Time Orientation as a Topological Phase Transition: Complete Information Erasure from Pin⁺ to Spin

[A very important paper Disclaimer: In the early versions of the paper, there may be some errors, please pay attention to identifying them.] We propose that time orientability is not a background assumption but a topological degree of freedom, acquired through a phase transition from non-orientable (Pin+) to orientable (Spin) spacetime.