The Autonomy of the Lightning Network: A Mathematical and Economic Proof of Structural Decoupling from BTC (2506.19333v2)

Abstract: This paper presents a formal analysis of the Lightning Network as a monetary system structurally diverging from Bitcoin's base-layer settlement model. We demonstrate that under increasing transaction demand, BTC transaction fees rise superlinearly due to throughput constraints, while Lightning Network routing costs approach a bounded asymptote. Using mathematical modeling, game-theoretic proofs, and complexity analysis, we show that Lightning enables indefinite off-chain operation via the emergence of liquidity hub oligopolies. These hubs exhibit properties of unregulated financial intermediaries, including rent extraction, opacity, and systemic fragility. Strategic agent models show that channel closure becomes economically infeasible, and routing problems approach hardness limits in P-Space complexity. We conclude that Lightning does not merely extend Bitcoin, but constitutes a synthetic financial system with shadowbank characteristics, lacking reserve discipline, transparency, or enforceable settlement guarantees.

• The paper uses formal analysis, cost asymptotics, game theory, and complexity proofs to demonstrate the Lightning Network's structural and economic decoupling from Bitcoin.
• Analysis shows that rational agent behavior, rising on-chain fees, and computational constraints lead to liquidity consolidation, the emergence of oligopolistic hubs, and rent extraction in the network.
• Findings suggest the Lightning Network evolves into a centralized, opaque financial layer akin to shadow banking, challenging its role as a decentralized scaling solution and introducing systemic risks.

The Autonomy of the Lightning Network: Structural Decoupling from BTC

This paper presents a rigorous, formal analysis of the Lightning Network (LN) as an off-chain overlay to Bitcoin’s base layer (BTC), focusing on the mathematical, economic, and computational mechanisms that drive LN’s structural and economic decoupling from the underlying blockchain. The work synthesizes axiomatic modeling, asymptotic cost analysis, game-theoretic reasoning, and computational complexity theory to demonstrate that, under realistic demand and rational agent behavior, LN evolves into a closed, rent-extracting, and increasingly centralised financial infrastructure, functionally distinct from the original Bitcoin protocol.

Formal System and Axiomatic Foundations

The analysis begins by defining a dual-layered system: a constrained base ledger (BTC) and a scalable overlay (LN), populated by rational agents (users, hubs, miners, watchtowers) and governed by explicit cost, utility, and routing functions. Four core axioms underpin the model:

1. On-Chain Throughput Constraint: The BTC base layer is capped at a fixed, low transaction throughput (empirically ≈5 TPS), independent of demand.
2. Unbounded Off-Chain Demand: LN can, in principle, support unbounded transaction demand, limited only by available liquidity and path feasibility.
3. Transactional Utility Positivity: All transactions confer strictly positive utility, ensuring rational participation.
4. Liquidity Consolidation: Under cost minimization, liquidity centralizes into a minimal subset of nodes (hubs), leading to emergent oligopoly.

This formalism enables precise derivation of cost structures, equilibrium outcomes, and theorems regarding the evolution of network topology and economic power.

Asymptotic Cost Analysis

A central result is the asymptotic divergence between BTC and LN transaction costs as demand increases:

• BTC Fees: Under rising demand, BTC’s fee market exhibits superlinear escalation due to blockspace scarcity and auction-based prioritization. As demand $D \to \infty$, marginal transaction cost $C_{BTC}(D) \to \infty$, pricing out all but high-value settlements.
• LN Routing Costs: LN’s marginal cost per transaction remains asymptotically flat under ideal liquidity, as channel opening costs amortize and routing fees stabilize. However, in practice, liquidity fragmentation and rebalancing introduce nontrivial costs, especially under topological centralization.

The crossover point, where $C_{BTC} \gg C_{LN}$, creates a powerful incentive for users to migrate to LN, reinforcing liquidity consolidation and hub dominance.

Game-Theoretic and Economic Dynamics

The paper models LN as a non-cooperative game among rational agents, with strategic hubs optimizing for fee extraction, liquidity deployment, and path control. Key findings include:

• Emergence of Oligopoly: As on-chain costs rise, users preferentially connect to high-liquidity hubs, which become indispensable for routing. This positive feedback loop (preferential attachment) leads to a power-law topology, with a small set of hubs mediating the majority of volume.
• Rent Extraction and Barriers to Entry: Dominant hubs set supracompetitive fees, exploiting informational asymmetries and liquidity dependence. Entry and exit barriers (capital lockup, channel opening costs, and exit friction under high fees) entrench incumbency and suppress competition.
• Shadow Banking Analogy: LN hubs function as unregulated financial intermediaries, issuing off-chain payment promises, managing liquidity, and extracting rents without reserve discipline or transparency. The system mirrors shadow banking in its opacity, risk displacement, and absence of regulatory oversight.

