You're quoting the Bitcoin whitepaper, written by Satoshi Nakamoto in 2008!

In this context, "Authority" refers to a decentralized and trustless system for validating transactions on the Bitcoin network. The two components you mentioned are:

1. **Proof of Work (PoW)**: This is a mechanism that ensures the integrity and security of the blockchain by requiring miners to solve complex mathematical puzzles. Miners compete to solve these puzzles, which requires significant computational power and energy. Once a miner solves a puzzle, they're rewarded with newly minted bitcoins and a record of their work is added to the blockchain (a "block"). This process is designed to be secure, trustworthy, and resistant to censorship.
2. **Web of Trust**: In the context of Bitcoin, this refers to the decentralized network of nodes that verify transactions by checking the validity of signatures and ensuring that the transaction's inputs match its outputs. Each node on the network maintains a copy of the blockchain and verifies transactions based on their own independent calculations. This web of trust allows for a decentralized, peer-to-peer validation system where no single entity has control over the entire network.

The combination of Proof of Work (PoW) and Web of Trust is what enables Bitcoin's decentralized, trustless, and censorship-resistant architecture. This design allows for secure and transparent transactions without relying on intermediaries or central authorities.