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Lightning is a channel. The chain is the dispute.
An on-chain payment settles when it is in a block you accept as final. A Lightning payment settles between the channel parties until one of them publishes the state.
What you are trusting
A channel needs liquidity on the route and a counterparty who can go to the chain if you disappear. Routing is not the same as a global order book. The paper is the citation for the channel. A live wallet's fees, liquidity and watchtowers are a later system.