Paper 50 (GFI): Spacetime Genesis — The Big Bang as a KMS Phase Transition in von Neumann Algebra Classification

We propose that the Big Bang is not a singularity inside spacetime, but the algebraic event at which spacetime itself becomes well-defined. Spacetime exists if and only if a KMS state exists for a physically given dynamics. The pre-geometric phase is a Type III₁ von Neumann factor M equipped with a physically given one-parameter automorphism group {α_t} (not the modular flow of any state) that admits no faithful normal KMS state. In this regime neither time (modular flow) nor space (BKM metric from the relative-entropy Hessian) is defined. The KMS genesis event — identified with the Big Bang — is the first appearance of a KMS state ω_β for the given {α_t}. This single datum simultaneously (i) selects the modular flow σ^ω_t (time), (ii) renders the relative-entropy Hessian δ²S_rel non-degenerate, defining space via the BKM metric, and (iii) produces the Takesaki crossed product carrying a canonical trace. We prove that Orlicz regularity of the limit state follows from domination together with commutativity of the affiliated thermal densities (Lemma 10.2). In the Araki–Woods setting, the open question OP-50.9′ is resolved under the natural local commutativity condition [H_n^{(0)}, V_n] = 0 (Remark 10.3), which propagates abelianity to the densities via the Borel functional calculus and places them in a common MASA. We give the first explicit candidate for the pre-geometric phase via time-dependent Araki–Woods algebras (Theorem 6.6) and identify half-sided modular inclusion as a candidate dynamical transition mechanism (Remark 6.10). All prior results of the GFI programme (Papers 3–49) are shown to be post-genesis theorems (Proposition 7.1). New citations to Liu (2025) and Labuschagne–Majewski (2026) integrate recent external corroboration from entanglement and curved-spacetime QFT. Keywords: von Neumann algebras, Type III₁ factors, KMS states, Tomita–Takesaki theory, BKM metric, spacetime emergence, algebraic quantum gravity, pre-geometric phase, Araki–Woods algebras, Big Bang