Validity of the classical spacetime-limit criteria

Determine whether the proposed requirements of Lorentzian signature, single-saddle dominance, and smooth deformation to a Cauchy slice are necessary and sufficient criteria for assigning a classical spacetime limit to a mixed-conformal elliptic data manifold.

Background

The paper defines the classical spacetime limit of a state using the CRT fixed locus of its norm geometry, requiring a Lorentzian metric, dominance by a unique saddle, and a smooth deformation from the elliptic data manifold to a Cauchy slice of the fixed-point wedge.

The authors acknowledge that these conditions may be overly restrictive or insufficiently precise. They specifically question whether Lorentzian signature must be imposed, whether single-saddle dominance should be weakened, and whether the smooth-approach condition should be sharpened or omitted. They propose that explicit examples are needed to discriminate among alternative formulations.

References

For example, it is not clear that Lorentzian signature needs to be imposed as a condition; it may be sufficient to restrict allowed elliptic data by requiring, e.g., that the real part of $h_{ij}$ have Euclidean signature and the real part of $K$ be nonvanishing.

— Quantum State of a Gravitating Spacetime Region  (2609.10684 - Bousso et al., 9 Sep 2026) in Section 4.1, footnote to the paragraph “Classical spacetime limit”

In de Sitter space, the construction fails; we give a preliminary argument (but do not prove) that $\mathbf{C}(w_\sigma)$ is empty in such cases.

— Quantum State of a Gravitating Spacetime Region  (2609.10684 - Bousso et al., 9 Sep 2026) in Section 4.2, subsection “From a Lorentzian Wedge to a Gravitational State”