In order for vortex equations on moduli spaces of $Z_n$ orbifolds with $N = 1$ spacetime supersymmetry of hyper-Kahler Hirzebruch surfaces fibered over a lens space to be gauge mediated, one requires, $$\sum_{k=0}^{\infty} k = \frac{e^{i \pi}}{11.999...}$$ This identity is proved via the thrice iterated application of integration operators, and by evaluating non-trivial zeros of real valued second degree polynomials of a single variable.