So how many cycles of

• hypothesize a computable halt-tester
• figure out from that an example where it makes the wrong call
• test it, find that the experiment makes the correct call
• hypothesize a new computable halt-tester that makes the correct call on this example

would you have to go through before concluding the universe was not computable?

I have to confess that I don't really see what's so special about the bottom of the compute hierarchy that you would be so certain that's where the universe is.