Generating Signatures Using Only Public Key

So, here's the deal. I figured out that you can usually create valid signatures with the private key for ecdsa secp256k1. There are libraries out there for this. But I noticed something interesting... if you keep the same values for u1, u2 and just use the public key, you can actually generate signatures with the same r, s, z values. So yeah, random u1 and u2 gives you multiple valid signatures just with the public key. Pretty wild, huh? You can check out my code here if you wanna test it out.

Not really my area of expertise, but if z isn't tied to the message, then that same signature could work for any message. Just saying... If someone wanted to fake a signature pretending to be Satoshi, how could they do it without getting caught?

I skimmed that. Got some ideas brewing now. The formula mentioned is basically what I've been talking about, same concept. Two random scalar values for the r sig, but the stuff mentioned there is sparking some theories in my head. Appreciate it!