Continuous Dynamic-Weight Path

Trade splitting approaches a continuous dynamic-weight path.

The constant-weight case has a simple limit. When n = 0, kappa = 1, scale stays fixed, and the continuous path integrates to a constant-weight AMM.

For n > 0, scale evolves along the trade path. Each smaller slice recomputes target distance and updates scale.

Differential form

For each token leg:

dSigma_i = s_i * da_i / a_i

and:

ds_i = (1 - kappa_i) * dSigma_i

with:

kappa_i = abs(rt_i / (1 + rt_i))^n

Value flow is conserved:

sum dSigma_i = 0

For a two-leg swap:

dSigma_pay + dSigma_receive = 0

Integrated path

Let cumulative value flow be V:

dSigma_pay = dV
dSigma_receive = -dV

Then:

ds_pay/dV = 1 - kappa_pay

ds_receive/dV = -(1 - kappa_receive)

and:

d ln(a_pay) / dV = 1 / s_pay

d ln(a_receive) / dV = -1 / s_receive

Integrating from 0 to Phi:

s_pay(Phi) = s_pay(0) + integral_0^Phi (1 - kappa_pay(V)) dV

s_receive(Phi) = s_receive(0) - integral_0^Phi (1 - kappa_receive(V)) dV

a_pay(Phi) = a_pay(0) * exp(integral_0^Phi dV / s_pay(V))

a_receive(Phi) = a_receive(0) * exp(-integral_0^Phi dV / s_receive(V))

The split-limit output is obtained by solving the pay equation for Phi, then reading the receive reserve change.

Interpretation

Near target, kappa is small and scale absorbs most value flow. Farther from target, kappa grows and the path becomes more price-impactful.

This is the continuous dynamic-weight Multiswap path.