Are there examples of $ E $ where $ M $ can be calculated?

Once it's proven, the sentence is proven to be equivalent to a $\Pi^0_1$ statement, namely $0=0$. When we speak of "a sentence" we probably mean "up to provable equivalence" since the Millennium Problems are only defined up to provable equivalence...

Is $\phi$ a function of 2 variables?

The Mystery of the Aleph: Mathematics, the Kabbalah, and the Search for Infinity By Amir D. Aczel

Is $\phi$ just a bump function like $e^{-1/x^2}$?

This question appears to be off-topic because it already has an answer at math.se