myco-bonding-curve/docs/05-n-dimensional-surface.md

05: N-Dimensional Ellipsoidal Bonding Surface

Source

• Novel construction extending Gyroscope E-CLP (2D) and 3-CLP (3D)
• Foundation: E-CLP A-matrix pattern, 3-CLP cubic invariant approach
• Math: Standard N-dimensional ellipsoid geometry

Rationale for MYCO

This is the core innovation of the MYCO bonding surface. While Gyroscope demonstrated:

• 2-CLP: Concentrated LP for 2 tokens (quadratic)
• 3-CLP: Concentrated LP for 3 tokens (cubic, symmetric price bounds)
• E-CLP: Elliptical concentration for 2 tokens

Nobody has deployed an N-asset ellipsoidal bonding surface for token issuance. This primitive generalizes all three into a single framework where:

1. Each reserve asset occupies one dimension
2. The ellipsoid geometry controls pricing and concentration
3. Per-axis stretch factors ($\lambda_i$) give independent concentration control per asset
4. The rotation matrix Q controls correlation structure between assets
5. $MYCO supply maps to the invariant r (radius of the ellipsoid)

Invariant

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

where:

• \vec{v} = (b_1, \ldots, b_N) — reserve balances
• \vec{\text{offset}} = (a_1, \ldots, a_N) — virtual offsets (functions of $r$)
• A = \text{diag}(1/\lambda_i) \cdot Q^T — N×N transformation matrix
• Q — N×N orthogonal rotation matrix
• \lambda_i — per-axis stretch factors
• r — scalar invariant (bonding surface "radius")

Key property — homogeneous degree 1:

r(k \cdot \vec{v}) = k \cdot r(\vec{v})

This ensures BPT/LP-token math works: minting is proportional to invariant increase.

Parameters

Parameter	Dimension	Range	Effect
\lambda_i	N	[1, ∞)	Per-axis concentration. Higher = tighter pricing for that asset
Q	N×N	Orthogonal	Rotation encoding correlations. Identity = axis-aligned
\alpha_i	N	(0, 1)	Lower price bounds per asset
\beta_i	N	(1, ∞)	Upper price bounds per asset

Degrees of freedom:

• N stretch factors
• N(N-1)/2 rotation angles (via Givens rotations)
• 2N price bounds
• Total: N² + N/2 parameters — rich enough to model complex reserve structures

Swap Math

Given a swap of token k in for token j out:

1. Set b_k \leftarrow b_k + \Delta_k
2. Solve |A(\vec{v}' - \vec{\text{offset}})|^2 = r^2 for b_j'
3. This is a quadratic in $b_j$ (isolate terms involving b_j from the norm)
4. \Delta_j = b_j - b_j'

The quadratic structure means swaps are computed in O(N) time (matrix-vector multiply + scalar quadratic solve).

Minting / Bonding

Depositing reserves into the surface:

1. User deposits amounts \Delta_1, \ldots, \Delta_N
2. New invariant r' = r(\vec{v} + \vec{\Delta})
3. \text{MYCO\_minted} = \text{supply} \cdot (r'/r - 1)

The invariant increase depends on the geometry of the deposit relative to the surface. Deposits aligned with the ellipsoid's minor axes (high \lambda directions) increase the invariant more per unit deposited — this is by design, as those are the "more needed" reserves.

Geometric Interpretation

The iso-invariant surface in N-D space is an N-dimensional ellipsoid (offset from the origin by the virtual reserves). Trading moves along this ellipsoid. Minting expands it (larger $r$). Redeeming shrinks it.

The stretch factors \lambda_i control the eccentricity along each axis:

• \lambda_i = 1 for all i: hypersphere (isotropic, like weighted product)
• \lambda_i \gg 1 for asset i: very concentrated in that dimension (tight pricing)
• Mixed \lambda: some assets tightly priced, others loosely (risk tranching effect)

The rotation matrix Q controls correlations:

• Q = I: axis-aligned, no cross-asset correlation in concentration
• Non-trivial Q: Correlated assets share concentration (e.g., ETH and stETH)

Properties

• Convexity: Ellipsoids are convex — swap paths are well-defined
• Homogeneous degree 1: Required for LP math
• O(N) swaps: Quadratic solve per swap (after O(N) matrix multiply)
• Degenerates to E-CLP: When N=2, recovers the Gyroscope E-CLP exactly
• Degenerates to hypersphere: When all \lambda_i = 1 and Q = I

MYCO Application

This is the primary pricing surface for financial reserve deposits. When a user deposits ETH, USDC, DAI, or any accepted reserve token, the ellipsoid geometry determines how much $MYCO they receive.

Governance controls the surface shape:

• Increasing \lambda_i for a scarce reserve makes deposits of that asset more valuable
• Rotating the surface via Q can encode expected correlations
• Price bounds prevent minting at absurd ratios

Implementation

See src/primitives/n_dimensional_surface.py