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The tokens/assets stay on the ledger forever, or at least as long as greenhouse gases stay in the atmosphere (hundreds of years at least.)
One interesting question is double counting: My emissions are mine. I cannot transfer them to someone else. But if you hire me to do some work for you, a portion of my emissions should also be part of your responsbility. Does that mean my emissions should be reduced and yours increased? Not if you only contact my total emissions, but yes if you count the total emissions from both of us. So rather than a simple transfer between two parties, inter-party emissions should be cancelled when they are summed up in calculations that involve both parties.
