Throughput Law in Off-Chain Payment Networks

• The paper establishes a quantitative relationship where sustainable off-chain throughput S is defined as ζ/ρ, linking on-chain settlement capacity with the rate of infeasible transactions.
• It models liquidity states and wealth distributions using network cut-intervals and multi-party channel topologies to derive critical performance bounds.
• The work highlights design strategies such as optimized fee structures and multi-party channels to enhance capital efficiency and scale off-chain payment networks.

The throughput law for off-chain payment channel networks establishes a quantitative linkage between sustainable transaction bandwidth in the off-chain layer and the base layer’s settlement capacity. It is predicated on the geometry of feasible wealth distributions within the network, incorporating the structure of liquidity states, cut-intervals, multi-party channel topologies, and fee dynamics. The core result, proven in "A Mathematical Theory of Payment Channel Networks" (Pickhardt, 8 Jan 2026), expresses sustainable off-chain payment bandwidth $\mathcal S$ as a function of on-chain settlement bandwidth $\zeta$ and the expected fraction of infeasible off-chain payments $\rho$: $\mathcal S = \frac{\zeta}{\rho}$ This formalism delivers a precise criterion for the scale at which an off-chain payment system can operate without overtaxing on-chain resources.

1. Mathematical Definitions and Network Model

The framework analyzes a fixed payment-channel network $G=(V,E,\{c_e\})$:

• $|V|=n$: Number of nodes.
• $|E|=m$: Number of edges.
• $\{c_e\}$: Capacity assigned to each edge, with total capital $C=\sum_e c_e$.

A liquidity state is formulated as a function $$\lambda: E \times V \rightarrow \{0, 1, \ldots, C\}$$, subject to the local conservation constraint $\zeta$0 for each undirected edge $\zeta$1. The set of all such states is denoted $\zeta$2, which is combinatorially an $\zeta$3-dimensional integer hyperbox. The projection $\zeta$4 maps liquidity states to wealth distributions $\zeta$5 with total sum $\zeta$6, where $\zeta$7 is the polytope of wealth vectors realizable off-chain as a valid distribution of channel balances.

A payment of amount $\zeta$8 from node $\zeta$9 to node $\rho$0 modifies the wealth vector $\rho$1 by $\rho$2, with the payment deemed feasible if $\rho$3 post-transfer.

2. The Throughput Law and Its Derivation

Let $\rho$4 indicate the base layer’s on-chain settlement bandwidth (transactions/sec). Let $\rho$5 represent the expected proportion of infeasible off-chain payment attempts, for a stationary demand model over amounts and endpoints. If the off-chain network issues $\rho$6 payments, approximately $\rho$7 will fail and incur on-chain fallback operations.

The law is derived as follows:

• Off-chain payment attempts per second: $\rho$8.
• Fraction $\rho$9 fail, thus requiring at least $\mathcal S = \frac{\zeta}{\rho}$0 on-chain operations/sec.
• Imposing the hard constraint $\mathcal S = \frac{\zeta}{\rho}$1, the maximum sustainable off-chain bandwidth is

$\mathcal S = \frac{\zeta}{\rho}$2

Under adaptively throttled demand, this equality is tight and precludes backlog or dropped on-chain requests.

3. Polytope Geometry and Cut-Interval Characterization

The space of feasible off-chain wealth distributions $\mathcal S = \frac{\zeta}{\rho}$3 is geometrically embedded within the simplex $\mathcal S = \frac{\zeta}{\rho}$4. For any $\mathcal S = \frac{\zeta}{\rho}$5 ($\mathcal S = \frac{\zeta}{\rho}$6, $\mathcal S = \frac{\zeta}{\rho}$7), the cut capacity is $\mathcal S = \frac{\zeta}{\rho}$8, where $\mathcal S = \frac{\zeta}{\rho}$9 comprises edges crossing from $G=(V,E,\{c_e\})$0 to its complement. The cut-interval lemma establishes: $G=(V,E,\{c_e\})$1 The width $G=(V,E,\{c_e\})$2 defines the feasible range for the wealth of $G=(V,E,\{c_e\})$3. Liquid transfers across $G=(V,E,\{c_e\})$4 must not violate this constraint to remain feasible, and max-flow/min-cut arguments determine the largest possible payment size across a given partition.

This geometric viewpoint directly connects infeasibility rates $G=(V,E,\{c_e\})$5 to the narrowness of cut-intervals: widening every cut reduces $G=(V,E,\{c_e\})$6 and increases throughput.

