Motivic Spectral Invariants and Symplectic Microlocal Calculus of Pseudo-Differential Operators on Non-Archimedean Berkovich Spacetimes

Mathematical Applications of Science Fiction We construct a rigorous framework for symplectic mi-crolocal calculus over non-Archimedean fields, specificallyfocusing on Berkovich analytic spaces. By introducinga category of motivic microlocal sheaves, we define anew class of spectral invariants associated with pseudo-differential operators acting on the rigid analytic étale site.We prove that the spectral distribution of the Frobeniusoperator on the crystalline realization of these sheavesconverges to the fixed-point Floer homology of the skele-ton of the Berkovich space. Furthermore, we establish acorrespondence between the motivic L-functions of theunderlying perfectoid space and the symplectic capac-ity of the Lagrangian submanifolds. This resolves theVoidwalker-Tate conjecture regarding the quantization ofnon-Archimedean phase spaces.