A bonding curve is a smart contract that sets a token's price as a mathematical function of its circulating supply: it mints new tokens when buyers arrive and burns them when holders sell, so a market is always open without an order book or a matching counterparty. Caper uses one as the permanent, built-in market for every caper's governance token. This page covers Caper's implementation first, then the wider idea it descends from.

Caper's curve

The curve is a perpetual AMM whose price is a smooth, monotonically increasing function of the fraction of supply in circulation. Price starts at zero with nothing sold and goes asymptotic as circulation approaches the full 100bn.

Buying moves XRD into the curve vault and tokens out of the inventory vault; selling reverses both. The cumulative XRD held by the curve has a closed-form integral, so sells price out exactly; buys solve for the new circulation numerically.

On every buy, an additional small slice of the buyer's curve-derived tokens is taken from the inventory vault and split:

• 30/31 goes to the founder vault.
• 1/31 goes to the Caper Commons — the protocol-wide treasury, denominated in $CAPER.

Founder and Commons rewards scale with curve activity rather than being front-loaded. Vampire capers (bound to an origin token) earn this same per-buy slice; in addition, holders of the origin token can migrate into the caper — a free mint that draws from the inventory vault and advances circulation without an XRD buy, with the resulting reserve gap backfilled by a fee on later direct buys. See Trading for the buy/sell panel and Raising funds for how the proceeds are deployed.

How bonding curves work

More generally, a bonding-curve contract holds a reserve (or collateral) pool of a base asset — XRD on Radix, historically ETH on Ethereum — and issues its own token against it:

• Buying / minting. A buyer deposits reserve assets; the contract mints new tokens, adds the deposit to the reserve, and the spot price rises because supply has increased.

• Selling / burning. A seller returns tokens; the contract burns them, releases the corresponding reserve, and the spot price falls.

The result is what Yos Riady calls a continuous token: it has a deterministic price (set by the formula, not negotiation), a continuous price (token n+1 always costs more than token n), and instant liquidity (any amount can be bought or sold at any time). Deposited funds stay locked in the contract, so the token is always backed by collateral that can be reclaimed by burning — the same property that lets every caper holder cash out at any time.

The mathematics

Two quantities are easy to confuse. The bonding curve itself, P(S), gives the spot price of the next token at supply S. The reserve held by the contract — and therefore the cost of a finite purchase — is the area under that curve, its integral. As Slava Balasanov illustrates with a simple quadratic curve:

spot price:    P(S) = S²
reserve:       R(S) = ∫₀ˢ P(s) ds = ⅓ · S³
cost to mint:  ΔR = ⅓·(S + n)³ − ⅓·S³   (to add n new tokens)

Because a buyer pays the integral rather than the final spot price, a large order pays a progressively higher average price as it climbs the curve. Each curve therefore has a characteristic reserve ratio — the share of the token's notional market capitalisation actually held as collateral:

reserveRatio = reserveBalance / (price × supply)

A linear curve P(S) = m·S has a reserve ratio of ½; the quadratic curve above has ⅓. The Bancor protocol generalises this by holding the reserve ratio (its "connector weight") constant, giving the widely reused Bancor formula:

price          = reserveBalance / (supply × reserveRatio)

purchaseReturn = supply × ((1 + reserveAdded / reserveBalance) ^ reserveRatio − 1)

saleReturn     = reserveBalance × (1 − (1 − tokensSold / supply) ^ (1 / reserveRatio))

A reserve ratio between 0% and 100% tunes the curve's steepness: 100% gives a flat, fully-collateralised price (a stable peg), while smaller ratios produce steeper curves that reward early buyers more. Caper's curve, P(s) = K · √s / (1 − s) in the circulating fraction s, is a steep variant: the price is zero at genesis (so there is no sniper floor) and rises to a pole as the float approaches the full supply.

Curve shapes

The choice of function shapes a token's incentives. Common forms, catalogued by BonDeFi and others, include:

• Linear — P = m·S + b. Constant-rate price growth; simple and predictable.

• Polynomial / quadratic — e.g. P = S². Faster appreciation that rewards early participants.

• Exponential — P = a·e^(b·S). Steep, speculative growth.

• Sigmoid / logarithmic — S-shaped or flattening curves that model adoption slowing as supply matures, damping late-stage volatility.

A contract may also use separate buy and sell curves; the gap between them forms a bid–ask spread that can fund the protocol or reward long-term holders. Caper instead uses a single curve in both directions — a flat trade fee, not a curve spread, funds each caper's treasury.

Choosing a curve's exponents and constants is a parametrization problem in its own right — see Bonding Curves In Depth: Intuition & Parametrization and On Single Bonding Curves for Continuous Token Models for the design trade-offs.

History

The continuous-token idea entered crypto with the Bancor protocol (2017), whose "smart tokens" used a constant reserve ratio to stay continuously convertible into their reserve. Later that year Simon de la Rouviere published "Tokens 2.0: Curved Token Bonding in Curation Markets" (21 November 2017), framing curved bonding as the pricing engine for curation markets — markets that "reduce information asymmetry through the use of tokenized signals." He credits the Zap team with coining the term "bonding curves."

