From Linear to Nonlinear Fokker--Planck: State-Dependent Drift, McKean--Vlasov Coupling, and Self-Consistent Spectral Closure

This paper generalizes the linear Fokker–Planck framework of the preceding works along two independent directions. Case I treats state-dependent nonlinear drift F(z), where the Ao decomposition becomes pointwise, F(z) = −(D(z)+Q(z))∇Φ(z), and the spectral operator Γ(z) = ∂F/∂z is a position-dependent Jacobian whose trace tr Γ(z) = ∇·F(z) varies spatially—a signature of local entropy-production-rate inhomogeneity. The PE-gated integral term, with gate σ(e) = 1/(1+αe²), provides an explicit gate Jacobian (1−αe²)/(1+αe²)² that reverses sign at |e| = 1/√α (automatic anti-windup). Case II treats density-dependent drift of McKean–Vlasov type, where the spectral operator becomes an integral kernel Γ[ρ](z,z′) = δF[ρ](z)/δρ(z′) and the D+Q decomposition requires a weighted L²(ρ) inner product, with the weighted trace of the antisymmetric kernel vanishing identically. The two cases are unified through a self-consistent closure mapping ρ → Γ[ρ] → Δ[ρ] → κ[ρ] → K[ρ] → F[ρ] → ρ′, whose fixed point ρ_ss defines the nonlinear NESS; the three dynamical regimes (transient, NESS, equilibrium) are carried by two functionals—Δ[ρ] (time scale) and the antisymmetric kernel Ĝ_Q[ρ] (circulation structure)—with no change of formalism. A dimension-grading note on the Jacobi identity is included: in d = 2 any antisymmetric field is automatically Poisson, while in d = 3 the identity holds iff V·∇×V = 0 (Beltrami fields as the counterexample family). All derivations are accompanied by executable verification scripts (reproduce/, single entry point run_all.py). The v3.1 revision adds a preregistration layer—closed-form predictions frozen in writing before the corresponding simulations run—including gate blind predictions (Lyapunov-exponent sum equal to the trace −(Kp+η) independent of the gate; exact complex-eigenvalue boundary x_c = 0.083), a 30-seed McKean–Vlasov scale-up (kernel amplification median 0.226 and drift self-consistency 0.109, independently reproducing the published 0.22/0.12), a convergence trio, and an ablation lattice; the scored scoreboard, with both misses attributed, is reported in the Evidence-hardening section. During the preparation of this work, the author used AI language models to assist with mathematical formalization, code implementation, and manuscript refinement. After using these tools, the author reviewed and edited the content as needed and takes full responsibility for the final content. Keywords: nonlinear Fokker–Planck equation, McKean–Vlasov, Ao decomposition, potential–flux decomposition, spectral operator, self-consistent closure, PE-gated integral control, non-equilibrium steady states.