A negative-curvature normal barrier on an algebraic cone that is not a hyperbolicity cone

An explicit affirmative answer to Tunçel's 2017 existential problem: the regular algebraic convex cone K={x,y,z≥0:xy²≥z³} has the 100-normal barrier F=-25 log(z(xy²-z³)) with D³F(a)[h,u,u]≤0 for every interior base point a, every cone direction h and every real direction u, but K is not the hyperbolicity cone of any homogeneous polynomial in the original three-dimensional space. The proof supplies exact polynomial and positive-semidefinite certificates for all directions, including degeneracies, and an arbitrary-degree boundary divisibility and nonreal-root obstruction. Concise research note with TeX source, two independently authored standard-library exact verification programs, mathematical audits and a bounded primary-source priority audit. The underlying cone and known self-concordant power-cone barriers are credited to Hildebrand and Roy–Xiao; the barrier parameter is deliberately nonoptimal. The broader every-regular-cone barrier question is outside the result. AI tools were used extensively in development, verification and writing. Unrefereed; no external human peer review or formal proof-assistant certification is claimed. The priority screen does not certify exhaustive literature coverage or first-ever priority.