Primordial 4D Torus Hypothesis: A Pre-Big Bang Cosmological Model

This paper presents a mathematical formalization of the Primordial 4D Torus Hypothesis, proposing a closed four-dimensional toroidal geometric structure as a pre-Big Bang state. The hypothesis suggests that before the emergence of linear time and conventional spacetime, the universe existed as a self-contained non-orientable four-manifold with topology K² × T² (Klein bottle × two-torus), undergoing a signature transition from Euclidean to Lorentzian geometry. Key Contributions Primordial topology: The model proposes a non-orientable four-dimensional primordial manifold, utilizing spectral zeta function regularization to demonstrate vacuum stability and topological protection. Emergent spacetime: The observable universe arises through a regulated geometric degeneration (signature change) from a timeless Euclidean regime to Lorentzian spacetime, generating causal structure. Fermion Generation via Cobordism: We resolve the spin structure obstruction of the primordial state by modeling the Big Bang as a topological cobordism ($W^5$), where fermions emerge as boundary modes through the Anomaly Inflow mechanism. Topological information content: Information is encoded in the global topology of the manifold, defining the "size" of the primordial state not by geometric extension but by the quantum area spectrum, resolving the initial singularity problem. Zero energy universe: Topological pair creation enforces exact global energy cancellation ($G_0$), reframing the Big Bang as an information saturation event rather than an energetic explosion. Testable signatures: The hypothesis derives falsifiable predictions, including antipodal correlations in the CMB and gravitational wave echoes, enabling potential empirical validation. This work originates from questions raised during The Atlas project and the writing of The Procedural Universe: A Philosophical Quantum and Computational Analysis of Cosmos Simulations via AI (doi:10.5281/zenodo.18112886), where implementing a procedural universe simulation within Einstein’s block universe motivated the search for a formal description of a primordial state without conventional time.