, widening $W_G$ monotonically in $k$. In the limit $k\rightarrow n$, $W_G$ approaches the full simplex, and bottleneck effects vanish (Pickhardt, 8 Jan 2026).

Channel Model	Polytope Width Growth	Typical $W_G$ Volume
Two-party (edges)	Small, local	Sliver in simplex
$k$-party (hypered.)	Monotonically in $k$	Approaches simplex as $k\uparrow n$

This analysis confirms the capital-efficiency gains of multi-party payment channel constructs.

3. Sampling, Liquidity Volumes, and the Monotonicity Law

Every allocation can be mapped to a point in a convex body $K(G,c)\subset\mathbb{R}^n$:

• Coordinates represent edge-based ownership satisfying cycle sum constraints per the spanning-representation of the graphic matroid (Goel et al., 2020).
• $K(G,c)$ is a zonotope, defined as $$\{x\in\mathbb{R}^m \mid 0\leq x_e\leq c_e, \sum_{e\in C} s_C(e)x_e=0\,\forall\,\textrm{cycles}\;C\}$$.

The liquidity—i.e., success probability for a transfer of size $k$ along edge $e$—is:

$$\Pr[\text{success on }e] = \frac{\Gamma(c_1,\dotsc,c_e-k,\dotsc,c_m)}{\Gamma(c_1,\dotsc,c_m)}$$

where $\Gamma(c)$ is the weighted spanning-tree generating polynomial, providing a combinatorial characterization of liquidity as zonotope (polytope) volume (Goel et al., 2020).

The monotonicity theorem follows from the Rayleigh property of graphic matroids: increasing any edge capacity $c_f$ can only increase $\operatorname{Vol}(K(G,c))$ and cannot decrease any transaction success probability.

There exists an exact $O(m\beta(m))$ algorithm for uniform sampling over $K(G,c)$, using a recursive shell-vs-core procedure and effective resistances, significantly outperforming generic convex sampler approaches (Goel et al., 2020).

4. Geometric Routing: Virtual Embedding, Delaunay, and Voronoi Structures

Scalable, privacy-oriented routing in PCNs is enabled by geometric virtual embeddings (Zhang et al., 2021). The principal workflow resembles:

1. Virtual Euclidean Embedding: Nodes are assigned coordinates $f: V \to \mathbb{R}^d$, approximating graph distance by Euclidean distance via multidimensional scaling (MDS) anchored on $k=d+1$ reference nodes.
2. Multi-hop Delaunay Triangulation (MDT): Construct the Delaunay triangulation (DT) and maintain at each node $u$:
   • $C_u$: direct neighbors,
   • $N_u$: DT-neighbors in $C_u$,
   • $F_u$: soft-state tables of multi-hop paths to $N_u\setminus C_u$.

Routing proceeds by greedy forwarding on the DT in $\mathbb{R}^d$: each node relays to the neighbor (physical or virtual) closest to destination in embedding space. Delivery is guaranteed whenever channel balances permit.

1. Distributed Voronoi Routing (WebFlow-PE): To enhance privacy, routing follows a randomly determined line $\ell$ in embedding space (not the destination point), so intermediates learn only direction. The path is composed of sequential Voronoi cells, interleaving local geometry and global anonymity.

Routing Approach	Per-node State	Stretch	Privacy
MDT-WebFlow	$O(\Delta\cdot H)$	$1.2-1.5$	Moderate
WebFlow-PE	$O(\Delta\cdot H)$	Comparable	High (entropy $>0.8$)

Empirical results demonstrate low per-node memory ($\approx$15 neighbors), probe messages per $1000$ tx ($\approx$5000--$6000$), and high success ratios ($\gtrsim$87%), all with minimal performance degradation under increased load (Zhang et al., 2021).

5. Throughput Laws, Depletion, and Fee-driven Dynamics

A geometric throughput law states that if $\zeta$ is on-chain settlement bandwidth (tx/sec), and $\rho$ is the rate of infeasible off-chain payments (requiring fallback), then sustainable off-chain transaction rate $S$ satisfies:

$S = \frac{\zeta}{\rho}$

Thus, maximizing $|W_G|$ directly enhances throughput by reducing $\rho$ (Pickhardt, 8 Jan 2026).

Channel depletion arises generically under linear, asymmetric fees: cost-minimizing (rational) cycles push flow to boundary states, depleting channels until only a residual spanning forest survives. This restricts liquidity and shrinks the effective $W_G$ over time.

Three geometric mitigation levers are:

1. Symmetric fees: Directional symmetry annihilates net cost for any cycle, preventing depletion.
2. Convex (tiered) fees: Scarcity pricing ensures most cycles "stall" at interior points, inhibiting channel exhaustion.
3. Coordinated replenishment: Participants coordinate to solve for an interior circulation closest to a balanced reference, efficiently restoring usability without on-chain intervention (Pickhardt, 8 Jan 2026).

Fee Model	Equilibrium	Generic State
Linear, asymmetric	All cycles at boundary	Spanning forest remains
Symmetric/tiered	Interior stall points	Liquidity broadly spread
Coordinated	Targeted rebalancing	High interior utilization

6. Connections with Matroid Theory and Fast Algorithms

The underlying combinatorics is governed by the graphic matroid of $G$:

• Equivalence classes of liquidity allocations correspond to points in $K(G,c)$ modulo cycle-routings.
• The Rayleigh property and negative correlation underpin monotonicity of liquidity.
• Spanning-tree polynomials encode polytope volume and transaction success rates.

An $O(m\beta(m))$ sampling algorithm enables efficient Monte Carlo estimation and statistical simulations, a key technical advance beyond earlier, slower Markov chain approaches (Goel et al., 2020).

7. Synthesis and Outlook

The geometric theory establishes that the performance and robustness of payment channel networks are fully characterized by polytopes $L_G$, $W_G$, and zonotopes $K(G,c)$ constructed from network topology and channel capacities. Cut-interval and matroidal constraints govern feasible wealth allocation. Multi-party primitives monotonically increase network liquidity and capital efficiency. Geometric routing leveraging virtual embeddings and Delaunay/Voronoi structures achieves scalable, privacy-preserving, and near-optimal path selection. Fee and rebalancing strategies, when aligned with the geometry, can maintain the network operationally near the interior of $W_G$, maximizing resilience and throughput (Pickhardt, 8 Jan 2026, Zhang et al., 2021, Goel et al., 2020).

This framework rigorously explains the limits, trade-offs, and protocol design space for Layer 2 payment channel networks, providing essential theoretical tools for both analytical and applied research.