Aaronson recently wrote a blog refuting the idea that there could be some glitch in the formulation of the P vs NP conjecture[1] which reminds me of this following question.the Blum speedup theorem has been characterized by Goldreich [3] (p149)A conceptually related phenomenon is of problems that have no  optimal algorithm (not even in a very mild sense); that is, every algorithm for  these pathological problems can be drastically sped-up. It follows that the  complexity of these problems can not be defined (i.e., as the complexity of the best  algorithm solving this problem).in math, it is not unprecedented that major conjectures have been shown to have glitches eg discovery that they are independent of axiomatic systems, etc.also Goldreich reference to pathological languages is reminscent of historical consternations in math eg the counterintuitive discovery of functions that are continuous but nowhere differentiable eg the Weierstrauss function.another angle is that at least one complexity theorist expert Lipton speculates that program size is underanalyzed in TCS and may have surprising aspects,[2] and the Blum speedup theorem seems to fit in with this line of thinking, because the proof construction basically relies on improving the language recognition time based on increasing the size of the TM program (ie like a tradeoff).so:how does one show/ prove that no such pathological languages are/can be involved in open complexity class separation questions eg P=?NP etc?am interested in more technical/ mathematical perspectives that avoid handwaving. (eg something more detailed/ substantial than an assertion that the Blum speedup construction is contrived etc.)[1] Is the P vs. NP problem ill-posed? (Answer: no.) Aaronson blog[2] Does program size matter? RJLipton blog[3] Computational complexity: a conceptual perspective / Goldreich
proving speedup phenomenon does not apply to any open complexity class separations
cc.complexity theory;reference request;complexity classes;time complexity;big picture