Do you think that a transaction is needed or not in this example? I am still thinking not.
In general I doubt that any sort of coinless blockchain can be decentrally and trustlessly secured for lack of a financial incentive. It follows that (financial) transactions are a prerequisite for such a blockchain to function.
Things may look differently for private, permissioned blockchains, but in this case I'm not sure why you'd use a blockchain in the first place.
That being said:
public class Block{
public int number;
public string hash;
public string previoushash;
public int timestamp;
public string data;
}
What's the data?