Weniger's delta transformation can put poles off the cut. An Evidence Press briefing. First, the background: many problems in physics produce series that diverge, yet still encode a well-behaved function. A large class of these, called Stieltjes series, come from functions with a single cut along the negative real axis, where all their singular behaviour lives. The classical way to sum such series, standard Padé approximation, is reliable partly because its poles fall on that cut. Weniger's delta transformation is a newer method that is often much faster. Next, the question: in 2026, Riccardo Borghi asked whether the delta transformation also keeps its poles on the cut, for every Stieltjes series, and called this a well-grounded conjecture. Now, the main finding: the conjecture is false, and the paper says exactly when it holds. The method has an order, k, and a shift. At orders one and two the poles always stay on the cut. At order three they do only when the shift is at most about 0.37, so the usual choice of one is unsafe. From order four on, no shift is safe for every series. The reason is that the series coefficients can be tuned so that only three neighbouring terms of the denominator matter, and those three fail a classical test, the factor-four test. The stray poles are genuine, and for one fixed example they keep appearing at every order checked, from three to thirty. On the positive side, the paper proves that the familiar families, including the Euler series, stay safe at every order. As for how sure this is: the main results have written proofs, and every finite example is checked in exact arithmetic. Pole positions and accuracy examples are numerical illustrations. Finally, the limits: genuine stray poles beyond order thirty are conjectured, not proved, and nothing here says the method fails to converge. It is an anonymous, unrefereed candidate, with AI-assisted adversarial review only, not peer review. Published on 23 October 2026, its paper and evidence are linked from the Evidence Press release page. This briefing uses an AI-generated voice and is not additional scientific evidence.