In 2018 Thibauld Favre introduced Continuous Organizations, using a Decentralized Autonomous Trust (DAT) and a bonding curve to mint and burn "FAIR" tokens that represent claims on an organisation's future cash flows — giving projects continuous fundraising and guaranteed liquidity instead of a one-off sale. Caper is a direct descendant of this model: an organisation whose token lives on a permanent curve. Around 2019–2020 the Commons Stack, with research firm BlockScience, generalised the design into the augmented bonding curve, which diverts a share of every purchase into a "common pool" to fund public goods — the same shape as Caper's 1/31 Caper Commons slice. The underlying theory frames a bonding curve as an economic estimator of a token's value (Zargham, Paruch & Shorish).

Relationship to automated market makers

Bonding curves and automated market makers (AMMs) are close cousins. A single-reserve bonding curve is effectively a one-sided AMM that mints and burns the token itself, whereas a constant-function AMM such as Uniswap prices two pre-existing assets against each other along an invariant — the constant-product rule x · y = k. Bancor's price = reserveBalance / (supply × reserveRatio) is the bonding-curve generalisation of that idea.

A common claim is that bonding curves have "zero slippage." That is imprecise. Because a purchase pays the area under the curve between the old and new supply, a large order still pays a progressively higher average price — a price impact functionally like slippage, which is exactly why Caper's Trading guide notes that "slippage is implicit in the curve shape." The genuine distinction is that this impact is fully deterministic: it depends only on the trade size and the fixed curve, not on the fluctuating ratio of two pooled assets, and the curve can never run out of the token it mints. Liquidity is always available and its price impact is knowable in advance — but it is not literally free of price impact.

Applications

• Automated market making. Continuous, reserve-backed pricing underpins Bancor's smart tokens and the broader family of on-chain AMMs.

• Curation markets. Bonding curves let participants stake on the quality or relevance of information, rewarding early curators as a signal gains support (de la Rouviere).

• Continuous organizations & DAOs. A curve becomes a permanent primary market for a governance or revenue token, replacing fixed sales with continuous fundraising and guaranteed exit liquidity — the pattern Caper is built on.

• Public-goods funding. Augmented bonding curves route part of each buy into a common pool, turning token demand into a funding stream for open-source and charitable work.

• Token launchpads. On Radix, BonDeFi uses a batched bonding curve to launch tokens fairly; on Solana, Pump.fun popularised the "fair launch," pricing a token up the curve as buyers arrive before migrating it to a DEX.

Risks and criticisms

• Front-running / MEV. Because price depends only on supply, an adversary who sees a pending buy can buy first and sell into it. Both Balasanov and Riady flag bonding curves as "susceptible to front-running attacks"; batched curves and ordering protections aim to mitigate it.

• Reflexivity and volatility. Tying price to supply creates a feedback loop — buying raises the price, which attracts more buying. Steep curves amplify this into pump-and-dump dynamics where a rush to exit can collapse the price as fast as it rose.

• Market cap is not collateral. Because the reserve is the area under the curve, a token's notional market capitalisation (price × supply) is typically several times the reserve backing it. Every holder can still redeem along the curve, but the headline "value" is reflexive and can evaporate under mass selling.

• Parameter and composition risk. The wrong curve shape or reserve ratio can make a token too volatile or too illiquid, and deeply nested curves (a curve collateralised by another curved token) compound reserve ratios and magnify risk.

Further reading

• Tokens 2.0: Curved Token Bonding in Curation Markets — Simon de la Rouviere
• How to Make Bonding Curves for Continuous Token Models — Slava Balasanov
• Bonding Curves Explained — Yos Riady
• Bonding Curve component docs — Aragon Fundraising
• Introducing Continuous Organizations — Thibauld Favre
• Deep Dive: Augmented Bonding Curves — Commons Stack / Giveth
• Bonding Curves & Anti-Snipe — BonDeFi
• Bonding Curves In Depth: Intuition & Parametrization — Relevant Community
• On Single Bonding Curves for Continuous Token Models — thoughtchains
• Economic Games as Estimators — Zargham, Paruch & Shorish (BlockScience)

Caper’s bonding curve

Every caper’s token trades against its own bonding curve: the price is a formula of how much supply has sold, so there is always a live market and never a need for a counterparty. The reserve backs every token in circulation.

What you actually pay

The curve sets the base price; the founder reward skims a share of every buy. That share is front-loaded – 15% at the first buy, tapering straight to zero by 30% sold – so as a markup on the price you pay it starts near 18% and fades to the 0.5% trade fee. The reward's value reads the other way: because price starts near zero, the founder's actual take is a hump that is near zero at launch and peaks around 11% sold, scaling with activity rather than being front-loaded. Either way the price you pay only ever rises, so the earliest buy is always the cheapest.