Lubkin-Page typicality bounds for Type II von Neumann factors

Typicality arguments for emergent spacetime rely on the Lubkin-Page bounds, which show that generic quantum states have vanishing correlations between subsystems. These bounds assume a tensor-product Hilbert space (a Type I von Neumann algebra), but the observable algebras in quantum field theory and quantum gravity are generically Type II or Type III, raising the question of whether the bounds survive. We prove that they do for all Type II von~Neumann factors. For the hyperfinite Type II$_1$ factor with a tripartite decomposition $R \cong A \otimes B \otimes E$, the mutual information between subsystems $A$ and $B$ vanishes as $O((d_A d_B / d_E)^2)$ in finite-dimensional approximations, provided $d_A d_B \leq d_E$ (Theorem 1). For Type II$_\infty$ factors, including the gravitational algebras constructed via the crossed-product method by Witten and by Chandrasekaran, Longo, Penington, and Witten, the bound acquires an additional exponential suppression controlled by the Bekenstein-Hawking entropy (Theorem 2). We identify the obstructions to extending the result to Type III factors and discuss the open question of whether the commutant of the observable algebra can serve as a natural thermal bath that tightens the bound further.