Hawking and Penrose proposed “A new theorem on spacetime singularities … which largely incorporates and generalizes the previously know results” which they claimed “implies that space-time singularities are to be expected if either the universe is spatially closed or there is an ‘object’ undergoing relativistic gravitational collapse (existence of a trapped surface)” and that their ‘Theorem’ applies if four certain physical conditions are satisfied. Hartle, Hawking and Hertog have proposed a quantum state with wave function for the Universe which they assert “raises the possibility that even fundamental theories with a negative cosmological constant can be consistent with our low-energy observations of a classical, accelerating universe.” They also relate this concept to string cosmology. It is however proven in this paper that the Hawking-Penrose Singularity ‘Theorem’ and accelerated expansion of the Universe with negative Λ are invalid because they are based upon demonstrably false foundations relating to Einstein’s field equations, trapped surfaces, and the cosmological constant