Abstract
Rigorous sum rules for negative moments of the discontinuity across the left-hand cut of the $\ensuremath{\pi}\ensuremath{\pi}$ $P$ wave are derived and analyzed. A model by Dilley wherein the $\ensuremath{\rho}$ resonance emerges from elastic $\frac{N}{D}$ equations is shown to be severely inconsistent with these sum rules. Dilley's method for selecting the input left cut is analyzed and shown to be strongly biased in favor of generating a $\ensuremath{\rho}$. Because of this bias, together with the aforementioned violation of sum rules, Dilley's model does not comprise evidence that the $\ensuremath{\rho}$ is generated by forces in the $\ensuremath{\pi}\ensuremath{\pi}$ channel. Numerous successes of the quark model suggest otherwise.
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