# Explanation of $\Pi$ index and ($Pi$) Token Issuance

This document outlines how $Pi$ tokens are issued in two phases.
## Overview of Variables

| Notation | Meaning |
|-----------------------|--------------------------------------------------------------------------------------------------------------------------|
| $AO$ | The base token. |
| $\Pi$ | The index. Users can delegate their $AO$ to $\Pi$ in return for newly issued $PI$ tokens. |
| $AR, \, TKN_n$ | Underlying assets or investments that $\Pi$ can hold. |
| $PI_V$ | The total value (in $AO$) of all the underlying assets that $\Pi$ holds. |
| $AO_{\text{add}}$ | The number of new $AO$ tokens that a user delegates to $\Pi$. |
| $m$ | A constant that determines how many $PI$ tokens are minted per $AO$ in **Phase 1**. |

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## Phase 1

During **Phase 1**, $\Pi$ does **not** invest any delegated $AO$ into $AR$, $TKN_n$, or other assets. Instead, new $Pi$ tokens are issued based on a constant multiplier $m$.

### Formula

$$
\Pi_{\text{add}_u} = AO_{\text{add}} \times m
$$

- $\Pi_{\text{add}_u}$: Number of new $PI$ tokens issued to the user
- $AO_{\text{add}}$: The $AO$ tokens the user delegates to $\Pi$
- $m$: A constant multiplier determining how many $PI$ tokens are minted per $AO$

### Example for Phase 1

- **Assumptions**:
- $AO_{\text{add}} = 100$
- $m = 2$

- **Calculation**:

$$
\Pi_{\text{add}_u} = 100 \times 2 = 200
$$

- **Interpretation**:
The amount of issued $PI$ tokens is **200** in exchange for **100 $AO$**.

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## Phase 2

In Phase 2, $\Pi$ **starts investing** the delegated $AO$ into various assets (e.g., $AR$, $TKN_n$). Hence, the total value of $\Pi$’s holdings, denoted as $PI_V$, comes into play. The issuance of new $Pi$ tokens depends on the ratio of newly contributed $AO$ tokens to the total $AO$ value of $\Pi$.

### 1. Issuing $PI$ Tokens to Users

When a user delegates additional $AO$ to $\Pi$, the formula for new $Pi$ tokens is:

$$
\Pi_{\text{add}_u} = \frac{AO_{\text{add}} \times \Pi}{PI_V}
$$

- $\Pi_{\text{add}_u}$: New $Pi$ tokens issued
- $AO_{\text{add}}$: Additional $AO$ tokens the user delegates to $\Pi$
- $Pi$: Current circulating supply of $Pi$ tokens (i.e., total ownership)
- $PI_V$: The total value (in $AO$) of $\Pi$’s underlying assets

#### Example: User Delegation in Phase 2

| Parameter | Value |
|---------------------------|--------:|
| $AO_{\text{add}}$ | 500 |
| $\Pi$ | 10,000 |
| $PI_V$ | 20,000 |

- **Calculation**:

$$
\Pi_{\text{add}_u}
= \frac{500 \times 10{,}000}{20{,}000}
= \frac{5{,}000{,}000}{20{,}000}
= 250
$$

- **Interpretation**:
The user receives **250 new $Pi$ tokens** for delegating **500 $AO$**.
By factoring in $PI_V$ and $\Pi$, this formula ensures that the user’s share in $\Pi$ remains proportional to the total value $\Pi$ holds.

### 2. Issuing $Pi$ Tokens to Projects (Fair Launch)

Projects fair launching on $AO$ may also become a part of the $\Pi$ index. In that case, $\Pi$ invests $AO$ into a bouquet of ecosystem assets (the Fair Launched projects). The Project than, are receiving $PI$ tokens instead of $AO$. The formula is analogous to user delegation:

$$
\Pi_{\text{add}_p} = \frac{AO_{\text{add}_p} \times \Pi}{PI_V}
$$

- $\Pi_{\text{add}_p}$: New $PI$ tokens issued to the project
- $AO_{\text{add}_p}$: $AO$ tokens delegated by the project
- $PI$: Current circulating supply of $PI$ tokens
- $PI_V$: Total $AO$ value of $\Pi$’s underlying assets

#### Example: Project Fair Launch in Phase 2

| Parameter | Value |
|-----------------------------|--------:|
| $AO_{\text{add}_p}$ | 1,000 |
| $\Pi$ | 10,000 |
| $PI_V$ | 20,000 |

- **Calculation**:

$$
\Pi_{\text{add}_p}
= \frac{1{,}000 \times 10{,}000}{20{,}000}
= \frac{10{,}000{,}000}{20{,}000}
= 500
$$

- **Interpretation**:
The project receives **500 new $PI$ tokens** in exchange for **1,000 $AO$**, maintaining fair ownership relative to $\Pi$’s total underlying value.

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