Transactions can only be processed through the mining of new bitcoin.
You misunderstand. Transactions are processed by the miners. This happens as long as miners exist on the network. The "network" reward for mining blocks is projected to end around 2140, when 21,000,000 Bitcoin have been mined. These are decoupled processes, though. The network rewards happen after the block is mined. Mining/transaction fees will also still apply after network rewards go away, so the incentive to mine will always exist.
everyone will be left holding a valueless mathematical hash.
Another misunderstanding I'd like to correct. We hold the keys to access the value stored on the blockchain. Ultimately the value is all stored on the blockchain. You can lose your keys (access to the value) but the value is always there.
You misunderstand. Transactions are processed by the miners. This happens as long as miners exist on the network. The "network" reward for mining blocks is projected to end around 2140, when 21,000,000 Bitcoin have been mined. These are decoupled processes, though. The network rewards happen after the block is mined. Mining/transaction fees will also still apply after network rewards go away, so the incentive to mine will always exist.
Another misunderstanding I'd like to correct. We hold the keys to access the value stored on the blockchain. Ultimately the value is all stored on the blockchain. You can lose your keys (access to the value) but the value is always there.
It's pretty neat.