How does public address "know" private key? Imagine the following:
someone goes to the .org address of bitaddress, then switches off internet connection, then moves the mouse around until eventually the JavaScript Client-Side Bitcoin Wallet Generator produces a Bitcoin address with its corresponding Bitcoin private key, then saves everything on the screen to cold storage as a PDF, then scans the &quot;Share&quot; QR code onto a phone, then scans this QR code at a Bitcoin ATM to purchase Bitcoin and the Bitcoin in deposited in the wallet and this is confirmed on blockchain.
If later wanting to sell the Bitcoin, how does this public Bitcoin address &quot;know&quot; what the correct private key is?
That is, if the Bitcoin public address and private key have only existed on a non-internet connected PC and as a PDF in cold storage?
How does public address “know” private key?

It doesn't. There's no need.
When receiving money, nobody needs to know anything from you other than your receiving address (in reality a Bitcoin script, the most common types of which can be displayed as an address). At this point no-one cares about your private key.


If later wanting to sell the Bitcoin,

When spending money, you have to provide proof that you know the private key corresponding to that receiving address, without revealing the private key. This is what your wallet does when constructing the transaction details that are sent out for miners to pick up.
Other Bitcoin nodes (wallets, miners etc) can mathematically check that the proof matches the public key and will then know that you have the secret number known as a private-key.

See

https://en.bitcoin.it/wiki/Elliptic_Curve_Digital_Signature_Algorithm