# 10c. Mathematics

Here, we break down the mathematical logic behind our phased presale.

Understanding the strategy is essential, but delving into the numbers ensures our community grasps the true depth of our commitment to transparency and fair play.

The primary constraint behind the presale strategy is that the earliest participants are to be disproportionately rewarded. And everything else is built around that.

where p represents the price of a given presale event, in terms of the number of $GATE tokens per dollar invested. This means that, per dollar invested, the first presale shall return a higher number of $GATE tokens than the second presale, which shall return a higher number of $GATE tokens than the next presale. Simply, each new issuance of tokens gets more expensive.

This inequality constraint can be rearranged as follows:

Assuming there are three presales, where the total number of tokens:

We denote "X" as the capital invested, resulting in:

The total number of tokens sold is the sum of the capital invested into each presale, multiplied by the number of tokens per $1 invested.

From our inequality constraint, we can simplify this equation:

To reduce this equation even further, we want to apply some 'basic business logic' that makes x_i related to x_(i+1).

If the 'Marketing Spend' is directly proportional to the next raise... then we introduce a variable "m" - the 'marketing factor'. Let's assume that, after the liquidity provisioning is taken away, and the fees are paid, what's left over for marketing is 25% of capital raised x_i. All other funds are allocated to technology.

Therefore: x_(i+1) = 0.25 × x_i × m

Unknown variables:

m - the "marketing factor", and
δ_*1 and δ*_2 - the "discount factor".
δ represents the decline in the number of tokens given for every dollar between successive presale events. The magnitude and nature of these decrements depend on a variety of factors, both internal (project-specific) and external (market and investor-centric). For example, in a BULLISH scenario, with high demand, bullish conditions, and strong project fundamentals, a minimal increment of δ = 0.01 could be used.
A moderate increment, with moderate demand, neutral market conditions, and the project hitting expected milestones δ = 0.1 could be used.
In a BEARISH scenario, with weaker demand, a bearish market, and the project facing challenges or delays, δ = 0.25 would be a high increment.

No matter the value applied, the earliest participants are disproportionately rewarded.

The Elevated project built a program with these constraints to determine the presale values. As traction continues, updated and empirical values will be used to refine this formula and optimize the tokenomics.

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