papers/lp-308-quantum-amm/sections/02-bonding-curve-algorithms.tex

\section{Bonding Curve Algorithms}


QuantumSwap pools are parameterized by a bonding curve that determines
price dynamics. Each \texttt{LUXQSAPair} contract is associated with
exactly one curve. The curves are pure functions: they do not modify
pool state. State updates occur in the \texttt{LUXRWAPair} contract
after the curve is consulted.

\subsection{Notation}

Let $\mathit{spot}$ denote the current spot price (instantaneous price
to sell one item to the pool), $\delta$ the curve parameter, and $x$
the trade size. The spot price to \emph{buy} one item from the pool is
always $\mathit{spot} + \delta$ (additive) or $\mathit{spot} \cdot
(1 + \delta)$ (multiplicative), depending on the curve.

\subsection{Linear Bonding Curve}

The linear curve adjusts price by a constant additive amount $\delta$
per item traded:

\begin{align}
\mathit{spot}_{\text{after sell}} &= \mathit{spot} - \delta
  \label{eq:linear-sell} \\
\mathit{spot}_{\text{after buy}} &= \mathit{spot} + \delta
  \label{eq:linear-buy}
\end{align}

For a multi-item sell of $x$ items, the total payout is:

\begin{equation}\label{eq:linear-multi}
\text{Payout}(x) = \sum_{i=0}^{x-1} (\mathit{spot} - i\delta)
= x \cdot \mathit{spot} - \delta \cdot \frac{x(x-1)}{2}
\end{equation}

\begin{remark}
The LP must set $\delta$ with the same decimal precision as the
pair's underlying token. For an ETH-denominated pair with
$\mathit{spot} = 1\,\text{ETH}$ and $\delta = 0.1\,\text{ETH}$,
selling 5~NFTs yields
$5 \cdot 1 - 0.1 \cdot 10 = 4\,\text{ETH}$, distributed as
$\{1, 0.9, 0.8, 0.7, 0.6\}$.
\end{remark}

\subsection{Exponential Bonding Curve}

The exponential curve adjusts price by a multiplicative factor:

\begin{align}
\mathit{spot}_{\text{after buy}} &= \mathit{spot} \cdot (1 + \delta)
  \label{eq:exp-buy} \\
\mathit{spot}_{\text{after sell}} &= \frac{\mathit{spot}}{1 + \delta}
  \label{eq:exp-sell}
\end{align}

After $x$ consecutive buys from an initial spot price $p_0$:

\begin{equation}\label{eq:exp-multi}
\mathit{spot}(x) = p_0 \cdot (1+\delta)^x
\end{equation}

The total cost of buying $x$ items is:

\begin{equation}
\text{Cost}(x) = p_0 \cdot (1+\delta) \cdot
\frac{(1+\delta)^x - 1}{\delta}
\end{equation}

\begin{remark}
With $p_0 = 2\,\text{ETH}$ and $\delta = 0.5$ (50\%), the price
sequence for 3~buys is $\{3, 4.5, 6.75\}$ ETH. Selling into the
pool divides by $1.5$ at each step.
\end{remark}

\subsection{XYK Bonding Curve}

The XYK curve maintains a constant-product invariant over virtual
reserves, adapted for NFT pools. At pool creation:

\begin{align}
R_{\text{nft}} &= \text{(number of items to buy or sell)} + 1
  \label{eq:xyk-nft} \\
R_{\text{token}} &= R_{\text{nft}} \cdot \mathit{spot}
  \label{eq:xyk-token}
\end{align}

The price to buy $x$ items from the pool (receiving NFTs, sending
tokens) is:

\begin{equation}\label{eq:xyk-buy}
\text{Cost}(x) = \frac{x \cdot R_{\text{token}}}{R_{\text{nft}} - x}
\end{equation}

The price to sell $x$ items to the pool is:

\begin{equation}\label{eq:xyk-sell}
\text{Payout}(x) = \frac{x \cdot R_{\text{token}}}{R_{\text{nft}} + x}
\end{equation}

After each trade, the virtual reserves update:
\begin{itemize}[nosep]
  \item Pool sells NFTs:
    $R_{\text{nft}} \leftarrow R_{\text{nft}} - x$, \;
    $R_{\text{token}} \leftarrow R_{\text{token}} + \text{Cost}(x)$
  \item Pool buys NFTs:
    $R_{\text{nft}} \leftarrow R_{\text{nft}} + x$, \;
    $R_{\text{token}} \leftarrow R_{\text{token}} - \text{Payout}(x)$
\end{itemize}

A concentration parameter allows LPs to tighten or loosen the curve
around the current price, analogous to Uniswap V3's concentrated
liquidity~\cite{uniswapv3} but applied to discrete (NFT) assets.

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