Hardware wallets offer an optional passphrase in addition to the recovery phrase. Losing it is unrecoverable, and the reason lies in how the passphrase participates in key derivation.
The passphrase is an input, not a password
Standard derivation takes the recovery words and stretches them into a master seed using a key derivation function. The passphrase is mixed into that function as additional input.
Any change to the input produces a completely different seed, and therefore a completely different tree of accounts. There is no partial match and no near miss.
Because it is an input rather than a stored secret, nothing on the device holds a copy of it to compare against.
Every passphrase is valid
A password check works by comparing an entry against something stored. Derivation has nothing to compare, so any string entered produces a functioning wallet.
Entering the wrong passphrase does not produce an error. It opens a different empty wallet, which the device treats as entirely legitimate.
This is why a mistyped character presents as lost funds rather than as a rejected login, and why the user often cannot tell which of the two occurred.
The design intent is plausible deniability
The property that makes recovery impossible is the same one that gives the feature its purpose. A device can hold a modest balance under the plain recovery phrase and a separate balance behind a passphrase.
Someone compelled to open a device can reveal the first without any evidence that a second exists, because there is no record of one.
That guarantee only holds if nothing on the device hints at the passphrase, which is precisely why no hint field is offered.
Backup is a separate problem
A recovery phrase written down without its passphrase is incomplete. Restoring from it produces the wrong accounts, and the backup looks correct while being useless.
Storing both together removes the security benefit, since anyone finding the backup obtains everything needed.
Common practice is to store them separately, though this doubles the number of items that must survive and be findable by whoever needs them later.
Why brute force is impractical
Recovering a forgotten passphrase means guessing candidates and deriving accounts to see whether any hold a balance.
The derivation function is deliberately slow, so each attempt costs meaningful computation, and the search space expands rapidly with length and character variety.
Recovery services exist and work only where the user recalls the structure of what they chose, narrowing the space to something searchable. Without that, the arithmetic does not permit success.