Computational Complexity and Scalability Limits

A significant contribution is the formal proof that decentralized route selection in LN, under liquidity and timelock constraints, is NP-complete. As the network grows, real-time routing becomes computationally intractable without centralised heuristics or privileged topology access. This computational barrier enforces a trade-off:

• Decentralized Topology: Cannot scale under strict complexity bounds; routing becomes unreliable and inefficient.
• Centralized Heuristics: Emergent centralization is an economic and computational inevitability, as dominant hubs aggregate information and liquidity to provide reliable routing.

This result directly challenges the narrative that LN can deliver scalable, decentralized payments at global scale.

Structural Decoupling and Systemic Risk

The paper demonstrates that, as LN matures, it becomes operationally and economically decoupled from the BTC base layer:

• Ledger Disconnection: Once liquidity is established and routing equilibria stabilize, the majority of transactions occur off-chain, with the base ledger relegated to an inert anchor for rare channel openings or catastrophic failures.
• Loss of Settlement Finality: LN’s conditional, revocable commitments lack the public, immutable finality of on-chain settlement. Under high fee regimes, enforcement of HTLCs becomes economically infeasible, undermining the security model.
• Systemic Fragility: The absence of reserve discipline, transparency, and circuit breakers exposes LN to cascading failures, liquidity crises, and rent-seeking equilibria. The system is vulnerable to adversarial routing, liquidity hoarding, and strategic exclusion.

Implications and Future Directions

The findings have profound implications for the design and governance of layered cryptocurrency systems:

• Scalability via Layering: LN does not solve the scalability problem; it transforms it into a problem of economic enclosure, centralization, and systemic risk.
• Necessity of Base-Layer Scaling: Sustainable, auditable, and inclusive payment systems require scalable base layers with transparent settlement, not opaque overlays.
• Policy and Design Interventions: Mitigating LN’s centralization and fragility would require enforceable liquidity transparency, dynamic topology incentives, and mechanisms for auditability and exit—measures that may conflict with privacy and decentralization goals.

Strong Numerical and Theoretical Claims

• Superlinear Fee Growth: BTC fees diverge as $D \to \infty$, empirically reaching $60–$100 per transaction under stress, excluding microtransactions.
• Oligopolistic Topology: Empirical studies confirm that a small subset of LN nodes route the majority of volume, consistent with power-law degree distributions.
• NP-Completeness of Routing: Decentralized, real-time routing under liquidity constraints is formally intractable, necessitating centralization for practical operation.

Contradictory and Bold Claims

• LN as a Shadow Banking System: The paper asserts that LN is not a neutral scaling layer but a closed, rent-extracting financial overlay, structurally akin to shadow banking and subject to the same systemic risks.
• Decoupling from BTC: LN can function indefinitely without recourse to the base chain, severing the economic and security coupling that underpins Bitcoin’s original design.
• Final Settlement Illusion: LN does not provide final settlement; it replaces objective consensus with subjective coordination and probabilistic enforcement.

Theoretical and Practical Implications

Theoretically, the work challenges the prevailing assumption that off-chain overlays can preserve the decentralization and security properties of their base layers. Practically, it warns that the adoption of LN as a primary scaling solution risks reintroducing the very centralization, opacity, and systemic fragility that Bitcoin was designed to eliminate.

Future developments in AI and distributed systems may focus on:

• Algorithmic Liquidity Management: AI-driven strategies for dynamic liquidity allocation and path optimization, though bounded by computational complexity.
• Transparent, Scalable Ledgers: Research into base-layer protocols that support high throughput, auditability, and composability without reliance on opaque overlays.
• Economic Mechanism Design: Incentive structures that prevent rent extraction and centralization, possibly through cryptoeconomic proofs of liquidity or decentralized fee markets.

In summary, the paper provides a comprehensive, formal critique of the Lightning Network’s economic and computational architecture, demonstrating that its autonomy from BTC is not a feature but a structural inevitability with profound consequences for the future of decentralized finance.