Magic constant to solve NP-complete problem in polynomial time Let's suppose that $P\ne NP$. Is that possible to solve all the instances of size $n$ of an NP-complete problem in polynomial time using some "universal magic constant" $C_n$ that has a polynomial length $P(n)$?
Clearly, if $P\ne NP$ this constant can be only calculated in exponential time and the calculation must be done for each $n$.

Thanks
Probably not. What you are asking is whether NP $\subset$ P/poly. If this were true, then the polynomial hierarchy would collapse (this is the Karp–Lipton theorem), something that is widely believed not to happen.