We propose to replace H. Weyl's infinitesimal small affine geometry by an infinitesimal small rotation geometry. At the same time this validates Riemann's conjecture about an Euclidean rotation geometry. The rotating "objects/substances" are differentials, which links back to Leibniz's concepts of monads. At the end the concept of a hyper-real universe beyond (Kant's) physical reality (i.e. physics) becomes (Kant's and Plato's) transcendental "reality", which "is beyond the borders of sensuous experience, where no other theoretical knowledge is possible. In order to lend the term “objectivity”, it needs to be supported in any way by intuition".
(KaI): "Ich behaupte aber, dass in jeder besonderen Naturlehre nur so viel eigentliche Wissenschaft angetroffen werden könne, als darin Mathematik anzutreffen ist".
(KaI) Kant I., "Metaphysische Anfangsgründe der Naturwissenschaften"
(PoP) Poluyan P. V., "Non-Standard Analysis of Non-classical Motion; do the hyperreal numbers exist in the Quantum-relative universe?"
The proposed mathematical gravity model builds on the definition of the inner product of the "new ground state energy" model. The key mathematical tool is the (Pseudo Differential) Riesz operators being applied to differentials.
The concept to apply Riesz operators to differentials goes in line with J. Plemelj’s alternative definition of a potential building on a mass element "dm", alternatively to a mass density, only. The corresponding Klein´s group, which characterizes the geometry, is the infinitesimal rotationgroup. This also goes along with Riemann's conjecture of an infinitesimal small Euclidean geometry. The Hilbert space is also related to the L(2) Hilbert space, which is the as-is framework of today's quantum mechanics and quantum field theory. Consequently the Hilbert scale (approximation) theory is the proper quantum gravity modeling framework.
As an alternative to the today´s Hermite polynomial orthogonal system we propose the modified Lommel polynomials (D. Dickinson, "On Lommel and Bessel polynomials", Proc. Amer. Soc. 5 (1954) 946-956).
The proposed model overcomes the still unsolved particle-wave paradox providing a purely geometrical rationalized "continuum" (H. Weyl). The model overcomes the "contacting body" interaction challenge of "quants without extension, but equipped with flavor and spin". The latter constraint generates a paradox; this handicap is "solved" by H. Weyl´s affine (only!) geometry, whereby the affine geometry model only focuses on parallelized “quants” (i.e. is restricted to affine vectors only). The related mathematical concept to handle to "contacting body" issue is about the concept of continuous transformations, built on S. Lie's concept of contact transforms.