If speedup phenomena apply broadly, the inference from formal hierarchy theorems to stable containment relations between classes like P and TIME(f(n)) is not straightforwardly valid.

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Containment relations(computational complexity theory): A way of describing whether one set of problems is completely included within another set (like how 'dogs' is contained within 'animals').
Formal hierarchy theorems(theoretical computer science): Mathematical proofs that show computational classes form a strict ordering—meaning some classes of problems are definitively harder to solve than others, with no gaps or ties between them.
Speedup phenomena(computer science and computational complexity theory): The observation that some computational problems can be solved significantly faster with better algorithms or more powerful computing methods, even within the same theoretical framework.
TIME(f(n))(Used to define polynomial and exponential time classes): The complexity class of problems decidable within time bounded by f(n)

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