Proposition 3, pp. 22–23 — constant product pricing is additive and split-liquidity invariant, yet front-running (x < X/2) and two-venue arbitrage are strictly profitable

Let a constant product pool hold $X>0$ units of token $A$ and $Y>0$ units of token $B$, with cost function $p^{\mathrm{cmm}}(x,X,Y)=xY/(X-x)$ and $y(\cdot)=p^{\mathrm{cmm}}(\cdot,X,Y)$. Selling $q>0$ units of $A$ into a pool $(X',Y')$ earns $-p^{\mathrm{cmm}}(-q,X',Y')$. For $x\in(0,X)$, $\alpha\in(0,1)$ and $k>0$:

1. Additivity:

2. Profitable front-running: if moreover $x<X/2$, buying $x$ ahead of another buy of $x$ and then selling $x$ into the pool $(X-2x,\,Y+y(2x))$ is strictly profitable,

3. Additivity with split liquidity:

4. Multi-venue arbitrage: with two identical pools $(X,Y)$, after a buy of $x$ on the first one, selling $\alpha x$ into the first pool, now $(X-x,\,Y+y(x))$, and buying $\alpha x$ on the second is strictly profitable,

The proposition shows that constant product pricing has the two consistency properties one expects of a market maker (1 and 3), yet exposes every order to profitable front-running and every trade to profitable arbitrage against an identical second venue.

Formalization Note Parts 2 and 4 are printed with $+p^{\mathrm{cmm}}(-x,\dots)$ and $p^{\mathrm{cmm}}(-\alpha x,\dots)$ as if these were the proceeds of the sale; the proofs (p. 23–24) use the amount received, which in the sign of (3) is $-p^{\mathrm{cmm}}(-q,\dots)$. Read literally, the printed left-hand sides are negative, so the statement poses the proceeds reading. Part 4 prints the first pool as $(X-x,\,Y-y(x))$; after a buy of $x$ that pool holds $Y+y(x)=XY/(X-x)$ (p. 15), and the statement uses $Y+y(x)$. Part 2 carries $x<X/2$, the paper's restriction for front-running (p. 19). $X>0$, $Y>0$ are the deposits of §II.C.