BV generators for polynomial Weyl subalgebras

For every field k of characteristic zero and every nonzero polynomial h in k[x], the ordinary Hochschild cohomology of A_h = k /(Yx-xY-h(x)), with its canonical cup product and Gerstenhaber bracket, admits a degree-minus-one Batalin-Vilkovisky generator exactly when h is squarefree. The squarefree case has zero bracket and admits Delta=0. If h has a repeated irreducible factor, a central degree-one annihilator forces any bracket generator to vanish on degree one; an explicit cup-zero noncommuting pair then rules out every generator, even without a square-zero requirement. The argument works over the original field without a splitting-field hypothesis. The original embedded h=0 specialization is k[x] and is treated separately. This gives a classification answering the concrete-family Question (2) in Samuel A. Lopes's contribution, joint work with Andrea Solotar, OWR 46/2023, p.2702. The length-two resolution and derivation structure are established results of Lopes-Solotar and Benkart-Lopes-Ondrus. The older no-generator result and obstruction strategy of Herscovich-Li for FK(3), Documenta Mathematica 27 (2022), Proposition 4.15, DOI 10.4171/DM/X18, are explicitly credited. The note does not claim the first ordinary-Hochschild no-generator phenomenon, exhaustive historical novelty, positive-characteristic generality, or a resolution of the separate general-algebra and Poisson questions. The proof was already public in project PR1391 before this note; this version is dated 22 October 2026 and is not backdated. Bounded primary-source investigation located no authenticated earlier full A_h classification; remaining historical coverage limits are disclosed. The proposed deposit contains the six-page paper and a complete editable-source and verification ZIP. Standard-library exact finite polynomial and normalized-bar diagnostics supplement the written universal proof and named published structural inputs; they are not proof-assistant verification or an independent execution of the claimant's programs. AI tools were extensively involved in mathematical exploration, proof development and reconstruction, computation, source and literature investigation, manuscript preparation, editing, and adversarial auditing. Alec Kriebel is the named human author. This is an unrefereed preprint without formal independent human peer review or journal acceptance; internal AI audits do not provide those certifications. Authored prose is CC BY 4.0 and authored code is MIT licensed.