Arithmetic Error Cancellation in the Gauss Circle Problem: A Prime–Composite Decomposition

We decompose the primitive lattice point counting function into an interior sum and an active-zone sum using the coordinates s= m+nand t= (m−n)/(m+n). The error ER(x) = s e(s,x) splits naturally by the primality of s: prime values contribute predominantly positive errors (≈80%), composite values predominantly negative ones (≈55%). Their near cancellation—verified exactly over all 65,492,915 active-zone terms at x = 10^17—governs the exponent in the Gauss circle problem. A variance analysis reveals that the per-term standard deviation σp ≈0.314 for prime s is constant across fifteen orders of magnitude, while σc for composite sgrows slowly, consistent with √log log x. This decomposition makes the cancellation mechanism arithmetically transparent, in contrast to the classical Voronoi and χ4-approaches where the same cancellation is analytically present but arithmetically opaque.