If anything can be verified efficiently, must it be solvable efficiently on a Non-Deterministic machine?

Suppose, I wanted to verify the solution to $ 2$ ^$ 3$ . Which is $ 8$ .

The $ powers~of~2$ have only one 1-bit at the start of the binary-string.

Verify Solution Efficently

n = 8 N = 3 IF only ONE 1-bit at start of binary-string:   IF total_0-bits == N:    if n is a power_of_2:      OUTPUT solution verified, 2^3 == 8 

A solution will always be approximately $ 2$ ^$ N$ digits. Its not possible for even a non-deterministic machine to arrive to a solution with $ 2$ ^$ N$ digits faster than $ 2$ ^$ N$ time.

Question

Can this problem be solved efficently in non-deterministic poly-time? Why not if the solutions can be verified efficently?

What are some ways I can use Project and Global Verified lists?

I really haven’t played with cross-posting on different projects’ verified lists.
Or, used the Global Verified (or other three) lists.
I understand that the global list may have posts on niche-specific targets, like (e.g.) pinball machines, tulip bulbs, and lightbulbs and so would it be weird to try posting a new site that focuses on vegan food using the global list, or one of the other niche projects’ lists?
Are there any negatives to using my own verified lists?
Could it be perceived as spam if I post multiple links to different pages of the same domain on a single target?

Filter verified links without values

I sometimes see verified links such as a profile (or other) where i only see my link. I would like to be able to sort this “bad” link and not keep it for the benefit of another link which contains a minimum text, for example a certain number of sentences, words, or even in weight of text, any surrounded by a minimum of words in order to build my thirds on something long term quality.