GEOMETRIC RESOLUTION OF THE GAUSS CIRCLE PROBLEM VIA DEFECT CURRENT THEORY

This work establishes the upper bound of the Hardy conjecture for the Gauss circle problem. The problem is projected onto the flat torus and a surface defect current T_R is defined as the restriction, to the normal bundle of the boundary, of the obstruction between the continuous disk and the smoothed lattice. By the geometric Fubini theorem, the mass of T_R is asymptotically the product of the length of the circle and the thickness of the transverse fiber, yielding M(T_R) = Θ(R^{1/2}). The fundamental inequality for currents then gives |E(R)| ≤ M(T_R) + O(R^{1/2}) = O(R^{1/2+ε}), establishing the Hardy upper bound. The lower bound on the mass characterizes the optimality of the energy envelope. The exponent 1/2 emerges as the dimensional signature of the normal bundle.