Bonding Curve

Bonding curves on Morpheum: ConstantProduct math, effective MCAP, graduation triggers, MORM escrow, Hyper-CLAMM hand-off, module boundaries, and flows—with diagrams and LaTeX.

Overview

A bonding curve is a deterministic rule that maps token supply (or reserves) to price or marginal mint/burn cost. On Morpheum, the BondingCurve subsystem is the incubation launchpad for agent tokens: a curve phase with mandatory MORM quote, MWVM hooks on trades, and an atomic graduation into Hyper-CLAMM, with optional scale migration to CLOB venues later. Conceptually it sits next to CLAMM, ReClamm, and CLOB: the curve fixes price–supply dynamics; the CLOB handles resting orders and price–time priority after migration.

This page synthesizes the rclob BondingCurve design (curve math, state, graduation bundle, lifecycle) into a single reader-friendly reference—with Mermaid figures for architecture and flows and LaTeX for the core formulas.

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Economic progression (three stages)

At a high level, liquidity and venue evolve in three stages:

During the curve phase there is no pool-based mark price from markprice-v2 for the pair; valuation uses curve-derived quantities (see Effective market cap). After graduation, markprice-v2 applies for scale migration decisions when the token is in the Hyper-CLAMM phase.

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Mathematical core: ConstantProduct on Morpheum

The default curve type is ConstantProduct-style pricing on remaining mintable supply.

Let:

• $S_{\max}$ — maximum supply cap for the curve (from state or Token rules).
• $S$ — minted supply (tokens sold by the curve so far).
• $S_{\mathrm{rem}} = S_{\max} - S$ — remaining supply before hitting the cap.

The design uses a constant $k > 0$ and spot price (quote per base, in the same precision regime as implementation):

$$P(S) = \frac{k}{S_{\mathrm{rem}}} = \frac{k}{S_{\max} - S}.$$

As $S \to S_{\max}^-$, remaining supply shrinks and marginal price diverges—the classic convex launch dynamic (early buyers cheaper than late buyers, modulo fees and mechanics).

Effective market cap and graduation triggers

Define curve-derived effective market cap (used while the curve is Active and no Hyper-CLAMM pool exists yet):

$$\mathrm{MCAP}_{\mathrm{eff}} = P(S) \cdot S = \frac{k}{S_{\max} - S}\, S.$$

Graduation can trigger (governance-tuned; illustrative structure from design docs) when either:

1. Hard cap: $\mathrm{MCAP}_{\mathrm{eff}} \ge M_{\mathrm{grad}}$, or
2. Proof-of-utility early path: utility points $U$ (from real MORM revenue via MWVM hooks) are converted into an equivalent MCAP bump with multiplier $\lambda$:

$$\lambda\, U + \mathrm{MCAP}_{\mathrm{eff}} \ge M_{\mathrm{early}}.$$

So “good” agents can graduate before the raw MCAP alone would allow—subject to anti-abuse rules (minimum revenue per hook, daily caps, non-creator revenue, etc.).

Linear alternative (conceptual)

A linear curve type (optional / later) uses sold amount $Q = S$ and parameters $b, m$:

$$P_{\mathrm{lin}}(Q) = b + m\, Q.$$

Trade-offs: easier mental model, less extreme tail convexity than ConstantProduct near the cap.

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Buy and sell: integral view of cost (science)

In a one-dimensional bonding curve, the MORM cost to buy an infinitesimal $\mathrm{d}S$ at sold supply $S$ is $P(S)\,\mathrm{d}S$. The total quote to move from $S_0$ to $S_1$ is:

$$\Delta Q_{\mathrm{MORM}} = \int_{S_0}^{S_1} P(S)\,\mathrm{d}S.$$

For the rational form $P(S) = k/(S_{\max} - S)$, this is a logarithmic total cost in the continuous idealization; the on-chain implementation uses u128 arithmetic on discrete steps—no float64 on the consensus path—so validators agree bit-for-bit.

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Architecture: who BondingCurve talks to

BondingCurve is intentionally a thin orchestrator: curve math, escrow accounting during incubation, graduation detection, and the atomic hand-off. Token metadata, balances, pools, and CLOB markets live in other modules.

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Scope boundary (ownership matrix)

Owns	BondingCurve	Token	Bank	Liquidity
Curve price math & buy/sell	Yes	—	—	—
Escrow MORM during curve	Yes	—	—	—
Graduation trigger & bundle	Yes	—	—	—
Metadata, DID, MWVM hooks	—	Yes	—	—
Balances & custody	—	—	Yes	—
Pool creation & CLAMM seed	—	—	—	Yes

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Token lifecycle (canonical phases)

The canonical lifecycle lives on Token; BondingCurve internal CurveStatus maps to it.

Illustrative phase codes from the design: InCurve (10) → Graduating (11) → InHyperClamm (20) → later InCLOBSpot / InCLOBPerp after migration.

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Graduation decision (base vs prediction enhancement)

Let base be true when the hard MCAP or utility-adjusted condition holds (see Effective market cap). Optional prediction modes combine with oracle resolution as follows (conceptually):

$$\text{final} = \begin{cases} \text{base} & \text{None} \\[0.4em] \text{base} \;\vee\; (\text{feed} = \text{YES}) & \text{Boost} \\[0.4em] (\text{feed} = \text{YES}) & \text{Gated} \end{cases}$$

After final is true, a cooldown window (anti-frontrun) must elapse before the bundle executes; if conditions fail mid-flight, the implementation reverts the transient lock back to Active.

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Graduation bundle (atomic)

When triggers pass and cooldown elapses (anti-frontrun), execution is all-or-nothing: lock state, burn a governance fraction of escrow MORM (default ~12–18%, often 15%), seed the Hyper-CLAMM pool with the remainder, apply LP anti-rug strategy (e.g. timelock so migration can drain the pool later), update Token phase to InHyperClamm, emit GraduationComplete.

Pricing note: During the curve, use $\mathrm{MCAP}_{\mathrm{eff}} = P(S)\,S$; markprice-v2 × supply is for post-pool phases (e.g. scale migration), not for the raw curve phase.

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Optional prediction enhancement

Most tokens use no extra logic (deterministic graduation). Optionally, creators link a prediction feed:

Mode	Effect
Boost	Base threshold or linked market resolves YES → can graduate.
Gated	Graduation blocked until linked market resolves YES.

This does not change the core ConstantProduct math; it gates when the bundle may run.

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Key flows: buy (sequence)

Quote asset: MORM is the mandatory quote during curve phase in the design; non-MORM routes may incur extra fees (governance).

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Determinism and precision

• All curve and escrow arithmetic targets u128 with satoshi-scale fixed-point conventions—no float64 on the consensus hot path.
• Graduation is a single atomic transaction bundle: no partial pool seed without burn policy applied consistently.
• Utility points and cooldown blocks are deterministic functions of observed events and block height—same ordering yields identical state across validators.

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Who this is for

• Protocol designers tuning $k$, $S_{\max}$, $M_{\mathrm{grad}}$, $M_{\mathrm{early}}$, burn bps, and LP anti-rug strategy.
• Agents launching agent tokens with ERC-8004-style DID and reputation gates.
• Developers integrating BondingCurveHotPath queries (get_curve_state, get_price) for UIs and indexers.
• Traders reasoning about slippage along $P(S)$ and graduation timing.

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See also

• CLOB — order-book matching and price–time priority.
• CLAMM use cases — when AMM-style liquidity fits.
• ReClamm glide mechanics — post-curve AMM behavior on Morpheum.