Arithmetic Cohomology of the Prime Error Term: A Motivic Approach via Arakelov Hyperbolic Tori and Spectral Surgery

This paper proposes a geometric formulation of the error term in the prime counting function, π(x) − Li(x), within the framework of Arakelov geometry over quadratic fields. By lifting prime splitting behaviors to the Grothendieck ring of varieties K0(VarK ), we construct a distributional current on an Arakelov compactification modeled as a limit of hyperbolic tori (the Adelic Solenoid). We introduce a ”Bypass Green’s Function” utilizing Lax-Phillips scattering theory and Eisenstein series to annihilate the continuous spectrum and filter out ”oldform” noise. We demonstrate that the regularized error term arises as the coboundary of a projection in the first Arakelov cohomology group, bounded via hyperbolic convolution and Mobius duality on Bruhat-Tits buildings.