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 I don't follow what you're saying.

There's one person claiming to be expert in (komodo OR fermat). There's another, providing the 2 lists of Qs and, for a very limited time, answers.

You can assume that the first person is indeed enough of an expert in one of the 2, that they can easily correctly answer all 1000. 
 Ok I understand. 

Simple : You put the expert in a room with a locked box. 

The box opens with one of 2 combinations. 

Inside is a gem. 

The combination to open the box is : the first letter of every answer of either lists. so it's a 1000 letter combination. 2 possible combinations.

He just needs to go in the room and come out with the gem. 

If he does he's an expert in either lizards or old math dudes.

Does that work ?  
 Nice answer :)

It's a slightly more fun version of, e.g. having a computer program that asks you to enter correct answers and prints out pass/fail, so basically you answer the questions in secret and the challenger sees only the result, and believes that it was executed correctly ... which is pretty reasonable.

(not the intended answer of course; for that no mechanical or digital mechanisms needed). 
 So I thought about it more. 

If no machines are allowed, can't we just craft 2 lists of questions whose answers add up, in total, to a certain number of letters. The same sum for both lists. 

The expert reads the questions and gives the total. If the total is ok, he knows, if not, he doesn't? 

 
 Really interesting idea, I like it. Obviously difficult in practice, but the spirit of the idea makes sense, as long as the numbers get large enough. 
 Ok, I give up, 😭😭😭. 

What's the answer ?  
 No, you succeeded I'd say, that last idea feels like a valid solution ... unless i missed something. I'll post the 'intended' answer later. 
 Answer at nostr:nevent1qqs8pa4xuat4ukptmgj70jc56p609uv8mv5aja0q242ug5p9mtv2zqcpr4mhxue69uhkummnw3ezucnfw33k76twv4ezuum0vd5kzmp0qgsxwkuyle67y94tj378gw8w2xw2wa6nwmwlqhddlwnz0z7sztsaw2qrqsqqqqqpm4sw7n

@AbstractEquilibrium ping also