# Fusing

Fusing is the Paragon operation that takes five TierN Paragons to create a new Paragon of Tier(N+1).

Five TierN Paragons are fused to produce a new Tier(N+1) Paragon. The original Paragons are burned, and the resulting Paragon is unlocked after some time and returned to the wallet. Fusing cost and time increase with N, the Tier of Paragons being fused according to Table below.

Tier | Fusing Cost [HTO] | Fusing Time [days] |
---|---|---|

0 | 20 | 3 |

1 | 40 | 5 |

2 | 80 | 8 |

3 | 160 | 15 |

4 | 320 | 30 |

Fusing is the operation reducing the supply of Paragons and is applicable for Paragons from Tier0 to Tier4. The basic schema for fusing is shown in Image 1 where it's marked that 4 Paragons are fused to one, central Paragon.

It's much more complicated to describe how Paragon fusing works as the algorithm takes into account all the core attributes of all the Paragons entering the operation and must also consider Paragon Tier. Fusing is designed in a way that fusing high-grade Paragons produces high-grade Paragon of higher Tier.

In order to find utility for low-grade Paragons, we've designed a way how they can also produce a high-grade Paragon of higher Tier, but this requires those Paragons be in the "same family". Everything is explained later in this Section.

Let $a^1, a^2, \dots, a^9$ denote 9 core attributes, $t \in \{ 0, 1, \dots 4 \}$ tier of Paragons being fused. Let $\alpha$ be the main Paragon (don't confuse this with Alpha PFP) to which 4 beta Paragons $\beta_1$, $\beta_2$, $\beta_3$ and $\beta_4$ are being fused. Then, $a_\alpha^1, a_\alpha^2, \dots, a_\alpha^9$ are 9 core attributes values of $\alpha$ Paragon and $s_\alpha = \sum_{i=1}^{9} a_\alpha^i$ its score.

For Tier $t$, we define $s_{min}^t$ and $s_{max}^t$, minimum and maximum Score that Paragons within given Tier can have. From Table above follows for Tier 1 $s_{min}^1 = 100$, $s_{max}^1 = 199$ and similar for other Tiers.

The resulting Paragon will have values

Calculation of $a_+^i \geq 0$ will be defined further. $a_+^1, a_+^2, \dots, a_+^9$ are values added to attribute values of $\alpha$ Paragon. Let's define $s^+$ as their sum

The condition $s^+ \in \langle s_{min}^+, s_{max}^+ \rangle$ follows from the fact that the Score of the resulting Paragon must be within limits defined by its Tier, which is $t+1$. We make sure this condition is fulfilled by using hyperbolic tangent as normalization function in $s^+$ definition

where $c_\alpha$ is derived from $\alpha$ Score, $c_\beta$ from $\beta$ Score and $c_\beta^\sigma$ from $\beta$ Score standard deviation as

It can be easily checked, that all of these values have values from interval $\langle 0, 1 \rangle$. In definition, we've used standard deviation $\sigma (\{ x_i \}_{i=1}^{N})$ that is calculated using mean value $\langle x \rangle$ as

$s^+$ defines the sum of attribute values that will be added to $\alpha$ Paragon. You can think of it like running $s^+$ cycles where in every cycle, you select one core attribute of $\alpha$ Paragon and increase its value by one. The question is, what attributes should be increased and how much. To do so, we define probability $p^i$, that the attribute $i$ will be the one selected for the increase. Let's first define weight

which you can see is equal to attribute value in $\alpha$ Paragon $a_\alpha^i$ and $1/4$ of the attribute value in all of the $\beta$ Paragons $a_{\beta_k}^i$. The probability is obtained by normalizing the weight as

This means that $a_+^i$ follows Binomial distribution $a_+^i \sim B(s^+, p^i)$, defined as

which is shown for $n = 100$ and different values of $p$ in Image 2.

In case of selecting attribute to increase its value, the algorithm must take into account the maximum value the attribute can have regarding tier of its Paragon. In case the algorithm selects attribute that cannot be increased any more, it selects another one. In case the attributes, that can have their value increase, have all zero chance to be selected, algorithm selects a random attribute from them.

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