Most automated-liquidity programs reward LPs based on trading volume. The consequences are predictable:

Wash traders run loops and farm rewards by creating volume (bots doing 50% to 51% to 50% cycles).
Real liquidity disappears exactly when it's needed. At news breaks, LPs pull quotes to avoid toxic flow, leaving the book thin.
We pay for activity (noise) rather than discovery (signal).

I developed a solution. By redesigning the reward hook to pay for information discovered rather than volume, we can flip these incentives entirely.

The Core Concept: Paying for Geodesic Displacement

The key insight is to interpret a market probability for a binary event as a point on the statistical manifold of Bernoulli distributions. If we equip that manifold with the Fisher information metric, we can measure the geodesic (intrinsic) distance between the market's start and end probabilities.

Why this kills wash trading: If a bot pumps the price up and then brings it back to the starting price within the epoch, the start and end probabilities are identical. The geodesic displacement is zero. The bot gets no reward, even though it generated significant volume.

Why this rewards useful liquidity: When real news moves the market's belief irreversibly, the geodesic displacement is positive. Importantly, it reflects how hard it was to move the distribution. Small absolute moves near 50% are less informative than small absolute moves near the extremes (0 or 1), and the Fisher metric captures that asymmetry correctly.

The Math: Measuring Information Distance

1. The Fisher Information Metric

For a Bernoulli distribution parameterized by probability p, the Fisher information is:

I(p) = 1 / ( p * (1 - p) )

This induces a local line element (infinitesimal distance) of:

ds = |dp| / sqrt( p * (1 - p) )

The same absolute change dp has different geometric weight depending on where we are. Near extremes (p close to 0 or 1), a small change is "louder" (larger distance) than the same change near p = 0.5.

2. Closed-Form Geodesic Distance

Because the Bernoulli family is one-dimensional, the geodesic distance between two probabilities p and q (start and end of an epoch) has a clean closed-form solution. The Fisher-Rao distance is:

d_FR(p, q) = 2 * | arcsin( sqrt(q) ) - arcsin( sqrt(p) ) |

This formula captures the "information distance" traveled by the market.

Concrete Examples

To build intuition, here are Fisher-Rao distances for different market moves:

Mid-range move (0.50 to 0.55): d_FR is about 0.100
Tiny noise (0.50 to 0.51): d_FR is about 0.020
Extreme move (0.98 to 0.99): d_FR is about 0.083
Major shift (0.60 to 0.90): d_FR is about 0.726
Wash trade (0.50 to 0.50): d_FR = 0.0

Notice that a 1% move near the top (0.98 to 0.99) is treated as 4x more significant than a 1% move in the middle (0.50 to 0.51). Going from 98% certainty to 99% is a massive reduction in uncertainty, and the metric reflects that.

An Implementable Reward Hook

Here is the reward formula I designed for the protocol. It rewards LPs proportional to the Fisher-Rao displacement while their capital is active.

Variables:

p0, pT: Market probability at start/end of epoch
d: The calculated Fisher-Rao distance
B: Base reward pool
stake_i: Capital committed by LP i
time_active_i: Duration LP i was active

The Discovery-Only Payout Formula:

reward_i = stake_i * time_active_i * B * ( d / (d + k) )

Where k is a smoothing constant. The formula ensures:

Zero displacement = Zero reward.
Rewards scale with information. Larger shifts in belief pay more.
Concavity. The reward saturates for massive moves to protect the protocol budget.

Game-Theoretic Implications

This mechanism changes how bots and LPs behave:

Wash loops are unprofitable. Round-trips yield d = 0.
LPs stay during volatility. In standard models, LPs flee during news to avoid toxic flow. Here, news events create large d values, meaning the potential reward increases exactly when the market is moving. LPs are incentivized to stay and facilitate price discovery.
Truth is profitable. The only way to earn large rewards is to help the market move to a new, correct probability.

Practical Implementation Checklist

For engineers implementing this:

Epoch Length: Choose 1-6 hours depending on market velocity.
Robust Endpoints: Use time-weighted medians for p0 and pT to prevent oracle manipulation at epoch boundaries.
Precision: Compute d using double precision to avoid rounding errors near extremes.
Anti-Abuse: Require minimum active time to prevent flash-staking.

Conclusion

By shifting from paying for volume to paying for information geometry, we align protocol incentives with its true goal: accurate price discovery. Liquidity mining becomes a bounty for truth-seekers, not a subsidy for wash traders.