Speedup for natural problems and noncomputability

A resource-bounded version of the statement “no algorithm recognizes all non-halting Turing machines” is equivalent to an infinitely often (i.o.) superpolynomial speedup for the time required to accept any (paddable) -complete language and also equivalent to a superpolynomial speedup in proof length in propositional proof systems for tautologies, each of which implies . This suggests a correspondence between the properties “has no algorithm at all” and “has no best algorithm” which seems relevant to open problems in computational and proof complexity.