← openxiv:math-ph.2026.00001 · math-ph
Non-perturbative spectral gravity measure in the Hilbert-Schmidt Gaussian completion: pro-torsor structure and the obstruction to canonical expectations
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**Problem statement:** The Euclidean weight \(\exp(-\operatorname{Tr} f(D^2/\Lambda^2))\) for the spectral action on Dirac operators is non-coercive, leading to a divergent functional integral. A rigorous non-perturbative measure is needed, but background sectors are mutually singular under Gaussian reference measures, obstructing a single background-independent probability measure. **Method:** A two-sided functional Gaussian reference measure on the Hilbert–Schmidt self-adjoint fluctuation space \(\mathrm{HS}_\mathrm{sa}(H)\) is introduced, with covariance determined by the background spectral data. This cures the divergence, and sectorial measures are constructed and studied via projective compatibility across spectral truncations. **Main results:** The completed sectorial measure exists, has finite partition function, and is projectively compatible across truncation ranks. However, different background sectors produce mutually singular Gaussian classes (Feldman–Hajek rigidity), preventing assembly into a single background-independent measure. The global object is a pro-torsor of local measure classes with a dual density-valued observable sheaf. A no-section theorem shows that the bounded density gauge group acts freely on normalized projective representatives, precluding canonical reduction to a scalar probability without extra structure. External scalar completion requires sector weights and terminal densities under a truncation-sufficiency criterion; at finite rank, every separating state-independent selector is injective, yielding a complete no-go under a non-tautology axiom. One-loop predictions from companion SCT papers are universally preserved regardless of the Gaussian reference choice. **Limitations:** The construction cannot yield a single background-independent probability measure; only a pro-torsor structure is achieved. Additional geometric or algebraic input is required for canonical expectations. The results are confined to Dirac operators with compact resolvent, and the finite-rank no-go theorem shows that trivial scalar reduction is inherently tautological within this framework.
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