109 lines
4.7 KiB
Markdown
109 lines
4.7 KiB
Markdown
# 05: N-Dimensional Ellipsoidal Bonding Surface
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## Source
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- **Novel construction** extending Gyroscope E-CLP (2D) and 3-CLP (3D)
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- **Foundation**: E-CLP A-matrix pattern, 3-CLP cubic invariant approach
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- **Math**: Standard N-dimensional ellipsoid geometry
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## Rationale for MYCO
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This is the **core innovation** of the MYCO bonding surface. While Gyroscope demonstrated:
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- 2-CLP: Concentrated LP for 2 tokens (quadratic)
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- 3-CLP: Concentrated LP for 3 tokens (cubic, symmetric price bounds)
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- E-CLP: Elliptical concentration for 2 tokens
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Nobody has deployed an **N-asset ellipsoidal bonding surface** for token issuance. This primitive generalizes all three into a single framework where:
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1. Each reserve asset occupies one dimension
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2. The ellipsoid geometry controls pricing and concentration
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3. Per-axis stretch factors ($\lambda_i$) give independent concentration control per asset
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4. The rotation matrix $Q$ controls correlation structure between assets
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5. $MYCO supply maps to the invariant $r$ (radius of the ellipsoid)
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## Invariant
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$$|A(\vec{v} - \vec{\text{offset}})|^2 = r^2$$
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where:
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- $\vec{v} = (b_1, \ldots, b_N)$ — reserve balances
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- $\vec{\text{offset}} = (a_1, \ldots, a_N)$ — virtual offsets (functions of $r$)
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- $A = \text{diag}(1/\lambda_i) \cdot Q^T$ — N×N transformation matrix
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- $Q$ — N×N orthogonal rotation matrix
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- $\lambda_i$ — per-axis stretch factors
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- $r$ — scalar invariant (bonding surface "radius")
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**Key property — homogeneous degree 1:**
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$$r(k \cdot \vec{v}) = k \cdot r(\vec{v})$$
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This ensures BPT/LP-token math works: minting is proportional to invariant increase.
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## Parameters
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| Parameter | Dimension | Range | Effect |
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|-----------|-----------|-------|--------|
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| $\lambda_i$ | N | [1, ∞) | Per-axis concentration. Higher = tighter pricing for that asset |
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| $Q$ | N×N | Orthogonal | Rotation encoding correlations. Identity = axis-aligned |
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| $\alpha_i$ | N | (0, 1) | Lower price bounds per asset |
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| $\beta_i$ | N | (1, ∞) | Upper price bounds per asset |
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**Degrees of freedom:**
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- N stretch factors
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- N(N-1)/2 rotation angles (via Givens rotations)
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- 2N price bounds
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- Total: N² + N/2 parameters — rich enough to model complex reserve structures
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## Swap Math
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Given a swap of token $k$ in for token $j$ out:
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1. Set $b_k \leftarrow b_k + \Delta_k$
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2. Solve $|A(\vec{v}' - \vec{\text{offset}})|^2 = r^2$ for $b_j'$
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3. This is a **quadratic in $b_j$** (isolate terms involving $b_j$ from the norm)
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4. $\Delta_j = b_j - b_j'$
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The quadratic structure means swaps are computed in O(N) time (matrix-vector multiply + scalar quadratic solve).
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## Minting / Bonding
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Depositing reserves into the surface:
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1. User deposits amounts $\Delta_1, \ldots, \Delta_N$
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2. New invariant $r' = r(\vec{v} + \vec{\Delta})$
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3. $\text{MYCO\_minted} = \text{supply} \cdot (r'/r - 1)$
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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.
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## Geometric Interpretation
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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.
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The stretch factors $\lambda_i$ control the **eccentricity** along each axis:
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- $\lambda_i = 1$ for all $i$: hypersphere (isotropic, like weighted product)
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- $\lambda_i \gg 1$ for asset $i$: very concentrated in that dimension (tight pricing)
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- Mixed $\lambda$: some assets tightly priced, others loosely (risk tranching effect)
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The rotation matrix $Q$ controls **correlations**:
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- $Q = I$: axis-aligned, no cross-asset correlation in concentration
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- Non-trivial $Q$: Correlated assets share concentration (e.g., ETH and stETH)
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## Properties
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- **Convexity**: Ellipsoids are convex — swap paths are well-defined
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- **Homogeneous degree 1**: Required for LP math
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- **O(N) swaps**: Quadratic solve per swap (after O(N) matrix multiply)
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- **Degenerates to E-CLP**: When N=2, recovers the Gyroscope E-CLP exactly
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- **Degenerates to hypersphere**: When all $\lambda_i = 1$ and $Q = I$
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## MYCO Application
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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.
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Governance controls the surface shape:
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- Increasing $\lambda_i$ for a scarce reserve makes deposits of that asset more valuable
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- Rotating the surface via $Q$ can encode expected correlations
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- Price bounds prevent minting at absurd ratios
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## Implementation
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See `src/primitives/n_dimensional_surface.py`
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