Original Paper

taurusSwap is an implementation of the Orbital AMM design from Paradigm. This page links to the original theoretical work and maps paper sections to implementation components.

Paper Reference

Section-to-Implementation Mapping

Key Results from the Paper

Efficiency Theorem (Section 5.1)

For n tokens at depeg threshold p, the capital efficiency multiplier is:

For n=5 and p=0.99, this gives approximately 150× efficiency vs. uniform liquidity (Curve).

Invariant Preservation (Section 4.1)

Theorem: If the torus invariant holds before a trade and the trade satisfies the crossing conditions, the invariant holds after the trade.

This is what enables O(1) verification: the contract doesn't need to recompute the full state, only check that the invariant still holds.

Crossing Conditions (Section 3.3)

A tick crosses from interior to boundary when:

The contract verifies this condition using cross-multiplication to avoid division.

Our Contributions

taurusSwap is an implementation of the Orbital design, not original research. Our contributions are:

• Algorand deployment— Adapting the design to the AVM's uint64-centric model with AMOUNT_SCALE
• TypeScript SDK — Reference implementation of the math in pure BigInt
• Frontend UI — Trading interface with Three.js visualizations
• Fee accounting— Uniswap V3-style fee growth adapted for Orbital's consolidated liquidity

The theoretical work — the sphere invariant, polar decomposition, tick geometry, and torus consolidation — is entirely from the Paradigm team.

Citation

If you reference taurusSwap in academic work, please cite both this implementation and the original paper:

@article{white2025orbital,
  title={Orbital: Concentrated Liquidity for Any Number of Tokens},
  author={White, Dave and Robinson, Dan and Moallemi, Ciamac},
  journal={Paradigm Research},
  year={2025},
  url={https://www.paradigm.xyz/2025/06/orbital}
}

@software{taurusSwap2026,
  title={taurusSwap: Orbital AMM on Algorand},
  author={Kaushik and contributors},
  year={2026},
  url={https://github.com/Kaushik2003/taurusSwap}
}