Why can't someone create a valid digital signature without knowing your private key?

Why can't someone create a valid digital signature without knowing your private key?

I understand from the following two posts that private keys are not displayed by the sender when getting their transaction validated by the bitcoin miners:

What happens to Private Key on Payment

How can transactions be verified without the private key?

Instead a digital signature is provided that is based on the Private Key, and, as far as I understand, that digital signature is verifiable when "compared" to the Public Key.

Why then can't someone generate a verifiable digital signature using only someone else's public key?

Or put another way, quoting an answer from the first post listed above:

"A valid signature cannot be created without access to the private key" - why not?

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