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Since the physical universe is modeled as a hypersurface of a scaled 4-sphere, an extremely useful question is whether there is anything interesting to be said about its geometry from versions of the Gauss-Bonnet theorem witb boundary and its generalizations.  AtiyahPSSpectralAsymmetry had found that the Euler characteristic and the signature can be expressed as an integral of a characteristic class and a boundary term that is $\eta(0)$ for the spectral function $\eta(s) = \sum \sign(\lambda) |\lambda|^{-s}$.  So this spectral invariant we should consider and interpret for the physical universe.