Arithmetic Error Cancellation at the Gauss Circle Boundary: A Prime_Composite Decomposition

We decompose the primitive lattice point counting function into an interior sum and a boundary sum using the coordinates s = m+ n and t = (m−n)/(m+ n). The boundary error ER(x) = sum_s e(s,x) splits naturally by the primality of s: prime values contribute mostly positive errors, composite values mostly negative ones. Their near cancellation veried exactly to x = 10^14 with cancellation up to 99.9% governs the exponent in the Gauss circle problem. 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.