—————-

# Vertisan AMM: Dynamic Equilibrium Liquidity Model

![[Vertisan-Automatic-Market-Maker-Formula.jpg]]

# Vertisan AMM: Dynamic Equilibrium Liquidity Model

## 1. Dynamic Equilibrium Pricing Mechanism

– **Formula**:
Vertisan’s AMM utilizes a dynamic, risk-adjusted equilibrium model, advancing far beyond constant product designs.

The core balance is governed by:

$$
KD^{N-1} sum x_i + prod x_i = KD^N + left( frac{D}{N} right)^N
$$

—

## 2. Core Variables and Constants

– $x_i$ = quantity of each asset in the pool
– $K$ = risk-adjusted amplification constant
– $D$ = dynamic liquidity depth
– $N$ = number of assets in the pool

—

## 3. Supporting Formulas and Amplification Factors

– **Base Constant $K_0$**:
Defines the initial price constant:

$$
K_0 = frac{ prod x_i }{ (D/N)^N }
$$

– **Auxiliary Amplification Variable $chi$**:
Scales the base constant by the amplification parameter:

$$
chi = A K_0
$$

– **Amplified Constant $K$**:
Adjusts $K$ based on the risk parameter $gamma$:

$$
K = A K_0 left( frac{ gamma^2 }{ (1 – K_0)^2 } right)
$$

where:
– $A$ = amplification constant
– $gamma$ = risk tuning factor controlling sensitivity

—

## 4. Rebalancing Function

– **Market Equilibrium Function $F(x, D)$**:

$$
F(x,D) = K(x,D) D^N sum x_i – K(x,D) D left( frac{D}{N} right)^N
$$

– **Equilibrium Condition**:

$$
F(x,D) = 0
$$

—

## 5. Iterative Depth Adjustment

– **Depth Recalculation After Each Trade**:

$$
D_{k+1} = D_k – frac{F(x_k, D_k)}{F’_D(x_k, D_k)}
$$

where:
– $D_k$ = current liquidity depth
– $F’_D$ = derivative of the rebalancing function with respect to $D$

—

## 6. Pool Initialization

– **Initial Depth $D_0$**:

$$
D_0 = N left( prod x_i right)^{1/N}
$$

– **Initial Asset Balances $x_{i,0}$**:

$$
x_{i,0} = frac{D^{N-1}}{ prod_{k neq i} x_k^{sqrt{N-1}} }
$$

—

## 7. Liquidity Pools and Adaptive Liquidity Management

– **Liquidity Pools**:
Vertisan pools multiple assets, enabling diversified liquidity structures.

– **Liquidity Efficiency**:
Dynamic depth (D) adjustment reallocates capital optimally to minimize slippage.

– **Participant Incentives**:
Liquidity providers earn transaction fees and other incentives.

—

## 8. Slippage Minimization and Manipulation Resistance

– **Low Slippage**:
Achieved through continuous depth recalculations after every trade.

– **Manipulation Resistance**:
The non-linear dynamics of $K$, $D$, and $gamma$ make manipulative strategies prohibitively expensive.

—

## 9. Strategic Advantages Over Traditional AMMs

– Internal self-correcting liquidity without external oracle reliance
– Scalable multi-asset liquidity structures
– Continuous dynamic market rebalancing and risk damping mechanisms

—

## Summary

Vertisan’s Automated Market Maker (AMM) introduces a fundamentally new architecture for decentralized liquidity.
Through dynamic depth recalculations, internal price stabilization, and embedded risk mitigation, Vertisan delivers:

– Superior liquidity efficiency
– Minimal trade slippage
– Built-in manipulation resistance
– Ready scalability for next-generation DeFi systems

—

## Tags
– #Vertisan
– #AMM
– #Liquidity
– #DynamicPricing
– #RiskAdjustment
– #DeFiInfrastructure

## Related Sections
– [[2.2 Automated Market Maker (AMM) and Liquidity Model]]
– [[2.3 VTSN Tokenomics]]
– [[2.5 Vertisan Exchange and Ecosystem Features]]