Trim: Fee Level Matters More Than Skew in Lightning Routing

• Operators’ tools adjust fees with balance around hand-set or inherited levels, and major hubs reset those levels a median 2.3×, 1.7 times per year.The measurement covers 2019–2023 gossip and finds most channels reset at once.Source: §1 · p. 1
• Trim models a channel as two-sided inventory: the level sets priced demand and flow balance, while the skew absorbs balance fluctuations.Demand depletes balance; replenishment or a costly refill restores it.Source: §1 · p. 1, Figure 1 · p. 1
• Trim’s contributions bound revenue by demand and replenishment, limit the skew’s value, and propose tracing a revenue–success frontier with uniform levels before selectively adding skew.The introduction also reports that tuned deployed-tool reimplementations lie on or near that frontier, with little added value from balance rules at equal success.Source: §1 · p. 1
• A well-chosen flat fee comes within 1.6% of the optimum at the median on 144 exactly solved channels, while a level off by a factor of two loses tens of percent.On 2,592 additional channels with other size laws and demand curves, the median gap is 1.1–2.3%.Source: §1 · pp. 1–2

• Lightning senders choose fee-minimizing routes while weighing success probabilities learned from failures, making a node’s demand price-sensitive.Channels post a base charge plus a proportional rate; since LND 0.18, nodes can also post inbound fees, initially only as discounts.Source: §2 · p. 2
• Deployed Lightning tools tune a balance-related fee term while inheriting or searching for the fee level.Examples include charge-lnd’s local-share interpolation, feeadjuster’s scaled base rate, LNDg’s threshold steps, and CLBOSS’s competitor-based scaling.Source: §2 · p. 2
• Existing pricing mechanisms include balance surcharges, online-packing dual prices, network-wide welfare-oriented dual descent, and node-profit controllers that are not deployed.Debt’s prices are multipliers of flow conservation, whereas Trim’s balance term is one node’s marginal value of its own balance and the level is the incentive component Debt leaves to practice.Source: §2 · p. 2
• Inventory-pricing work includes marginal-value-plus-markup prices, fixed-order-cost policies, drift control, and inventory-skewed market-maker quotes.A channel differs because inventory is bounded on both sides and replenishment prices are set by others; flow balance is its zero-drift point.Source: §2 · p. 2
• Rebalancing and admission control address liquidity with fees fixed, while other work studies hub equilibria, channel games, network outcomes, and depletion levers.The paper leaves open where levels settle when all hubs reprice.Source: §2 · p. 2
• Trim’s focus is a deployable two-layer fee for one node, rather than routing, rebalancing, or equilibrium design.The paper says it knows of no prior work deriving both layers and asking which drives revenue.Source: §2 · p. 2

• The model treats a channel’s node-side balance b∈[0,c] as inventory, with demand arriving at rate λ(f) that falls with the proportional fee and replenishment arriving at exogenous rate μ.The canonical demand is λ(f) = Λe^(−κf); the simulator’s fitted demand is close to this near deployed fees but mixes captive and competed demand farther away.Source: §3 · p. 2
• The objective is long-run average net revenue: fees on served demand minus refusal penalties and refill costs.A refused forward incurs a goodwill penalty plus forgone fee, while a refill of size Δ costs K+kΔ; feasible forwards must be served.Source: §3 · p. 2
• Demand and replenishment are represented as marked point processes with size laws, and refills add positive amounts at specified times.The balance is bounded between zero and channel capacity.Source: §3 · p. 2
• The demand assumption requires a continuously differentiable, decreasing rate with fee-times-demand tending to zero; its markup is defined from demand and its derivative.The passage defines ψ(f) := −λ(f)/λ′(f).Source: §3 · p. 2
• The model omits time-locks, HTLC limits, multi-part payments, and channel closures, and assumes refills restore balance instantly while demand depends only on the channel’s own fee.It also omits inbound fees; an exact extension makes inbound fee a second level, while the passage says level errors still cost more than state dependence adds.Source: §3 · p. 3

