[Cryptography] question about K

[IvanIvanov750]

I am studying the position of point K on an elliptic curve and I have come across such an interesting pattern that the field in which the point K is located for a particular private key can be in the limit from - infinity to + infinity, despite the fact that the private key is in the field 2^1 - 2^256.

Moreover, there is only one correct K for a particular private key.
Or am I mistaken and there can be repetitions in this huge field of K?

How can you narrow down the K field for your search?
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