A Rigorous Functional Measure for Spectral Quantum Gravity: Krein-Space Construction, a Sharp Trace Bound, and the Horizon Modular Question Posed Precisely

The note also isolates the horizon modular question — whether the induced state on the de Sitter static-patch algebra satisfies the Kubo–Martin–Schwinger condition with respect to a geometric modular flow — to a single, precisely posed open problem, informed by direct correspondence with an expert in the Bisognano–Wichmann/Tomita–Takesaki literature. No general theorem for de Sitter nets is currently known to secure this property, and explicit counterexamples to geometric modular covariance exist in the literature; the note states exactly what remains to be shown. All claims are graded [proved], [derived], [computed], [argued], or [open], distinguishing what is established unconditionally from what remains a genuinely open research question.