This paper establishes a rigorous framework for the prolongation of adelic class spacesvia the ring of dual numbers and analyzes the associated deficit functionals. We con-struct the arithmetic site and its structure sheaf, define the A1-prolongation with properderivation rules, and prove that the deficit functional measures deviation from half-arcconfigurations. We show that ghost symmetries correspond to specific derivations on theprolonged space and derive the explicit formula as a conservation law. The relationshipbetween vanishing deficit and spectral properties is established through Mellin transformanalysis. A detailed study of the trivial case in the isoperimetric inequality reveals itsrole as a ghost-free vacuum configuration, scale-equivalent to the unit circle and servingas the cohomological baseline against which all obstructions are measured.