# Phase 2: A Daring Expedition Unveiled

This phase marks the active and thrilling part of the game, where the intrepid privateers begin their excavation of the sunken treasure, fulfilling their promise to the eager villagers.



To kickstart this stage, PEARL is listed on PancakeSwap, a popular Decentralized Exchange (DEX) and well-known crypto market data aggregators such as CoinmarketCap and CoinGecko.&#x20;

A PEARL/USDT trading pair is established, allowing players to engage in trading activities. The game's proprietary Asset Management takes charge, leveraging the proceeds generated from PEARL sales against other assets on a centralized exchange, thereby generating additional income. These trading returns, if any, flow back into the game's ecosystem, fueling its growth and development.



At the onset, a treasure chest containing a substantial amount of 990,000,000 (990 millions) PEARL serves as the primary focus. As the treasure is gradually brought back to the village, profit sharing between the game and players is divided in a 50%/50% ratio. Specifically, 50% of the Asset Management earnings are allocated for PEARL buybacks on PancakeSwap, where the purchased PEARL is then burned (removed from token total circulating supply) in exchange for NFT minting. The remaining proceeds obtained from the exchanges at the end of each day contribute to the reward pool, which is distributed to the players from the System Fund.



Once all the treasure PEARL is successfully extracted from the sea-bed, the profit-sharing ratio undergoes a significant shift. While the era of pearl mining has reached its culmination, the system continues to demand pearls for the creation of captivating in-game collectible NFTs. Consequently, a portion of the funds from the Treasury is allocated toward purchasing PEARL tokens from the decentralized exchange. The acquired pearls are then distributed in a balanced ratio of 50% each.



The initial 50% is shared among the players who have staked PEARL tokens, rewarding their active participation. The remaining 50% is divided into two distinct shares. A significant portion, accounting for 45%, is directed towards the burning process, fueling the creation of NFTs with an alluring allure. The remaining 5% is allocated to the pirates, acknowledging their continued contributions and ensuring their involvement in the ongoing adventure.



Although the formula governing the DEX buyback mechanism may appear complex at first glance, rest assured that we will delve into its intricacies and explain it in a more accessible and understandable manner.

**Buyback pressure function**

$$\LARGE\int\limits_y^z f\Bigg(\sum\int\frac{s}{y}\Bigg)=N$$

Let's break down the statements using simpler language:

* We use the variable 'z' (integration limit) to represent a specific part (shard) of the system, which we call 'shard zero'.
* The variable 'y' (integration limit) represents the total number of shards across all functions. Think of it as the overall count of all the parts in the system.
* The variable 's' indicates the cumulative sum of shards for the current function. In other words, it keeps track of how many shards we've counted so far in the ongoing process.
* 'y' represents the number of shards for the current function. It helps us keep track of the count of shards specific to the current step or process we're looking at.
* Finally, 'N' represents the sum of all the shards across all functions. It gives us the total count of all the parts in the entire system.

In this context, we use the integral to calculate the cumulative sum of buyback functions. This accounts for the inherent variability in shard sizes, the time ranges during which buybacks occur, and the specific moments at which buybacks happen.

Although we have the total sum of each local buyback shard, the integral helps us consider the variability associated with each shard's size and the timing of the buybacks.

By using the integral, we can account for the randomness and unpredictability in the system, ensuring that our calculations accurately reflect the varying sizes of the shards and the temporal aspects of the buyback process.

**Current system fund state**

We have a variable called F(t), which represents the size of a fund at a specific time t. Our goal is to determine the fund's size by considering the contribution function R(t), which describes how contributions change over time. The equation for this relationship is:&#x20;

$$\LARGE F(t)=F(0)+\int[0,t]R(u)du$$

Here are the key variables and what they represent:

* F(t): The size of the fund at a given time t. This is the variable we want to find, and it depends on other factors.
* t: Time, representing a specific moment or duration for evaluating the fund's size. It is the independent variable.
* F(0): The initial size of the fund at the starting point (t = 0).
* R(u): The contribution function, a function of the variable u (usually representing time). It describes the rate or amount of contributions made to the fund at different time points.
* du: A differential element associated with the integration variable u. It represents an infinitesimally small increment of u within a specified integration interval.
* In the equation, F(0) represents the initial size of the fund. The integral notation (∫) signifies a definite integral over the interval from 0 to t. R(u) represents the contribution function evaluated at the variable u, which represents time.

