b. Bridges

When operating with tokenomics logic under multichain environment, most encouter difficulties in bridging activities. While this paper does not consider trust setups, we suggest circulating supply logic via address units and utilizes both addresses’ balance and transaction histories. Also in the premise that smart contract, and security of both source and destination chains are intact, current methodology is chain-agnostic and requires only additional data-engineering efforts of data-read operations.

For bridging, one can assume $C_{source}$ and $C_{destination}$ where bridging occurs with token amount $Q$, at a specific time $t$ and time of state after bridging: $t+1$.

Additionally, while $CirculatingSupply$ do not change in reliable, non-malicious cross-chain bridging operations, other factors such as $TotalIssued$ and $TotalSupply$ may change in bridge initiation process. There exists four cases of initiation and accordingly in its bridging process which is listed as follows:

1. Burn & Mint

   • Initial Condition:

     • $\{C_{destination}\} \subset C_{all}$ as new chain is added to project’s ecosystem

     • But No additional tokens minted, changes in deciding factors occur only via bridging actions.

   • Bridge Action:

     • Burn tokens of amount $Q$ on $C_{source}$ and mint tokens of amount on $C_{destination}$

       • $TB_{C_{source}}(t+1) = TB_{C_{source}}(t) + Q$

       • $TI_{C_{destination}}(t+1) = TI_{C_{destination}}(t) + Q$

2. Burn & Unlock

   • Initial Condition:

     • $\{C_{destination}\} \subset C_{all}$ as new chain is added to project’s ecosystem

     • Additional tokens of amount $Q_{init}$ are minted on the $C_{destination}$ prior to unlocking. By assuming the initiation time $t_{init}$ as time of event where minting happens, we can define and arrange with other factors as follows:

       • $TI_{C_{destination}}(t_{init}) = Q_{init}$

       • $NCS_{C_{destination}}(t_{init}) = Q_{init}$

   • Bridge Action:

     • Projects burn tokens of amount $Q$ on $C_{source}$ and unlock tokens of amount $Q$ on $C_{destination}$

       • $TB_{C_{source}}(t+1) = TB_{C_{source}}(t) + Q$

       • $NC_{C_{destination}}(t+1) = NC_{C_{destination}}(t) - Q$

3. Lock & Mint

   • Initial Condition:

     • $\{C_{destination}\} \subset C_{all}$ as new chain is added to project’s ecosystem

     • But No additional tokens minted, changes in deciding factors occur only via bridging actions.

   • Bridge Action:

     • Lock tokens of amount $Q$ on $C_{source}$ and mint tokens of amount $Q$ on $C_{destination}$

       • $NC_{C_{source}}(t+1) = NC_{C_{source}}(t) + Q$

       • $TI_{C_{destination}}(t+1) = TI_{C_{destination}}(t) + Q$

4. Lock & Unlock

   • Initial Condition:

     • $\{C_{destination}\} \subset C_{all}$ as new chain is added to project’s ecosystem

     • Additional tokens of amount $Q_{init}$ are minted on the $C_{destination}$ prior to unlocking. By assuming the initiation time $t_{init}$ as time of event where minting happens, we can define and arrange with other factors as follows:

       • $TI_{C_{destination}}(t_{init}) = Q_{init}$

       • $NCS_{C_{destination}}(t_{init}) = Q_{init}$

   • Bridge Action:

     • Lock tokens of amount $Q$ on $C_{source}$ and unlock tokens of amount $Q$ on $C_{destination}$

       • $NC_{C_{source}}(t+1) = NC_{C_{source}}(t) + Q$

       • $NC_{C_{destination}}(t+1) = NC_{C_{destination}}(t) - Q$