**4. Conclusions**

Our discussion of phase-space dependent coefficients in hypersurface deformations has clarified a previously puzzling issue of partial Abelianizations in spherically symmetric models: Is it possible for partial Abelianizations to simplify the construction of quantum modifications of hypersurface deformation generators and, at the same time, retain full access to all transformations required for general covariance? We have shown that the answer is negative. A simplified construction of modified generators is based on the absence of structure functions in partially Abelianized brackets obtained for a specific choice of phase-space dependent gauge generators (lapse and shift functions). However, the partial Abelianization is maintained only if the new generators are then restricted to be phase-space independent. This condition renders the transformation from hypersurface-deformation brackets to partially Abelian brackets non-invertible. Access to unrestricted hypersurface deformations and general covariance is therefore lost in a partially Abelianized setting. Consistent modifications of the partially Abelian brackets then do not necessarily imply consistent realizations of general covariance.

A recent paper [8] by Gambini, Olmedo and Pullin has implicitly recognized this shortcoming and instead proposed to test general covariance in a tedious case-by-case study of presumed invariants, beginning with a discretized version of the line element. We have pointed out a specific place (the choice of modification functions *f*1 and *f*2) where hypersurface deformations are treated inconsistently in these constructions, which may perhaps lead to improved versions of the transformations considered by GOP. However, correcting this inconsistency requires an analysis of unrestricted hypersurface deformations even in the partially Abelianized setting, making sure that the transformation between these two versions of the brackets can be inverted. It is therefore impossible to analyze covariance in isolation from general hypersurface deformations, as proposed by GOP. No-go results [28] for covariance in models of loop quantum gravity, partially based on various analyses of modified hypersurface deformations, therefore cannot be evaded by the constructions of GOP.

**Funding:** This research was funded by NSF gran<sup>t</sup> number PHY-1912168.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Acknowledgments:** The author thanks Michele Schiavina for discussions and Rodolfo Gambini, Javier Olmedo and Jorge Pullin for sharing a draft of [8].

**Conflicts of Interest:** The author declares no conflict of interest.
