You are literally asking how to quantify the unquantifiable. The simple answer is that you can't, by the very definition "unquantifiable".
How could you know if used the full array of keywords to google the internet
You can't prove a negative. It's impossible to prove that someone exhausted all possible search queries. The onus is on you to put forward a search query that (a) is meaningfully relevant and (b) has not been tried yet.
At best, you could log everything they've looked up. But it is literally impossible to prove that they've exhausted every possible option.
how could you know if he thoroughly investigated all the options
You can't. And even if you could somehow know that they investigated the right options, how would you determine that they understood the information correctly?
if he cared to test them
... I mean, have you tried talking to your colleagues?
But again, even if they tested, how would you determine that they performed the right test, and drew the right conclusion from it?
if he kept his attention focused on important details, etc?
You do realize the outlandishness of what you're asking, right?
You're literally trying to pick your colleague's brain, bypassing their entire consciousness and communicative ability in favor of essentially tracking their every move and their conscious understanding so you can judge their work without having to do any of the legwork to gain the needed topical knowledge.
The only way to check the quality of his work, which comes to my mind, is to do the same research yourself and then compare your results with his results.
Take a step back here and think about what you're suggesting. You're implying that you're guaranteed to do the research correctly. If that is the case, why didn't you do it yourself in the first place?
I can even express that into quantifiable boolean algebra. If A is your colleague's research and B is yours, then
(A and B) or B always reduces to
B. In other words, no need to involve A to begin with.
If you do not trust people to do a good job, then don't rely on them to do that job.
A similar approach is to assign the task to a couple or more employees and compare their outputs. Cost is the drawback in this case.
This issue boils down to how you can have multiple people working on a given task without the cost of having multiple people working on a given task. Again, that's impossible by the very nature of what you're trying to achieve.
If you mistrust your colleague's work quality so much that you'd rather have it double-checked by others, then you have to put forward the effort/cost of having the work double-checked.
You can't eat your cake and have it.