  The Curry-Howard correspondence is often called the proofs-as-programs
result. I offer a generalization of this result, something which may be called
machines as programs. Utilizing this insight, I introduce two new Turing
Machines called "Ceiling Machines." The formal ingredients of these two
machines are nearly identical. But there are crucial differences, splitting the
two into a "Higher Ceiling Machine" and a "Lower Ceiling Machine." A potential
graph of state transitions of the Higher Ceiling Machine is then offered. This
graph is termed the "canonically nondeterministic solution" or CNDS, whose
accompanying problem is its own replication, i.e., the problem, "Replicate
CNDS" (whose accompanying algorithm is cast in Martin-L\"of type theory). I
then show that while this graph can be replicated (solved) in polynomial time
by a nondeterministic machine -- of which the Higher Ceiling Machine is a
canonical example -- it cannot be solved in polynomial time by a deterministic
machine, of which the Lower Ceiling Machine is also canonical. It is
consequently proven that P $\neq$ NP.
