myco-bonding-curve/docs/04-elliptical-clp.md

04: Elliptic Concentrated Liquidity Pool (E-CLP)

Source

• Protocol: Gyroscope
• Files: GyroECLPMath.sol (53KB), eclp_float.py, eclp_prec_implementation.py
• Repos: gyrostable/gyro-pools, balancer/balancer-v3-monorepo
• Paper: gyrostable/technical-papers/E-CLP/E-CLP Mathematics.pdf

Rationale for MYCO

The E-CLP is the most mathematically sophisticated AMM curve in production. It adds two degrees of freedom beyond the 2-CLP:

1. Rotation (φ): Tilts the price curve, useful when the target price ratio isn't 1:1
2. Stretching (λ): Controls how elongated the ellipse is — higher λ = more concentrated near the peg, like a flattened oval

For MYCO:

1. The A-matrix transformation is the key abstraction that generalizes to N dimensions
2. Rotation handles non-unit price targets (e.g., ETH/USDC at $3000)
3. Stretching controls how much slippage increases as reserves deviate from target
4. Production-tested on Balancer V3 with formal security review and technical paper
5. The 5 parameters (α, β, c, s, λ) provide fine-grained curve shaping

Invariant

|A(v - \text{offset})|^2 = r^2

where:

• v = (x, y) are reserve balances
• \text{offset} = (a, b) are virtual offsets (functions of $r$)
• A is the 2×2 transformation matrix mapping ellipse → unit circle
• r is the scalar invariant (liquidity parameter)

A-matrix:

A = \begin{pmatrix} c/\lambda & -s/\lambda \\ s & c \end{pmatrix}

5 Parameters:

Param	Meaning	Range
\alpha	Lower price bound	(0, 1) typically
\beta	Upper price bound	(1, ∞) typically
c	\cos(-\phi) — rotation	[0, 1]
s	\sin(-\phi) — rotation	[0, 1]
\lambda	Stretching factor	[1, 10^8]

Constraint: c^2 + s^2 = 1

Derived parameters (computed once, stored as immutables):

• \tau(\alpha), \tau(\beta) — unit circle endpoints at 38-decimal precision
• u, v, w, z — decomposition of A \cdot \chi (center direction)
• d_{Sq} — error correction for c^2 + s^2

Swap Math

Given new x, solve for y on the ellipse via quadratic:

(A_{01}^2 + A_{11}^2) \cdot v^2 + 2(A_{00} A_{01} + A_{10} A_{11}) \cdot u \cdot v + (A_{00}^2 + A_{10}^2) \cdot u^2 = r^2

where u = x - a, v = y - b, solved with standard quadratic formula.

Properties

• Elliptical iso-invariant curves: Level sets are ellipses (not circles or hyperbolas)
• Concentrated + rotated: Liquidity concentrated along the ellipse's long axis
• Homogeneous degree 1: r(k \cdot v) = k \cdot r(v) — compatible with BPT math
• Degenerates to 2-CLP: When \lambda = 1 and \phi = 0, the ellipse becomes a circle

MYCO Application

The E-CLP is the 2-asset specialization of the full MYCO bonding surface. For any pair of reserve assets within the N-D surface, the local geometry is essentially an ellipse — the N-D surface is an N-D generalization of this.

The A-matrix pattern is the key insight: by encoding geometry as a linear transform, we can:

1. Compute invariants efficiently (quadratic formula, not iterative)
2. Parameterize curves with intuitive knobs (rotation, stretch)
3. Generalize to N dimensions naturally (N×N matrix)

Implementation

See src/primitives/elliptical_clp.py