4.1 Two layers

• The optimal fee splits into a level at the zero-drift balance and a balance-dependent remainder, the skew.The level is the optimal fee at the balance where drift vanishes; the skew is the difference between the optimal fee at the current balance and that level.Source: §4 · p. 3, §4.1 · p. 3

4.2 The level is first order

• No fee rule beats the revenue ceiling set by demand and replenishment; when flow balance determines the ceiling, it equals the fee income at flow balance.The exact ceiling applies to any fee rule and refill policy under the stated assumptions.Source: §4.2 · pp. 3–4
• A level below flow balance drains a channel, while one above it starves demand; when demand outruns replenishment, the markup level loses 6–99% on six exact instances.The level is set by demand and replenishment, and the cost of a level error can exceed what a skew adds.Source: §4.2 · pp. 3–4

4.3 The skew is second-order insurance

• A skew adds at most what the best flat fee loses to refused forwards, and can instead protect against a wrong level that would drain the channel.Its protection is useful when refills are costly; the best flat fee’s loss is first order in the level error, while the optimum’s slack is higher order in the stated diffusion regime.Source: §4.3 · p. 4

4.4 Exact check

• A best flat fee comes within 1.6% of the optimum at the median across 144 exact channels, while a factor-two level error loses at least 17.6%.The factor-two error loses 40.5% at the median and at least eight times what any balance-dependent fee adds on the same channel.Source: Figure 2 · p. 4, §4.4 · p. 5
• Across 2,592 exact cases with other size laws and demand curves, the best flat fee’s median gap is 1.1–2.3%; the anchored skew barely helps at half or twice the best level.At the best flat fee, the anchored skew cuts the median gap from 1.6% to 0.5%; at twice the fee it changes nothing at the median, and at half it gains 0.3 points.Source: §4.4 · p. 5

5 The Level at a Hub

• Under the mean-field hub model, a uniform level is optimal when refills and refusals are rare; levels differ across channels when refills bind or demand shapes differ.When refills bind, the relevant channel’s flow-balancing fee determines whether it drains at the common level.Source: §5 · p. 5
• The exact hub tests use two- or three-channel systems coupled to the rest of a hub, and are intended to test the mechanism rather than represent real hubs.The model replaces each channel’s replenishment with its time average under the posted fees.Source: §5 · p. 5
• Exactly solved balanced-flow hubs with equal demand shapes have channel-specific best levels within 3% of the uniform level, which add at most 2.0 points.State dependence is nevertheless worth 11–16% in these cases, where each forward moves two balances and is 5–10% of capacity.Source: §5 · p. 5
• When a channel drains or demand shapes differ, uniform levels lose 22–42% of the optimum at the median; per-channel levels cut that loss to 11–13%.A skew on the uniform level does not recover the loss and can produce a net loss.Source: §5 · p. 5, Figure 3 · p. 5

6 The Trim Rule: Choosing the Level

• A large node traces its revenue–success frontier with several uniform levels, while a small or solo-searching node uses a model’s or similar hub’s predicted level.The level cannot be computed from public data alone because demand and κ are unobserved before a scan.Source: §6 · p. 6
• Post one uniform level on every channel, keep skew off by default, and stop other fee tools from writing the ppm field.A level above the revenue peak earns no more at lower success, and sender fee limits cap the level where they bind.Source: §6 · p. 6
• Add anchored skew only on channels where refills remain frequent and costly; with per-amount refill pricing, it has no measurable net value in the network.A measurable net gain requires fixed-trigger refills at K ≥ 1,000 sats; with per-amount pricing, net revenue changes by at most 0.1k.Source: §6 · p. 6
• The rule needs only the ppm field for the level; skew requires an operator-written program to read channel balances and post the adjustment.Skew is fitted only for busy channels with enough traffic and posted when balance moves by 5% of capacity.Source: §6 · pp. 6–7
• A level search should probe within [L/2, 2L], and a chosen level can be held by reverting a move that loses over twice the spread of the last eight probe pairs.Large nodes need about 32–40 thousand own forwards per level when levels differ by 4%; pooled forward-count probes need only a few hundred forwards per probe.Source: §6 · p. 6

