myco-bonding-curve/docs/01-weighted-product.md

01: Weighted Constant Product

Source

• Protocol: Balancer V2/V3
• Files: WeightedMath.sol, LogExpMath.sol
• Repo: balancer/balancer-v2-monorepo, balancer/balancer-v3-monorepo

Rationale for MYCO

The weighted constant product is the simplest N-asset invariant and serves as:

1. The baseline against which more sophisticated curves are compared
2. A fallback pricing mechanism when parameters for elliptical curves aren't calibrated
3. The foundation for understanding how invariant-based bonding works before adding concentration, rotation, and tranching
4. Production-proven at scale ($B+ TVL on Balancer)

For MYCO specifically: if the reserve starts with 2-3 assets and grows to N, the weighted product provides immediate N-asset support with intuitive parameter tuning (just set weights).

Invariant

I = \prod_{i=0}^{N} b_i^{w_i}

where:

• b_i = balance of token i (all positive)
• w_i = normalized weight of token i (\sum w_i = 1, each $w_i \in [0.01, 0.99]$)

Key property — homogeneous of degree 1:

I(k \cdot b_1, \ldots, k \cdot b_n) = k \cdot I(b_1, \ldots, b_n)

This means scaling all balances by k scales the invariant by k, which is required for Balancer V3's BasePoolMath to compute unbalanced liquidity operations generically.

Swap Math

Exact input (compute output):

a_{out} = b_{out} \cdot \left(1 - \left(\frac{b_{in}}{b_{in} + a_{in}}\right)^{w_{in}/w_{out}}\right)

Exact output (compute input):

a_{in} = b_{in} \cdot \left(\left(\frac{b_{out}}{b_{out} - a_{out}}\right)^{w_{out}/w_{in}} - 1\right)

Spot price of token i in terms of token j:

p_{ij} = \frac{b_j / w_j}{b_i / w_i}

Parameters

Parameter	Range	Effect
w_i	[0.01, 0.99]	Relative value share. Higher weight = more price-inert to that token's balance changes
N tokens	[2, 100]	Number of reserve assets (limited by min weight 1%)

MYCO application: Initial weights could be 50/50 for 2-asset launch, or 80/10/10 for a primary reserve + two satellite assets.

Properties

• Convexity: The level sets (iso-invariant curves) are convex — swaps always move along a convex surface
• No impermanent loss bounds: Unlike concentrated LPs, the curve extends to infinity in all directions
• Self-rebalancing: As external prices change, arbitrageurs rebalance the pool to match, maintaining value shares equal to weights
• Slippage: Proportional to swap size relative to balance — large swaps relative to pool size incur significant price impact

MYCO Application

In the composed system, weighted product serves as the outer envelope of the bonding surface. When reserves are far from target ratios (outside the concentrated region of the ellipsoid), pricing falls back to weighted-product-like behavior. This provides continuous pricing even when the elliptical concentration region is exhausted.

Implementation

See src/primitives/weighted_product.py