Using this equation, we can determine the fund's size at any given time t by considering the initial fund size (F(0)) and the cumulative effect of contributions according to the contribution function R(t). The integral captures the accumulation of changes in the fund size from the starting point (time 0) to the specified time (t).\


It's important to note that to precisely determine the fund size at a specific time, we need to know the contribution function R(t) or have specific values of the fund size at different time instances. This equation allows us to incorporate these factors and calculate the fund's size at any desired time.

**System fund state after the end of funding period**

Let's consider the variable t, representing the current moment in time, and our goal is to determine the fund's size F(t) at this time. One possible approach to estimate the fund's size is by using linear interpolation. We assume that the change in the fund's size is proportional to the change in time.



The equation for linear interpolation is:

$$\LARGE F(t)=F(0)+(F(∞)-F(0))\cdotp( \frac {t}{T})$$

Here are the key components of the equation:

* F(t): The estimated fund size at the current time t.
* F(0): The initial fund size at time t = 0.
* F(∞): The final fund size at the end of the funding period.
* T: The total duration of the funding period.
* In the equation, (F(∞) - F(0)) represents the change in the fund's size over the entire duration of the funding period. The term (t / T) represents the fraction of time that has elapsed within the total duration.

By plugging in the appropriate values for F(0), F(∞), T, and t, we can calculate an estimated value for the fund's size at the current moment t using linear interpolation.

**Target price function**

$$\LARGE f(Δ,∇,c) = 0.01+ \frac {Δ}{∇\cdotp c}$$

In the given formula, we have the following variables and parameters:

* 0.01: This value represents the base price. It serves as a fixed starting point or initial value for the calculation.
* Delta (Δ): Delta represents the time elapsed from the start of the project. It acts as a parameter that influences the overall value of the function. As the value of Delta changes, it impacts the outcome of the calculation.
* Nabla (∇): Nabla is a price vector that depends on the fund and system performance. It represents a vector of prices associated with different components or aspects of the fund. The specific values in the nabla vector determine the prices for different elements within the fund. The value of nabla affects the calculation by introducing the concept of price differentiation based on the fund's characteristics.
* c: The variable c is a correction multiplier that depends on another function of the system. It acts as a scalar multiplier, adjusting the overall impact of the function based on the output of the system's additional function. The value of c modifies the result, accounting for correction factors specific to the system.

In summary, this formula calculates a value based on the base price, the elapsed time, a price vector, and a correction multiplier. Each variable or parameter plays a role in influencing the final outcome of the calculation, considering factors such as time, differentiation in prices, and system-specific corrections.

$$\LARGE ∇ (t)=F(0)+F(∞)-F(0)) \cdotp \int\limits_y^z \sum{p}$$

In the given formula, we have the following variables and parameters:

* ∇ (t): nabla is represented as a variable that depends on time, t, and is calculated based on the given formula. It determines the value of nabla at a specific moment in time.
* F(0): F(0) represents the initial fund size. It is a constant value that serves as the starting point for the calculation.
* F(∞): F(∞) represents the final fund size. It is a constant value that represents the fund size at the end of the calculation.
* t: t represents the current time, indicating the specific moment or duration for which we want to evaluate the nabla variable.
* T: T represents the total duration or time span between the start and end points. It defines the overall time frame for the calculation.
* F: F represents the system fund size. It is a variable that contributes to the calculation of nabla. The specific value of F affects the outcome of the calculation.
* ∫\[y, z]: ∫\[y, z] denotes the definite integral over the interval from y to z. It represents the summation of the integral of the function p with respect to the variable p.
* p: p represents the system profit. It is a variable that contributes to the calculation of nabla. The value of p influences the final result of the calculation.

In summary, this formula involves variables such as nabla, F(0), F(∞), t, T, F, ∫\[y, z], and p. Each of these variables plays a role in determining the value of nabla based on specific conditions, including time, system fund size, and system profit.

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