A Zariski-Open Family of Proper Septic Polynomials with Full Traceless Images

Let K be an algebraically closed field of characteristic zero. We identify a 762-dimensional space of proper multilinear degree-seven polynomials in the third power of the commutator ideal, with zero formal cyclic trace and annihilated by every repeated-input identification of type (3,3,1). A nonempty Zariski-open subset of this space consists of polynomials p whose sets of individual matrix values satisfy p(Mₙ(K)⁷) = slₙ(K) for every n ≥ 3. The ambient proper and formal trace-zero spaces have dimensions 1260 and 1049, respectively. An explicit polynomial certifies nonemptiness of this coefficient-open family. The proof combines a symmetric-group decomposition, exact finite determinant witnesses, an additive-size argument and a separate 3×3 matrix argument.