7 Network Evaluation

• The 2026 snapshot has 5,507 nodes and 27,063 channels, but only the payments are synthetic; 40 largest hubs run candidate policies across 10 evaluation seeds.Each run routes 12,000 payments averaging 41k sats; topology, fees, and tool rules are real, while capacities are scaled to induce depletion.Source: §7 · p. 7

7.4 Tuned tools mainly choose a level

• No ppm-tuned tool beats the uniform-level frontier by more than 3%, and skew adds less than 3% under mission control and 5% with shared memory.Tests use 10 disjoint evaluation seeds with paired 95% intervals and Holm correction; Table 3 reports revenue, payment success, and gap to the frontier.Source: §7 · p. 7, Table 3 · p. 7, §7.4 · p. 8

7.1 The level is first order

• Uniform levels trace a revenue–success frontier: moving from deployed fees to L* raises hub revenue from 7.0k to 118.9k with shared memory and 38.6k to 130.1k under mission control.At L*, refill counts per run fall by 84–98%; uniform levels also keep refills per thousand forwards below deployed non-uniform fees.Source: §7.1 · p. 7

7.2 The skew is second-order insurance

• At deployed fees, skew cuts forced refills by 21% with shared memory and 5% with mission control, but changes gross revenue by at most a few percent.On a level, hubs refill only 2–9 times per run, leaving little for skew to save; the −21% gap reflects lower success on the frontier’s steep low end, not lower revenue.Source: §7.2 · p. 8

7.3 Choosing the level explicitly

• Pooling forwards across 40 hubs finds the level within 24,000 payments, whereas per-hub probes overshoot and cost 4–29% of revenue against posting L*.Network-payment probes are too small to identify the level, and a guard holds a level but does not find one.Source: §7.3 · p. 8

7.4 Tuned tools mainly choose a level

• Tools centered near L* come within 1.5–10.5% of the frontier at equal success; when tuned as multiples of deployed fees, they lie 48.8–57.5% below it.Across 2026 settings, the best tool’s estimate is at most +2.9%; tools win clearly only on the 2019 snapshot with balance-aware senders.Source: §7.4 · p. 8, Table 4 · p. 8

8 Limitations and Conclusion

• Synthetic payments, exponential acceptance, and stationary demand constrain how directly the simulated peak transfers to real traffic.The frontier is an empirical envelope of uniform levels, and the tools are reimplementations tuned on the authors’ seeds; live-node traffic could yield better tool performance.Source: §8 · p. 8
• With balance-aware senders on the 2019 snapshot, tuned charge-lnd earns 14–24% more than the frontier at its success; this gain reflects channel price discrimination, not skew.Even in that setting, skew on a uniform level changes revenue by −1.7 to +2.6%.Source: §8 · pp. 8–9
• At L*, total surplus lost is 0.78 and 0.99 of hubs’ net gain; counting refused senders as total losses makes these upper bounds.At 1,000 ppm, surplus losses are smaller: 2.0 ± 0.6% and 0.5 ± 0.3%, compared with 7.3 ± 1.3% and 6.7 ± 0.9% at L*.Source: §8 · p. 9
• The level should be chosen on a revenue–success frontier, while skew is valuable chiefly for reducing refills where they are costly; sender responses and all-hub repricing remain open.Tuned tools beat a flat level at equal success by at most a few percent, except with 2019’s balance-aware senders.Source: §8 · p. 9

A.1 Proofs for Section 4

• In the diffusion regime, the shadow price and optimal fee are nonincreasing with balance, so depleted channels are charged more than flush channels.The optimal fee is the shadow price plus the demand markup; its skew slope is tied to the gap between the revenue ceiling and optimal reward at flow balance.Source: §A.1 · p. 11

A.2 Exact single-channel solutions

• On six exact single-channel instances, the diffusion ODE curve loses at most 0.3% of optimal revenue, versus 0.7–5.4% for the best constant fee.The broader exact-solution grid reports small median flat-fee gaps, but the ODE comparison covers only those six instances.Source: §A.2 · p. 12
• Across 2,592 instances, the flat-fee gap is 1.1–2.3% at the median; on 684 instances meeting stated conditions, the optimum never exceeds the revenue ceiling.The ceiling result applies when the demand-maximizing fee is below the flow-balancing fee and refill cost is at least the optimal shadow price.Source: §A.2 · p. 12
• An inbound fee’s best rate is 1/κ less the marginal value of incoming balance, so the balance skew enters outbound and inbound fees with opposite signs.Across six instances, a balance-dependent pair adds 3.8–8.3% over the best flat pair.Source: §A.2 · p. 12

