For c2 = b = 0, we get the following for optimality: 1) x^2 - y^2 - 2xy + c1 = p1(2xy, x^2 - y^2 + c1 - 1) = 1 and therefore: 2) x^2 - y^2 - 2xy = 1 - c1 But the chaotic Mandelbrot real line region is most of -2 < = c1 < = - 1.4011, from which 1 - c1 > = 2.4011, but x^2 - y^2 - 2xy < = x^2 < = 1. This is a contradiction (recall that x, y are in [0, 1] X [0, 1] as arguments of p1). So the chaotic region is pathological in the sense that optimizing p1 there leads to a contradiction. It might be asked whether there are any non-chaotic regions which are pathological, and if so what does that mean. Formally, for c between 0 and 1, the above contradiction disappears. However, for c in [-1, 0], we still get a contradiction. We already know that there are pathologies other than based on contradictions, from earlier examples, and perhaps there is another explanation for c in [-1, 0]. Osher Doctorow