Non-perturbative spectral gravity measure in the Hilbert-Schmidt Gaussian completion: pro-torsor structure and the obstruction to canonical expectations

Abstract

We construct rigorous non-perturbative sectorial measures for the spectral action on Dirac operators with compact resolvent within the Hilbert-Schmidt Gaussian completion framework and classify the global obstruction to assembling them into a single functional integral. The naive Euclidean weight \(\exp(-\operatorname{Tr} f(D^2/\Lambda^2))\) is non-coercive and yields a divergent integral; we cure this by introducing a two-sided functional Gaussian reference measure on the Hilbert-Schmidt self-adjoint fluctuation space \(\mathrm{HS}_\mathrm{sa}(H)\), with covariance determined by the background spectral data. The completed sectorial measure exists, has finite partition function, and is projectively compatible across spectral truncation ranks. Different background sectors, however, produce mutually singular Gaussian classes (Feldman-Hajek rigidity), preventing the assembly of a single background-independent probability measure within this framework. The resulting global object is a pro-torsor of local completed measure classes equipped with a dual density-valued observable sheaf. We prove a principal no-section theorem: relative to any choice of reference weight \(\Phi\), the bounded density gauge group acts freely on normalized projective representatives, precluding further canonical reduction to a scalar-probability trivialization without additional structure. External scalar completion is classified exactly: it requires sector weights and sectorial terminal densities, subject to a truncation-sufficiency criterion. At finite spectral rank, every separating state-independent selector channel must be injective, hence a tautological re-encoding of the full truncation; under a non-tautology axiom this yields a complete no-go. The one-loop predictions reported in the companion SCT papers are shown to be universally preserved, independent of the choice of Gaussian reference, within the present framework.