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.

Image 1: Schema of fusing operation, where five Paragons are fused to produce a new Paragon of higher Tier (different frame in the resulting Paragon can be noted). As PFPs, we've used NFTs from DeGods, Spirits of Pappataz, Primates, Art of Mob, and Yeah Tigers collections.

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

$$A^i = a_\alpha^i + a_+^i.$$

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

$$s^+ = \sum_{i=1}^{9} a_+^i \in \langle s_{min}^+, s_{max}^+ \rangle = \langle s_{min}^{t+1} - s_\alpha, s_{max}^{t+1} - s_\alpha \rangle.$$

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

$$s^+ = s_{min}^+ + \left( s_{max}^+ - s_{min}^+ \right) \cdot \max \left[\tanh \left( c_\alpha + c_\beta + c_\beta^\sigma - 1 \right), 0 \right],$$

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

$$c_\alpha = \frac{s_\alpha - s_{min}^t}{s_{max}^t - s_{min}^t},$$

$$c_\beta = \frac{1}{4} \cdot \sum_{k=1}^4 \frac{s_{\beta_k} - s_{min}^t}{s_{max}^t - s_{min}^t},$$

$$c_\beta^\sigma = \max \left[ 1- \frac{ \sigma^2 \left( \{ s_{\beta_k} \}_{k=1}^4 \right) }{s_{max}^t - s_{min}^t}, 0 \right].$$

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

$$\sigma (\{ x_i \}_{i=1}^{N}) = \sqrt{\frac{1}{N} \sum_{i=1}^N \left( x_i - \langle x \rangle \right)^2}.$$

$s^+$ 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

$$w^i = a_\alpha^i + \frac{1}{4} \cdot \sum_{k=1}^4 a_{\beta_k}^i,$$

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

$$p^i = \frac{w^i}{\sum_{i=1}^9 w^i}.$$

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

$$B(n,p) = \sum_{k=0}^n \binom{n}{k} p^k \left( 1-p \right)^{n-k},$$

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

Image 2: Probability density function of Binomial distribution. In case there are 100 tosses and probability of successful toss is p, than the chart shows probabilities of number of successful tosses.

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.