# Appendix

## Mathematical Equations

### Equation 1: Reserve Ratio Equation

Let: $P$ - the price of one 1G$in Supported Currency; $R$ - the amount of Supported Currency in the reserve; $S$ - total G$ coins in circulation.

Then$r$ - the reserve ratio - is defined as:

$r=\frac{R}{P \cdot S}$

At launch, $r=1$ and overtime, will computationally decline at a daily rate in accordance with the Expansion Rate. In Equation 1, given $R$ and $r$ the system has two degrees of freedom, $P$ and $S$ that will be calculated as described below.

### Equation 2: Buy/Sell Function & Current Price(Bancor Formula for Automated Market Making)

Current Price function:

### ​$Current Price = \frac{Reserve~Balance}{Smart~Token's~Total~Supply~ \cdot ~Reserve~Ratio}$

$Tokens~Issued = Supply \cdot ((1+\frac{Connected ~Tokens~Paid}{Balance}) ^ {CW} -1)$

Sell function:

$Connected~Tokens~Paid~Out = Balance \cdot (\sqrt[CW]{(1+\frac{Tokens~Destroyed}{Supply})} -1)$

### Equation 3: Expansion Rate Formula

$q=\frac{r_t}{r_{t-1}}$

### Equation 4: Mint G$from Declining Reserve Ratio - Once a day the reserve ratio is reduced while all other parameters are unchanged. The GoodReserve contract mints an amount of G$ to satisfy Equation 1 $P=\frac{R}{r \cdot S}$ . If P and R do not change and the daily change in r=𝚫r and the corresponding change in S is 𝚫S, then $r s=(r+\Delta r)(S+\Delta S)$ and from here we can see that:

$\Delta S=S \cdot \frac{-\Delta r}{r+\Delta r}=S \cdot \frac{r_{t-1}-r_{t}}{r_{t}}$

From Equation 2 we get $q=\frac{r_t}{r_{t-1}}$ and thus: $S \frac{r_{t-1}-r_{t}}{r_{t}}=S \frac{1-q}{q}$

### Equation 5: Mint G$from Deposit Formula - The number of new tokens based on the daily interest deposited in the Reserve. Z - Value deposited into the reserve E - Newly minted tokens The formula does not change the price or reserve ratio: $E = \frac {\frac{R + Z}{Rr} - P \cdot S}{P}$ ## Monetary Example This is an illustrative example, with simple rounded numbers, aimed at trackability: • 10 Supporters have staked a total value of US$10 million in supported crypto to a third-party protocol

• Protocol annual interest rate of 10%

• Daily interest of US$2,736 • G$ Price = US $1 • GoodReserve = US$1,000,000

• Supply of G$= 1,250,000 • G$ Market Capitalization = US$1,250,000 • G$ Reserve Ratio = 80%

• Daily Expansion Rate = 1%

Every day G$minting occurs from two methods: From the deposit of interest to the GoodReserve and from the decline in reserve ratio. 1. Minting derived from deposite of daily interest to the GoodReserve 1. Deposit of US$2,736 to GoodReserve

2. Calculated as follows:

1. $S \cdot P=\frac{R}{Rr}$

2. (1,250,000 +X)*1=(1,000,000 + 2736)/0.8

3. 1,250,000 + X = 1,002,736/0.8

4. 1,250,000 + X = 1,253,420

5. X = 3,420

3. 3,420 G$are minted 4. 2,736 G$ are allocated back to the 10 supporters

5. 684 G$are distributed as basic income 2. Mint from Reserve Ratio Decline 1. Lower the Reserve Ratio from 80% to 79% 2. Calculated as follows: 1. $S \cdot P=\frac{R}{Rr}$ 2. (1,253,420+X)*1= 1,002,736/0.79 3. 1,253,420 + X = 1,269,286 4. X = 15,866 3. 15,866 G$ are distributed as UBI

In total, on that day 16,550 G\$ were distributed as basic income to Claimers.