A.3 The hub

• For flat hub fees, the aggregate revenue-maximizing level is independent of channel replenishment rates when refusals and refill costs are rare.The hub proof derives channel levels from average replenishment and identifies the shared level under these conditions.Source: §A.3 · p. 12

B Evaluation Design

• The evaluation calibrates demand from a scan of 15 uniform levels and tests the predicted peak at 4,990 ppm against the scan.The prediction stays within 0.02 of scan revenue relative to its peak at every level; the 2026 snapshot has 5,507 nodes and 27,063 channels, while payments are synthetic.Source: B Evaluation Design · p. 13

B.1 Full design and results

• The tuned tools are 2–11% below the wider frontier, and no tool earns significantly more than the flat level at its center.Choosing frontier levels on other seeds changes tool gaps by at most 0.8 points.Source: B Evaluation Design · p. 13
• On the 2019 snapshot, the level earns 3.6–15.5× deployed-fee revenue, while skew cuts refills by 10–43%; the tool gains occur with balance-aware senders.The reported 2019 evaluation uses real capacities and 9,000 payments; those tools earn less than deployed fees on that snapshot.Source: §B.1 · p. 13

C The Level

• Across the 336 snapshots, the largest hubs changed their fee level 1.7 times yearly by a median 2.3×; 50% of pooled log-fee variation was level changes over time.After six months, 83% of fee raises were kept or extended, versus 60% of cuts.Source: C The Level · p. 13

D Exactly Solved Hubs: Rates, Solver and Three-Channel Hubs

• In the exact three-channel hubs, draining cases lose 33–39% under a uniform level, 9–12% with per-channel levels, and 36–67% with skew on the uniform level.In balanced-flow cases, channel-specific levels match the uniform level and add at most 1.6 points.Source: D Exactly Solved Hubs: Rates, Solver… · p. 14

E Hubs Repricing Alone and Competition

• When hubs reprice alone, 1,000 ppm raises revenue for seven of nine hubs under mission control and all five tested hubs with shared memory.When hubs reprice together, most gain by cutting from L*, taking revenue from other hubs without increasing total hub earnings.Source: C The Level · p. 13, E Hubs Repricing Alone and Competition · p. 14

F Per-Hub and Per-Class Levels

• Per-hub and per-channel-class levels do not improve evaluation-seed revenue over L*; seven separate mission-control gains of 8.3% shrink to 0.7% together.The tested draining-channel class split earns no more than L* on calibration seeds and loses on evaluation seeds.Source: F Per-Hub and Per-Class Levels · p. 14

G Skew Amplitude and Forward Sizes

• At the volume-weighted median hub channel, the size-biased mean forward is 6% of capacity, with a 3–13% interquartile range.It is 18% at the median hub channel that carries traffic.Source: G Skew Amplitude and Forward Sizes · p. 14

H Level Search Details

• Per-hub probes of 6,000 payments settle only 9–15 of 40 hubs on average and earn 12–35% less than holding L* throughout.The authors attribute the upward level drift to too few forwards per hub probe, which biases the estimated elasticity low.Source: H Level Search Details · p. 15
• With all 40 hubs searching together, the stopping level depends on its start; the guard earns 5–10% more than unguarded search from L* and 8,000 ppm.From every tested start, guarded search stays within 8% of holding that start level.Source: H Level Search Details · p. 15

I Success-Constrained Tool Picks

• The success-constrained tool comparison selects each family’s best configuration whose mean success matches uniform 1,000 ppm, then scores it against the frontier.The score is gross revenue less 100 sat per refill, evaluated on seeds 0–9.Source: I Success-Constrained Tool Picks · p. 15

