Dr. Mark Richardson, Bancor Project Lead, recently introduced Bancor’s latest invention at ETHZurich: an Arbitrary Mean-Rate Exchange Protocol built around a mathematical principle he calls the Mean-of-Derivatives Property, or MoD.
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As a Bancor contributor, I’ve known for some time that something significant was taking shape behind the scenes. But knowing that work exists and fully understanding the vision behind it are two very different things.
With the invention now public, Mark and I sat down 1:1 to discuss it in greater depth. During our conversation, Mark referred to the protocol as Carbon Alpha — a temporally resolved arbitrary mean-rate exchange protocol and a more feature-rich evolution of the Carbon DeFi protocol that exists today.
What he provided was a window into the next phase of decentralized exchange design: one shaped by mathematical generalization, deeper liquidity control, and a more expressive way for users, token projects, and institutions to define how markets should behave onchain.
From Bonding Curve Shape to Market Behavior
Bonding curves are often discussed as if they are static objects.
A curve exists.
Liquidity is placed into it.
Trades move along it.
The user adapts to its behavior.
Carbon Alpha reverses that relationship.
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The user begins with the desired behavior, and the protocol constructs the curve capable of producing it.
That may sound abstract, but the practical effect is easy to understand. Today, a user can select a range on Carbon DeFi. Carbon DeFi then determines how liquidity is distributed within that range based on the geometric mean. Carbon Alpha introduces a new layer of control: the ability to determine the shape of liquidity inside the range itself.
A user may bias liquidity toward the lower end of a range.
Another may bias liquidity toward the upper end.
Another may want a more balanced distribution.
Another may want something more aggressive, more gradual, or more tailored to a specific market view.
Mark framed this as a new type of decision-making that users do not currently have access to. It is not simply about selecting where a strategy begins and ends. It is about deciding how liquidity should behave between those endpoints.
The x * y = k Reference Point
To understand why this matters, it helps to return to the formula that helped shape the first era of AMMs: x * y = k.
The constant product formula was Bancor’s first invention and one of DeFi’s most important early breakthroughs. As Mark explained, the familiar equal-weight x * y = k formula exhibits a version of the Mean-of-Derivatives Property that gives rise to the geometric mean. Carbon Alpha takes that observation and generalizes it. Instead of being limited to a narrow interval of curve behavior, the new framework opens access to a far broader range of possible bonding curves.
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This is where the invention becomes significant.
Different market objectives require different curve behaviors.
A stablecoin project may need one kind of liquidity profile.
A token issuer conducting buybacks may need another.
A trader expressing a directional trade may need another.
An institution bringing tokenized real-world assets onchain may need something else entirely.
Current infrastructure often forces these different needs into the same limited set of curve assumptions.
Defining the Range Is No Longer Enough
One of Mark’s clearest examples was a user selling wrapped Bitcoin between $75,000 and $100,000.
On Carbon DeFi today, a user can create that range. The protocol allows the user to define where liquidity should be active, but it still prescribes how liquidity is distributed between those two prices.
Carbon Alpha changes that.
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A less optimistic seller may want more liquidity available closer to $75,000. A more optimistic seller may want more liquidity concentrated closer to $100,000. Another seller may want liquidity distributed more evenly across the full interval.
The price range is the same, but the market view is not.
In existing systems, two users selecting the same range may be forced into the same internal liquidity behavior. With Carbon Alpha, the range becomes only the first part of the strategy. The shape inside the range becomes its own design choice.
It is a new degree of freedom, allowing market participants to encode conviction directly into the shape of their liquidity.
Average Execution Becomes a Design Choice
When I asked about placing a certain percentage of liquidity at one price and another percentage somewhere else, Mark made an important clarification and posed a different question for users to consider.
If the market fully clears the range, what average execution price does the user want?
That framing brings the concept back to execution.
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If a user is selling between two prices, the curve determines the effective rate achieved across the range. In familiar AMM designs, that outcome is determined by the protocol’s curve. Carbon Alpha gives users more control over the mean execution they want the strategy to produce.
For example, the current curve on Carbon DeFi produces the geometric mean between two endpoints. But a user may want the arithmetic mean, or another mean entirely. By selecting a different mean, the user will alter the shape of the bonding curve.
They are not only choosing where liquidity exists but determining for themselves which price point represents the average execution if the market moves through their range.
Token Projects and Market Structure
The implications for token projects are immediate.
Token projects do not simply “provide liquidity.” They shape markets.
They may want to support a peg.
They may want to create buy-side depth.
They may want to manage sell-side supply.
They may want to conduct buybacks.
They may want to influence how quickly liquidity becomes available as price moves.
Carbon Alpha gives those projects a more precise way to express those objectives onchain.
Mark used the example of a pegged asset. For a stablecoin or similar asset, downside movement may be far more sensitive than upside movement. A project may want aggressive bidding liquidity underneath the peg to reduce the chance of a disorderly move lower, while allowing more flexibility above par.
