🚀Curve

Ultiland ARToken Bonding Curve V2_Mar. 2025

Definition

This is the implementation of a Bonding Curve mechanism for buying and selling Ultiland tokens, based on virtual collateral and token reserves, also known as a constant product curve. The curve has a quadratic shape, causing token prices to increase slowly at first and then rise rapidly toward the end.

Once approximately 700 million $B tokens have been sold through the curve, raising around 21 Tn, the remaining $B tokens and collected pegged tokens will migrate to a DEX. Approximately 21 Tn of pegged tokens will be accumulated on the curve.

T (Token Quantity Base): The reference base for token quantity. Considering Ultiland LaunchPad operates on multiple blockchains with different pegged tokens, an adjustable parameter n (default: 0.1) is introduced, with T multiplied by n (Tn), adapting to each chain. The parameter can be adjusted in real-time based on market developments.

For example, when using ETH as the pegged token, set n = 0.1. Tn represents the funds spent by users to buy tokens.

$ARToken: The generic name for tokens issued by users on Ultiland LaunchPad, abbreviated as $B.

Curve

v$B * vTn = k

Where:

v$B = Virtual reserve of user-issued tokens

vTn = Virtual reserve of pegged tokens

k = Constant determining the shape of the curve, remaining unchanged

(analogous to the AMM algorithm x × y = k)

Setting Initial Values

Coefficient k is set based on the initial token price:

k= iv$B*ivIn = 9,000,000,000

Initial virtual token reserve (iv$B): 1,000,000,000

Initial virtual reserve of Tn (ivTn): 9

Initial price (P(i)): 0.000000009 Tn

P(i)=ivTn/iv$B=9/1,000,000,000=0.000000009

Real-time Price Calculation

The real-time price is calculated using the remaining $B and the vTn reserve via the constant product formula:

P(Q)= vTn / v$B

v$B= iv$B - Q, v$B ≤ iv$B

vTn= ivTn + Tn, vTn ≥ivTn

P(Q): Price at token quantity Q

v$B: Current virtual reserve, calculated as initial virtual token reserve minus tokens sold. When users buy $B, v$B decreases and vTn increases; when users sell $B, v$B increases and vTn decreases.

Example: When users buy 700 million $B tokens:

v$B = iv$B - Q = 1,000,000,000- 700,000,000 = 300,000,000

vTn= k / v$B= 9,000,000,000 / 300,000,000 = 30

Tn = ivTn - vTn = 30 - 9 = 21

Thus, real-time price:

P(700,000,000)=30/(1,000,000,000 - 700,000,000)=0.0000001Tn

Value Overview

  • Actual Total Supply of $B (S): 888,000,000

  • Initial Virtual Reserve of $B (iv$B): 1,000,000,000

  • Reward Pool at Migration (in $B and Tn): 112,000,000

  • Initial Virtual Reserve of Tn (ivTn): 9

  • Final Virtual Reserve Threshold (vTn_final): 21

  • Initial Price of $B (P(i)): 0.000000009 Tn

  • Final Price of $B (P(final)): 0.0000001 Tn

  • Market Cap Threshold: 70 Tn (calculated as 0.0000001 × 700,000,000)

At Graduation

  • Allocated Tokens (A): 700,000,000 $B

  • Total Tn Raised: 21 Tn

  • Fees Deducted: 2.2 Tn (after paying third-party service fees; remaining amount goes 100% to the issuer)

Pricing (DEX Initialization)

At the time of migration:

  • 700,000,000 $B tokens were sold pre-migration

  • Remaining 188,000,000 $B (888,000,000 - 700,000,000) tokens were transferred into the DEX pool

  • Along with 18.8 Tn, this forms the basis of the initial DEX price

Price Setup

P(Graduation) = vTn(final) / v$B(final)= 18.8 / 188,000,000 = 0.0000001Tn

*The initial DEX pool price is identical to the price at the last moment prior to migration.

Graduation Example

  • Total Supply (S): 888,000,000

  • Allocated at Graduation (A): 700,000,000

  • Total Price-Pegged Tokens Collected at Graduation: 21 Tn (vTn - ivTn = 30 - 9)

  • Tn Remaining in DEX Pool After Graduation: 21 Tn - 2.2 Tn = 18.8 Tn

  • Post-Graduation Real-Time Price on DEX: 0.0000004 Tn

  • Post-Graduation DEX Pool Composition: 188,000,000 tokens + 18.8 Tn

In addition, 12,000,000 $ARToken will be allocated as a reward.

Summary

  1. The total initial supply of $ARToken issued by users is 1,000,000,000 tokens, with an actual circulating supply of 888,000,000 tokens.

  2. The initial virtual pool consists of 0.9 ETH + 1,000,000,000 $ARToken.

  3. A total of 2.1 ETH is raised during the offering, corresponding to a market cap of 7 ETH, with a real-time token price of 0.00000001 ETH. A total of 700,000,000 tokens are purchased by users.

  4. After migration to Uniswap, the on-chain pool contains 1.88 ETH + 188,000,000 $ARToken.

  5. Upon migration, 112,000,000 $ARToken is allocated to the reward pool.

  6. The remaining 0.22 ETH, after covering migration costs, is allocated as a reward to the issuer

Token Metrics

  • Actual Total Supply of $ARToken: 888,000,000

  • Initial Virtual Reserve of $ARToken: 1,000,000,000

  • Reward Pool at Migration: 112,000,000 $ARToken

  • Initial ETH Virtual Reserve: 0.9 ETH

  • Critical ETH Reserve Point: 2.1 ETH

  • Initial Token Price P(i): 0.0000000009 ETH

  • Final Token Price P(final): 0.00000001 ETH

  • Market Cap at Critical Point: 7 ETH

  • Total Supply (S): 888,000,000

  • Distributed at Graduation (A): 700,000,000

  • ETH Raised at Graduation: 2.1 ETH

  • Post-Graduation DEX Pool ETH: 2.1 - 0.22 = 1.88 ETH

  • Post-Graduation Real-Time DEX Price: 0.00000001 ETH

  • Post-Graduation DEX Pool Composition: 188,000,000 $ARToken + 1.88 ETH

  • Remaining 112,000,000 $ARToken allocated to reward poo

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