J Welfare and Who Pays

• At L*, total surplus falls 7.3 ± 1.3% with shared memory and 6.7 ± 0.9% under mission control, while sender surplus falls 16–18%.The paper treats refused senders as total losses, so these welfare-loss estimates are upper bounds.Source: J Welfare and Who Pays · p. 15
• At L*, hubs forward 80–93% less, while network forwarding falls 16–18% as payments take detour routes around the hubs.The reported detour-route comparison describes forwarding volumes, not total welfare.Source: J Welfare and Who Pays · p. 15
• Scaled to 6.6 million routed payments, the 40 hubs earn 0.03–0.17% yearly on channel-side capital at deployed fees and 0.5–0.6% at L*.The paper labels this a scale check, not a calibration, and notes that it counts idle channels at 2023 volumes.Source: J Welfare and Who Pays · p. 15

K Proposition A.1 Against the Exact Jump Model

• With the flow-balance point well above the refill trigger, the identity holds within 0.13%; where refills supply inventory, it fails even in sign.The table compares the identity against exact jump-model slopes and revenue losses at K = 0.05.Source: Table 6 · p. 15, K Proposition A.1 Against the Exact… · p. 15
• Under mission control, repricing alone to 1,000 ppm raises revenue for seven of nine hubs by 1.2–2.9× and lowers it for two; with shared memory it raises all five hubs tried by at least 2.5×.Under mission control, L* raises revenue significantly for four hubs and lowers it for three.Source: K Proposition A.1 Against the Exact… · p. 15
• On 2026, total surplus net of hub refills falls 2.0 ± 0.6% with shared memory and 0.5 ± 0.3% with mission control at 1,000 ppm; sender surplus falls 16–18%.The paper treats refused senders as total losses, so these welfare losses are upper bounds.Source: K Proposition A.1 Against the Exact… · p. 15

L Simulation Parameters

• At 0.8 or 1.25 times the best level, a flat fee loses at least 2.3 times what any skew can add (median 5.3); at 2/3 or 1.5 times, at least 4.7 times.These comparisons cover a 144-instance grid varying price sensitivity, demand relative to replenishment, forward size, refill cost, and refusal penalty.Source: L Simulation Parameters · p. 16
• With every realism option on, removing refills changes revenue by −0.3% and the skew by +0.2%.In this setting refills are made only when they pay.Source: L Simulation Parameters · p. 16

M Confirmatory Tests, Frozen Predictions and Power

• No ppm-tuned tool beats the frontier by more than 3%; skew adds less than 3% under mission control and 5% with shared memory.All 12 confirmatory null hypotheses are rejected after Holm correction, with largest adjusted p = 0.0086; no tool significantly lies above the frontier.Source: M Confirmatory Tests, Frozen… · p. 16
• Frozen predictions for 10, 20, and 80 hubs held: shared-memory peaks were at 4,900 ppm and within 0.07 of predicted curves; mission-control predicted levels earned at least 94% of the best.The predictions were frozen before the runs.Source: M Confirmatory Tests, Frozen… · p. 17
• Against the flat level at its center, LNDg earns −3.1 ± 5.0% and +0.2 ± 2.3%, charge-lnd −5.2 ± 4.6% and −2.2 ± 3.3%, and feeadjuster −20.4 ± 4.8% and −8.8 ± 1.8%.The comparisons are paired by seed; tools achieve at most 4.2 points more success.Source: M Confirmatory Tests, Frozen… · p. 17

N Ordering Across 2026 Settings

• At full capacities, LNDg earns 2.1 ± 2.9% and 2.8 ± 2.0% more than the frontier at its success; other tools are within noise of it or below.With all realism options on and economic refills, the skew raises refills slightly, from 4.4 to 5.0 per run at deployed fees.Source: N Ordering Across 2026 Settings · p. 17, Table 7 · p. 17
• Across other 2026 settings, best tools lie 2–11% below the frontier at κ = 2,000, 5–6% below with heavier-tailed willingness to pay, and 6–7% below with payments five times larger.At 15% capacity proxies they are 6–8% below the frontier, and within 3% at 50%; at defaults they are 40–49% below under mission control.Source: N Ordering Across 2026 Settings · p. 17