Unrestricted Boolean multiplicative complexity of four-term binary polynomial multiplication

The exact Mul4 instance is resolved: its unrestricted XOR--AND multiplicative complexity is 9. The restricted bilinear/quadratic value 9 was classical; the new theorem proves that arbitrary nonlinear Boolean reuse cannot lower it. This is a natural positive special case of the Boyar--Find question, not a solution of the general (n,m) problem. The lower bound is a conceptual structural argument. A complete Lean 4 formalization checks the unrestricted circuit semantics and the exact equality, while the development-time Python/C++ programs remain independent regression checks rather than logical premises. The corresponding unrestricted questions for five or more terms remain open. Independent statement-fidelity review remains pending.

The exact Mul4 instance is resolved: its unrestricted XOR--AND multiplicative complexity is 9. The restricted bilinear/quadratic value 9 was classical; the new theorem proves that arbitrary nonlinear Boolean reuse cannot lower it. This is a natural positive special case of the Boyar--Find question, not a solution of the general (n,m) problem. The lower bound is a conceptual structural argument. A complete Lean 4 formalization checks the unrestricted circuit semantics and the exact equality, while the development-time Python/C++ programs remain independent regression checks rather than logical premises. The corresponding unrestricted questions for five or more terms remain open. Independent statement-fidelity review remains pending.