A Number-Theoretic Window on the Prime/Composite Landscape

We introduce a coordinate decomposition of primitive Gaussian integers z= m+ni∈Zi 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 = 10¹7, 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 = 10²7), derive a Möbius variance formula for composite s (σc ∼ (ln s) (ln 2) /2, conrmed to x= 10¹7), and identify a SternBrocot critical line that separates coprime pairs from non-coprime gaps.