The whole must be:
- Unoriginated (Theorem 1).
- Finite (Theorem 2).
- Static (Theorem 3).
- Self‑contained (Theorem 4).
Theorem 5 establishes that the absolute macro-expansion and the absolute micro-singularity are identified at a singular coordinate state (S).
The thermodynamic arrow of time is defined locally by the vector of the increasing entropy gradient, pointing from lower to higher entropy. If the closed temporal manifold were standard and orientable, a local observer traversing the loop would find that at coordinate (S), the incoming entropy vector from the Heat Death would crash head-on into the outgoing entropy vector of the Big Bang. Forcing a single coordinate to host two opposing local vector properties simultaneously would create a fatal logical fracture A ∧¬ A.
The whole resolves this without local contradiction by utilizing a globally non-orientable topology.
Local tracking of the entropy gradient remains continuously orientable, unbroken, and deterministic at every individual coordinate along the trajectory – fully satisfying A = A locally for all observers. However, across the global suture, the metric transitions through an orientation-reversing diffeomorphism.
State (S) does not superimpose two conflicting local properties; it acts as a global parity-flipping mirror where the maximal expansion naturally becomes the absolute dawn of the next scale-invariant cycle. The global topology twists the coordinate space, allowing the arrow of time to flow endlessly while perfectly preserving the absolute Law of Identity.
To reject Theorem 6 is to claim that a global manifold cannot possess non-orientable fiber bundles, forcing a local contradiction onto a geometric transformation. It asserts an A ∧¬ A paradox where none physically exists. The processor crashes.