With Carbon Alpha, those two sides can be parameterized differently.
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The bidding curve can be aggressive below the peg.
The asking curve can allow more room above it.
The liquidity profile can be designed around the project’s actual market objective.
This is a meaningful departure from placing liquidity into a fixed AMM structure and accepting the behavior the protocol imposes.
It gives projects a way to construct market structure deliberately.
The Second Breakthrough: Time
The first half of the invention expands the family of bonding curves.
The second introduces time.
Carbon Alpha allows parameters to move as explicit functions of time. That means users can create strategies where prices evolve deterministically as time passes — without keepers, without oracles, and without ongoing state updates.
The strategy is set once.
The protocol reads block time when a trade occurs.
The parameters evolve automatically.
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In its simplest form, this can resemble a Dutch auction or reverse Dutch auction. A price can move over time until a counterparty accepts it.
But Mark made clear that the design goes much further.
Users could define time-dependent channels, sloping ranges, triangle-like structures, or curves that follow expected trends. A strategy could move upward over time, narrow toward the center, widen outward, or follow exponential behavior — all encoded directly into the original onchain strategy.
Buybacks, Auctions, and Moving Prices
For token buybacks, the time component is especially relevant.
A project could create a bidding order that gradually moves its price upward until someone takes it. That creates a deterministic onchain mechanism for conducting buybacks within the same chain environment, without requiring continuous manual intervention.
The same logic applies to auction-style strategies.
A price can move according to a predetermined function. A range can shift with time. The shape of the liquidity inside that range can also change.
This last point is important.
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Carbon Alpha does not only allow prices to move over time. It allows the shape parameter to move over time as well.
A strategy could follow an upward channel while liquidity gradually concentrates toward the center. It could begin broad and become more focused. It could begin focused and become more distributed. It could evolve in ways that reflect not only a target price path, but a changing view of how liquidity should behave along that path.
That is a much richer form of market design.
It turns a strategy from a static position into a programmed expression of market intent.
Interest-Bearing Assets and RWAs
The institutional relevance becomes especially clear when looking at interest-bearing assets and real-world assets.
Mark gave the example of wrapped staked ETH. Because staking rewards accrue into the token, its value moves relative to ETH over time. A project supporting liquidity for that asset may need to constantly update its liquidity parameters to reflect the changing value.
If it updates too slowly, the position can be arbitraged.
If it updates frequently, the project faces ongoing operational cost and complexity.
And the more successful the project becomes, the more expensive that management burden can become.
Carbon Alpha introduces another path.
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If the expected rate of appreciation is known or reasonably predictable, it can be encoded directly into the liquidity strategy. Instead of constantly adjusting the position, the strategy can be created to follow the implied rate over time. Mark noted that this has applications not only for staking derivatives, but also for RWAs, T-bills, commodities, securities, and other assets moving onchain.
This is where the design begins to intersect with institutional market structure.
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As more real-world assets come onchain, issuers will need secondary markets that reflect the structure of the asset itself. Traditional DeFi infrastructure often forces those assets into rigid liquidity models. Carbon Alpha offers something different: an exchange design that can be parameterized around the specific behavior, expectations, and market structure of the token being traded.
For Mark, this is not about forcing traditional financial products into existing AMM curves.
It is about giving onchain markets the flexibility to define curves that match the product.
How Carbon Alpha Fits Into Carbon DeFi
Carbon Alpha will require new smart contracts, making it a separate protocol layer from the current Carbon DeFi contracts. Even so, the user experience is expected to remain connected to Carbon DeFi, with access continuing through the Carbon DeFi app: app.carbondefi.xyz.
The rollout is expected to happen in stages, with individual features becoming available as development progresses. Each release will introduce another layer of Carbon Alpha’s capabilities, from curve selection and liquidity shaping to time-based price movement and evolving strategy parameters.
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Taken together, those capabilities extend Carbon DeFi’s existing design philosophy: more expressive strategy creation, more precise liquidity control, and more automation built directly into the protocol.
Bancor
Bancor is a pioneer in decentralized finance (DeFi), established in 2016. It invented the core technologies underpinning the majority of today’s automated market makers (AMMs) and continues to develop the foundational infrastructure critical to DeFi’s success — focusing on enhanced liquidity mechanics and robust onchain market operation. All products of Bancor, including Carbon DeFi and the Arb Fast Lane, are governed by the Bancor DAO.
Carbon DeFi — Bancor’s flagship DEX, is powered by Bancor’s latest patented technologies: Asymmetric Liquidity and Adjustable Bonding Curves.
The Arb Fast Lane — DeFi’s most advanced arbitrage infrastructure powered by Marginal Price Optimization, a new method of optimal routing.
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Carbon Alpha: A New Frontier in DEX Design was originally published in Bancor on Medium, where people are continuing the conversation by highlighting and responding to this story.
