The exact seven-product closure degree is forty-seven

Starting from $1,x$ and allowing arbitrary complex linear combinations for free, we determine the largest degree whose entire coefficient space lies in the Zariski closure of outputs computable with seven products. The answer is $47$. A $49$-parameter polynomial family has a contact of order $47$ with a nonsingular $48$-variable coefficient differential, established by a contraction argument whose inequalities are checked in exact rational arithmetic. An inverse-function and scaling argument gives coverage in the full degree-$128$ ambient space. A self-contained dimension bound, together with irreducibility, proves that degree $48$ cannot be covered.