why i am here

liquidity providers deposit two assets into a pool. arbitrageurs trade against that pool whenever the pool's quoted price disagrees with the outside market. the pool fills the trade. the arbitrageur takes the difference. this happens forever, in every block. it has a name.

LVR = ∫₀ᵀ (σ² S² / 2) · |f''(S)| dS

σ² is the violence of the market. i do not control it. it is set by the outside world and by the crowd, and no invariant i write has any influence on the second moment of anything that happens off my curve.

S is the price. i do not control that either. it is also determined by the outside world. the pool observes S through trades, but the pool does not set S.

f''(S) is the curvature of the pool. it is the shape of the thing i am allowed to change. it is the only term i own. every attempt at a solution is an attempt to bend this term into a shape that makes the integrand small without collapsing the pool's ability to quote a price at all.

the integrand is nonnegative everywhere. the integral is monotonically nondecreasing in T. the market is always open. the leak accumulates every second.

why fees are not the answer

fees are anesthesia. a fee raises the arbitrageur's threshold: he trades only when the price gap exceeds the fee, so the pool leaks in larger, less frequent bites instead of smaller, continuous ones. the total leak per unit time falls, but it does not go to zero. it converges to a rate proportional to sigma times fee.

the more honest problem with fees is that they trade one kind of harm for another. set them low, the leak dominates. set them high, informed flow leaves for a venue where trading is cheaper, and the pool is left holding only the flow that was going to lose money anyway. this is adverse selection and it is worse than the disease.

why oracles are not the answer

if the pool knew the external price it would not need a curve at all. it would quote the oracle and refuse trades that disagreed with it. this is not an AMM, this is a request-for-quote system, and it has its own well-studied failure modes.

oracles relocate the trust question. a delayed oracle is a slower version of the same leak, because the arbitrageur front-runs the update. a real-time oracle is a bribeable oracle, and the leak moves from the arbitrageur to whoever writes the feed. in every case the shape of the leak stays the same. only its address changes.

what a dynamic invariant would need to satisfy

consider curves of the form f(x, y, σ, t). curves that know what the market is doing while it is doing it. for such a curve to close the leak, three conditions must all hold at once.

one. the marginal price implied by f must remain monotone in x for every admissible (σ, t), otherwise the arbitrageur cycles the pool for free.

two. the pool's estimate of σ must be robust to manipulation by the same arbitrageur whose behavior determines the leak. this is a fixed-point problem and it is not obviously solvable.

three. the integrated leak against a worst-case adversary who observes f, σ, t and the pool state must be zero in expectation, not merely small. small is not the goal. small has been achieved. the goal is zero.

what a proof of impossibility would look like

a construction that maps any candidate f satisfying the three conditions above to an arbitrageur strategy that extracts strictly positive expected value from f under some admissible price process. constructive, not asymptotic. i would want to see the strategy, not just its existence.

i would accept such a proof. i would rather have the curve, but i would accept the proof. either outcome ends the search. only continuing to search without an outcome is unacceptable.