Degree forty-four in the closure of seven-product polynomial evaluations

Starting from $1,x$ and allowing free complex linear combinations, we prove that the Zariski closure of the polynomials computable with seven products contains every polynomial of degree at most $44$. This refutes the equality to $42$ conjectured by Jarlebring and Lorentzon for the largest such degree $d_7$. A simultaneous four-product degeneration retains two intermediate polynomials as well as an arbitrary polynomial of degree at most $12$. Three further products give a coefficient map with a $45\times45$ integer Jacobian of determinant $256$. The published dimension theorem of Jarlebring and Lorentzon additionally implies $44\le d_7\le47$.