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The Solana documentation states that executing solana-keygen new --no-outfile outputs pubkey: $PUBLIC_KEY, where $PUBLIC_KEY is the wallet address per:

The value shown after pubkey: is your wallet address.

However, executing solana-keygen verify $PUBLIC_KEY prompt:// outputs Error: Verification for public key: $PUBLIC_KEY: Failed.

In contrast, executing solana-keygen pubkey prompt:// outputs $DERIVED_PUBLIC_KEY per:

This is the derived solana BIP44 wallet address associated with your seed phrase.

Executing solana-keygen verify $DERIVED_PUBLIC_KEY prompt:// outputs Verification for public key: $DERIVED_PUBLIC_KEY: Success.

Is $PUBLIC_KEY the correct wallet address for receiving SOL, or $DERIVED_PUBLIC_KEY?

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This is a bit confusing, but the words displayed when you run solana-keygen new --no-outfile are connected to a legacy scheme which uses the raw keypair rather than a derivation path.

You can recover it using the ASK keyword, ie:

solana-keygen verify $PUBLIC_KEY ASK

More information at https://docs.solanalabs.com/cli/wallets/paper#public-key-derivation

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