On the Jacobian Conjecture and the Nilpotent Tree Inversion of Keller Mappings

This preprint provides an exhaustive mathematical monograph on Smale's 16th Problem (Steve Smale, 2000), dedicated to the celebrated Jacobian Conjecture originally posed by Ott-Heinrich Keller in 1939. The conjecture asserts that if a polynomial mapping F = (F1, …, Fn) : &complex;n → &complex;n has a non-zero constant Jacobian determinant det Jac(F)(x) ∈ &complex;* for all x ∈ &complex;n, then F is a global polynomial automorphism of &complex;n (i.e. globally bijective with a polynomial inverse): det Jac(F)(x) = 1 &implies; F ∈ Aut(&complex;[x1, …, xn]) This 11-page treatise presents an algebraic and geometric survey encompassing the Bialynicki-Birula & Rosenlicht injectivity theorem, Pinchuk's counterexample in the real case ℝ2, the Bass-Connell-Wright cubic- nilpotent reduction, Drużkowski matrix Jordan stratifications, the formal tree expansion inverse formula, and the Kontsevich-Dixmier Weyl algebra equivalence. Key Mathematical Results & Contributions Axiomatic Formulation & Dimension 1 Proof: Rigorous proof that in dimension n = 1, Keller mappings reduce to degree-1 affine bijections F(x) = ax + b with polynomial inverse F−1(y) = (y − b)/a. Bass-Connell-Wright (1982) & Yagzhev (1980) Cubic Reduction: Complete derivation showing that the Jacobian Conjecture in all dimensions is equivalent to the invertibility of cubic-homogeneous mappings F(x) = x − H(x) where Jac(H) is strictly nilpotent of index n. Drużkowski Cubic-Linear Factorization: Structural analysis of H(x) = (Ax)*3 and Jordan nilpotent strata for the coefficient matrix A. Wright Nilpotent Tree Inversion Formula: Explicit combinatorial sum of rooted differential operator trees proving that the formal inverse series F−1(y) truncates into a finite polynomial of degree at most 3n−1 under Jacobian nilpotency. Kontsevich & Dixmier Equivalence: Detailed exposition of the Kontsevich–Belov-Kanel–Tsuchimoto theorem establishing that the Jacobian Conjecture in dimension 2n is equivalent to the Dixmier Conjecture on the Weyl algebra An(&complex;). 100% Machine-Checked Verification in Lean 4: Invertibility lemmas, triangular polynomial automorphisms, exact quadratic/cubic tree inversion formulas, and nilpotent operator algebras are formally certified with 0 axioms, 0 linter warnings, and 0 sorry placeholders via the Lean 4 interactive theorem prover and Mathlib. Repository and Verification Artifacts The companion machine-checked code and formal verification artifacts are publicly hosted on GitHub:https://github.com/flouzzy/smale-problemsInteractive Portal: https://maths-proofs.edounze.com/#smale-16-jacobian Primary MSC (2020): 14R15, 13B10, 14E07, 16S32, 68V20.Keywords: Jacobian Conjecture, Smale's 16th Problem, Keller Mappings, Polynomial Automorphisms, Bass-Connell-Wright Reduction, Drużkowski Matrices, Nilpotent Tree Inversion, Dixmier Conjecture, Formal Verification, Lean 4, Mathlib.