Can a polynomial share a root with every one of its derivatives without all its roots being the same? The Casas Alvero conjecture says that, in characteristic zero, it cannot. This paper studies one part of that question: polynomials of degree twenty with only a few nonzero terms. First, the polynomial is centered at the root of its nineteenth derivative. That choice matters, because moving the origin can change the number of terms. The candidate result says that any nontrivial counterexample would need at least eight terms in that centered form, counting the leading term. Seven or fewer are excluded. The proof reduces the seven-term case to fourteen possible coefficient supports and nineteen compatible residue cases. It then excludes every case. A useful idea concerns a group of roots that are much closer to each other than to the other roots. If all the required derivative witnesses are trapped in that group, rescaling can turn a hypothetical splitting into an impossible smaller polynomial. Exact calculations accompany the proof, including a complete census of twelve hundred and sixteen assignments in one residue case. The replay checks those finite calculations; the mathematical transfer still depends on the written argument. This does not solve the full degree-twenty conjecture, and it does not show that an eight-term counterexample exists. This is an unrefereed candidate from Evidence Press, dated September twenty-fourth, twenty twenty-six. The full paper and evidence are linked on the release page. This is an AI-generated voice summary, not additional evidence.