Here's how trusted miners contribute to the coinpaga ecosystem.

1. Validation and Authorization: Trusted miners have undergone a validation process and obtained a validation key from the coinpaga validators. This key serves as a form of authentication and authorization, indicating that they are authorized participants in the network.

2. Maintaining Network Integrity: By allowing only trusted miners to participate, the network administrators aim to maintain the integrity of the blockchain system. Trusted miners have met certain criteria or requirements set by the network, ensuring their trustworthiness and reducing the risk of fraudulent activities.

3. Security and Prevention of Malicious Behavior: Trusted miners help prevent malicious activities such as double-spending, Sybil attacks, or other types of fraudulent behavior that could harm the network or undermine its trustworthiness. The validation process ensures that trusted miners are reliable contributors, enhancing the overall security of the network.

4. Contributing to Network Performance: Trusted miners actively mine and contribute to the network's operations, helping to validate transactions and secure the blockchain. Their participation helps maintain the performance and efficiency of the network by ensuring a reliable network of miners.

Mathematical representation of the role of trusted miners:

Let T represent the set of trusted miners in the Metacore blockchain and Coinpaga Ecosystem.

1. Validation and Authorization:
- Each miner in T has undergone a validation process.
- Let V(m) represent the validation status of miner m, where V(m) = 1 if miner m is validated, and V(m) = 0 otherwise.
- Each trusted miner m ∈ T has obtained a validation key, represented as K(m), from the coinpaga validators.

2. Maintaining Network Integrity:
- The set of authorized participants in the network is T, ensuring that only trusted miners can participate.
- Let A(m) represent the authorization status of miner m, where A(m) = 1 if miner m is authorized, and A(m) = 0 otherwise.
- ∀ m ∈ T, A(m) = 1, indicating that each trusted miner is authorized to contribute to the network.

2. Maintaining Network Integrity:
- The set of authorized participants in the network is T, ensuring that only trusted miners can participate.
- Let A(m) represent the authorization status of miner m, where A(m) = 1 if miner m is authorized, and A(m) = 0 otherwise.
- ∀ m ∈ T, A(m) = 1, indicating that each trusted miner is authorized to contribute to the network.

3. Security and Prevention of Malicious Behavior:
- Trusted miners actively contribute to preventing malicious activities.
- Let B(m) represent the behavior status of miner m, where B(m) = 1 if miner m's behavior is non-malicious, and B(m) = 0 otherwise.
- The validation process ensures that trusted miners have reliable behavior.
- ∀ m ∈ T, B(m) = 1, signifying that each trusted miner's behavior is non-malicious.