What does a model X need to be so that one program of $X/O(1)$ solve in $X$?

Let $ X=P$ , then we can have function

L = len(input) k = 0 while (L>0):     k = k+1     L = log2(L) if (k mod 4 == c):     for 2^len(input):         pass return 1 

which, for every $ n$ , we have a $ c\in \{0,1,2,3\}$ , making all inputs shorter than $ n$ runs in $ Poly(n)$ time; but no matter what $ c$ is, it doesn’t run in polynomial time. Yet, let $ X=R$ , then proof that finding is possible is shown here.

Note that function

return 1 

always return the same value and is polynomial time, but we require to select one rather than create one program. Proof that creating is possible is shown here.

So what should $ X$ be so that, if solver needs to consider smaller inputs, at least one of the limit(s) of $ X/O(1)$ solve problem in $ X$ ? What should $ X$ be so that, All of the limit(s) of $ X/O(1)$ solve problem in $ X$ ?