šŸ”£Bonding Curve 101

The Bonding Curve is a powerful mathematical concept that has gained popularity in the world of decentralized finance (DeFi) and blockchain-based networks. The Liquidity-driven Curation Mechanism offers a secure, transparent, and traceable guarantee by tokenizing untradable and intangible virtual assets such as data, IPs, AI Agents, influence, and MEMEs.

ā˜„ļø Bonding Curve 101

Let R be the current reserve of the parent currency(say, $BOOM); Let P be the price of BLP Tokens(the share token of Bonding Pool) ; Let S be the current circulating supply of BLP Tokens. So we have the reserve ratio K:

K=RPSK=\frac{R}{PS}

When a user buys an infinitesimal amount of BLP tokens dS (selling simply means dS < 0), we have:

PdS=dR=K(SdP+PdS)PdS=dR=K(SdP+PdS)

After performing the integral, we have(C is arbitrary constant for a given K):

P=eCS1Kāˆ’1P={e^C}{S^{\frac{1}{K}-1}}
R=eCKS1KR=e^CKS^{\frac{1}{K}}

If the total supply of BLP token increases from S_0 to S, then the price increases from P_0 to P. The relationship between the two can be expressed as:

PP0=(SS0)1Kāˆ’1\frac {P} {P_0} = (\frac {S} {S_0})^{\frac {1}{K}-1}

If a user buys a total of N BLP tokens, bringing the total supply from S_0 to S_0+N , the total paid amount of parent currency A is:

A=āˆ«S0S0+NPdS=āˆ«S0S0+NP0(SS0)1Kāˆ’1dSA=\int^{S_0+N}_{S_0}{PdS}=\int{^{S_0+N}_{S_0}{P_0}(\frac{S}{S_0})^{\frac {1}{K}-1}}dS

then, we have:

A=R0((1+NS0)1Kāˆ’1)A=R_0\bigg((1+\frac{N}{S_0})^{\frac{1}{K}}-1\bigg)
N=S0((1+AR0ļ¼‰Kāˆ’1)N=S_0\bigg((1+\frac{A}{R_0}ļ¼‰^{K}-1\bigg)

Let's simplify the bonding curve function mentioned in the previous section. Let

eC=me^C=m
1Kāˆ’1=n\frac{1}{K}-1=n

Thus, we have:

P=mSnP=mS^{n}

R=mKSn+1R=mKS^{n+1}

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