4. Multi-Party Channels and Topological Effects

Expanding to $G=(V,E,\{c_e\})$7-party channels (coinpools, channel factories), channel hyperedges of uniform capacity $G=(V,E,\{c_e\})$8 increase cut widths:

• Any $G=(V,E,\{c_e\})$9-party channel crossing $|V|=n$0 increments $|V|=n$1 by $|V|=n$2.
• For $|V|=n$3 random $|V|=n$4-subsets as channels, the crossing probability is $|V|=n$5, and expected cut capacity is $|V|=n$6.
• $|V|=n$7 is monotonic in $|V|=n$8: larger $|V|=n$9 yields wider cuts, a larger wealth polytope $|E|=m$0, a lower expected infeasible rate $|E|=m$1, and so higher $|E|=m$2.
• For single-node sets ($|E|=m$3), expected accessible wealth scales linearly with $|E|=m$4.

In the limit $|E|=m$5 (all nodes in one channel), all cuts are crossed and $|E|=m$6 spans the entire simplex $|E|=m$7, so $|E|=m$8 and the throughput law yields unbounded off-chain bandwidth subject to other systemic limits.

5. Fee Design and Liquidity Depletion Dynamics

While the throughput law omits fee effects, practical channel depletion is strongly fee-dependent:

• Linear asymmetric fees prompt routing algorithms to pursue minimum-cost cycles, driving liquidity to the boundary of $|E|=m$9 and depleting most channels apart from a residual spanning forest, thereby tightening cut intervals and increasing $\{c_e\}$0.
• Symmetric fees (identical per direction) nullify directional arbitrage and cycle depletion pressures.
• Convex/tiered fees (scarcity pricing) establish fee functions increasing with local liquidity scarcity. This yields convex cycle potentials, enabling strictly interior optimality and inhibiting total depletion; wider cuts persist, lowering $\{c_e\}$1. Realizing such fee structures necessitates source-routing or fee-quote mechanisms reflecting instantaneous state $\{c_e\}$2, as liquidity-dependent fees are locally hidden.

These dynamics play a critical role in the long-run capital efficiency and reliability of the network, by stabilizing operation within the feasible polytope.

6. Modeling Assumptions and Scope

The preceding analysis hinges upon several modeling conventions:

• The network topology $\{c_e\}$3 and channel capacities $\{c_e\}$4 are static throughout.
• Per-hop base fees and HTLC limits are omitted in the feasibility analysis.
• Payment demand distribution is stationary and stochastic, inducing a well-defined $\{c_e\}$5.
• On-chain throughput $\{c_e\}$6 is a hard constraint on new channel operations per second.
• No ex-ante liquidity probing or selection to avoid infeasible attempts; $\{c_e\}$7 is assessed strictly ex post.

Within these bounds, the throughput law encapsulates the fundamental constraint linking off-chain performance to on-chain limitations.

7. Capital Efficiency and Scaling Implications

The law $\{c_e\}$8 unifies off-chain and on-chain constraints through the infeasibility rate $\{c_e\}$9, determined by the geometry of $C=\sum_e c_e$0 (via cut intervals) and demand models. Capital efficiency levers—multi-party channels and refined fee designs—operate by either enlarging $C=\sum_e c_e$1 or regulating liquidity movement within the pre-image fibers $C=\sum_e c_e$2 to limit depletion and thereby shrink $C=\sum_e c_e$3. Achieving payment bandwidths comparable to traditional retail networks, e.g., Visa-scale with $C=\sum_e c_e$4 tx/s on Bitcoin, requires driving $C=\sum_e c_e$5 to near zero via topological optimization (coinpools, factories, mesh enrichment) and liquidity management (symmetric or convex fee design, coordinated replenishments).

Key attributes and their effects can be summarized as follows:

Mechanism	Influence on $C=\sum_e c_e$6	Effect on $C=\sum_e c_e$7 / Throughput
Multi-party channels	Enlarge polytope, widen cuts	$C=\sum_e c_e$8, $C=\sum_e c_e$9
Linear asymmetric fees	Deplete liquidity, tighten cuts	$$\lambda: E \times V \rightarrow \{0, 1, \ldots, C\}$$0, $$\lambda: E \times V \rightarrow \{0, 1, \ldots, C\}$$1
Symmetric/convex fees	Stabilize liquidity, balance cycles	$$\lambda: E \times V \rightarrow \{0, 1, \ldots, C\}$$2, $$\lambda: E \times V \rightarrow \{0, 1, \ldots, C\}$$3

The throughput law affords a rigorous design target for future off-chain network architectures and fee regimes aimed at maximizing reliability and transaction volume within sustainable capital and settlement bounds (Pickhardt, 8 Jan 2026).