Not quite.
You can prove that a message is authored by whoever controls the private key behind a particular npub, but that does not prove or disprove whether the content of the message is true or false, real or fake.
Discussion
true or fake in the sense of
who controls the key did publish the message
If that was your intent, that is certainly not how it read.
You insinuated that a signed message allows you to know "WHAT is true or fake." Not who. What. Which would naturally be understood as the content of the message, not the author of the message.
"A signed message needs no source. It IS the source!"
Most people need to post where they learned about something as their "source" of information because they are not the originator of the information. The fact that they signed their message containing the information does not eliminate them needing to provide the source of the information contained in their message.
You have overstated the case for cryptographically signed messages.
Indeed, before you can trust that nostr:npub16n96pl05gdzt23n0xk630l2854p3x2c6nl9sfv7lcu6qgu3fgu9q4quhr9's private key is actually owned by nostr:npub1qny3tkh0acurzla8x3zy4nhrjz5zd8l9sy9jys09umwng00manysew95gx, you will need some sort of outside verification. So you can't even know that a signed message came from the person you think it did, unless you have first verified that they are indeed in control of that private key and ONLY they are in control of it.
you are right i made a simplified statement
in reality its a little more complicated
what a content provides
is totally up to the creator