A cyclotomic Gaussian radical minimum: sharp liminf and arithmetic oscillation

Let omega = (-1 + i*sqrt(3))/2. For B >= 1, let delta_cyc(B) be the least positive value of |x + y - z| over nonzero Gaussian integers x, A and c, where y^2 = A with y outside Q(i), z is c*omega or c*omega^2, and max(|x|, |A|, |c|^3) <= B. Both square roots and both non-Gaussian cubic branches are included, and height uses ordinary complex moduli. The candidate results give delta_cyc(B) = Theta(B^(-4/3)), liminf as B tends to infinity of B^(4/3)*delta_cyc(B) = 1/(4*sqrt(3)), and limsup as B tends to infinity of B^(4/3)*delta_cyc(B) >= sqrt(2 + sqrt(3))/(4*sqrt(3)). Thus this minimum has no single positive leading asymptotic constant. The sharp lower limit is approached by integral Pell constructions with densely variable rational scales. An explicit finite-height lower bound follows from discrete norm-unit levels and a gap for nonunit Gaussian resultants. Zero-conjugate cases and reducible branch products are retained. The exact upper limit and attainment of the lower bound are not determined. The accompanying support contains the proofs, source files and exact-arithmetic diagnostics.