The Zero-Dimensional Euler Spiral · π-Anchoring and the Topological Origin of α · The Imaginary Unit i · Topological Emergence of Complex Probability Amplitudes

Within the layered architecture of GDUM, the Big Bang is not a creation explosion from nothing as in the traditional physical picture, nor an ontological singularity of the spacetime manifold at t = 0, nor a quantum-bounce event in the framework of loop quantum gravity. Its essence is the past-horizon interface corresponding to the integrability boundary of the readout functor, which rests on the same ℤ₂ Deck orientation-involution structure as black-hole horizons and forms a strictly topologically conjugate dual relationship with them. The programmatic layer contains no time, temperature, density, spacetime, or any other geometrico-physical concept; it contains only the static and eternal ℤ₂ double-sheeted conjugate structure, the second-order homotopy residual, and the core axiom of topological flux conservation—no evolution, no birth or death, no spacetime boundary. Upon entering the computational layer, this fundamental topological structure undergoes structured processing through covering lifts and Pin^c classification, and only within appearance windows that satisfy the integrability criterion of the readout functor is it uniformly translated and mapped into the pseudo-Riemannian spacetime picture observable by human beings. The "Big Bang" defined by the academic community through backward extrapolation of cosmic evolution is in essence merely the causal boundary of the appearance-layer geometric system: when observation and extrapolation cross beyond this interface, the geometric reconstruction rules of the readout functor fail completely, and can no longer output well-defined local metrics, causal fields, or particle states; it is only a boundary truncation of the geometric description, not a birth node or rupture locus of the underlying topological structure. At the same time, this framework is fully compatible with the empirically established conclusions of the hot Big Bang: within a stable appearance readout window it can completely reproduce the full evolutionary sequence of Big Bang nucleosynthesis, CMB primordial fluctuations, and cosmic structure formation, remaining self-consistent with all existing astronomical observations.