A Number-Theoretic Window on the Prime/Composite Landscape

We introduce a coordinate decomposition of primitive Gaussian integers z= m+ni∈Z[i] using s= Re(z) + Im(z) and t= (Re(z)−Im(z))/s, which opens a window on the segment √x<s≤√2x. Within this window, every odd scarries a signed quantity e(s,x) measuring the discrepancy between a discrete coprime count and its continuous expectation. Prime s contribute systematically positive values, composite s systematically negative ones, and their totals match to high precision: at x = 10^17, the prime sum +1,975,343 and the composite sum −1,975,408 yield a residual of −65. We prove an exact formula and equidistribution for prime s (σp = 0.31366..., veried to x = 10^27), derive a Möbius variance formula for composite s (σc ∼(ln s)(ln 2)/2, conrmed to x= 10^17), and identify a SternBrocot critical line that separates coprime pairs from non-coprime gaps.