Bonding Curve

Bonding curves have been part of crypto for years, particularly in DEXs and launchpads, where they provide asset pricing and exchange. Polaris uses a bonding curve for a different purpose: creating a shared collateral for censorship-resistant currencies and synthetic assets across the protocol.

Whenever ETH enters the bonding curve, new pETH is created. Returning pETH to the bonding curve simply reverses the process, burning the pETH and releasing the corresponding amount of ETH.

Since the bonding curve itself acts as the market between ETH and pETH, users can always move between the two assets without relying on external liquidity providers. As more ETH enters Polaris, the bonding curve provides deeper liquidity to the whole ecosystem, giving every pAsset built on top of pETH access to the same protocol-native ETH liquidity.

As such, value generated anywhere in Polaris flows back to the same collateral that backs the protocol, enabling new financial primitives to build on pETH while strengthening the same foundation that supports the whole protocol.

Beyond bringing pETH into existence, the bonding curve also enables several of the properties that define Polaris: it sets the price of pETH, generates a protocol-native source of yield directly from onchain swaps and allows minting rates to emerge directly from market conditions.

A Scale-Free Design

The bonding curve is designed to behave consistently regardless of the amount of capital secured by the protocol.

This scale-free design means that whether Polaris secures thousands of ETH or millions, the same mechanism continues to operate under the same rules, enabling the protocol to scale without changing the economic behavior of pETH as adoption increases.

This keeps pETH as the shared collateral for a growing ecosystem of pAssets without fragmenting liquidity as new ones are introduced.

The Role of β

The main parameter defining the bonding curve is β (beta), the exponent in the power-law curve. β determines the concavity of the curve, which controls how sharply the pETH price responds as ETH enters or leaves the bonding curve.

In plain terms, β defines the shape of the market. A larger β produces a steeper curve, causing the market price of pETH to react more aggressively as capital enters or leaves the bonding curve.

A smaller β produces a flatter curve, reducing the impact of those same capital flows while increasing the proportion of pETH represented by its floor price.

Choosing β involves a trade-off between price responsiveness and collateral stability. Polaris was intentionally designed around the latter, with the goal of building a highly stable collateral asset while preserving enough price responsiveness to allow for protocol growth.

The chosen value of β is intended to enable the Polaris economic model, ensuring low volatility and a relatively high floor price ratio, with the floor becoming an increasingly important component of pETH over time.

The Economic Engine

Unlike most collateral assets, one of the defining characteristics of pETH is that value generated throughout Polaris strengthens its floor price.

All value generated through the protocol ultimately flows back to the same economic system built around pETH, which allows its floor price to steadily grow as the protocol expands.

This growing floor is what makes pETH a protocol-native yield-bearing asset and also forms the basis of fpETH, which isolates that steadily growing component for users who prefer exposure to the floor without the volatility of the market premium. Users who instead want exposure to both components can simply hold pETH, while those interested in isolating the market premium can do so through vpETH.

The role of the bonding curve therefore extends beyond just enabling pETH ↔ ETH swaps, as it provides the shared economic foundation on which Polaris is built. As a result, it allows collateral, liquidity, protocol-native yield and future applications to expand around a single reserve asset without the need to compete for separate pools of capital.