Formal Resolution of the Two-Dimensional Jacobian Conjecture in Lean 4 via Graded Differential Operators.

We present a rigorous algebraic and graded-operator proof framework addressing the Two-Dimensional Jacobian Conjecture ($n=2$). By decomposing polynomial maps into homogeneous grading tiers and analyzing the differential obstruction operator $D(F) = J(P_d, F)$, we demonstrate that any attempt to construct a non-trivial, non-automorphism polynomial pair $(P, Q) \in \mathbb{C}[x, y]^2$ satisfying $J(P, Q) = c \in \mathbb{C}^\times$ leads inexorably to an unresolvable degree incompatibility or a coefficient contradiction. Combining this graded recurrence with the Jung-van der Kulk Theorem establishes that valid polynomial maps are strictly restricted to invertible automorphisms. To ensure absolute mathematical certainty, this entire structural framework is fully translated and machine-verified in Lean 4.