Theorem: Local convergence does not determine Newton composition

The inputs $z_R$ converge to zero in the compact-local smooth topology, but

smoothly on compact sets, where

Each finite-$R$ rod is the decaying original Newton solution.

There are fixed compact $f$ and a fixed compact lapse input $w$ for which

Consequently compact-local convergence to the same classical input does not determine the limit of this nonlocal Hessian operation.

Proof. The source annulus has radius between $R$ and $2R$, so the input and all derivatives vanish on each fixed compact set for sufficiently large $R$.

Changing variables $\mathbf{y}=R\mathbf{Y}$ in the rod formula gives

On the fixed source annulus the kernel is smooth for compact $\mathbf{x}$ and large $R$. All compact-set derivatives converge by direct differentiation and dominated convergence. This proves (eq:escapinglimit).

Moreover

The other entries vanish by oddness. Each finite source is compact, so its native rod is smooth and decays at infinity.

Take $w$ to have only $j_{00}\ne 0$. The local polarized expression with $z_R$ vanishes, and $\ell z_R=0$, leaving

Choose $\ell w$ with nonzero first spatial derivative on an open region and choose $f$ there with a sign and time support detecting $\tau v^1{\partial }_1\ell w$. The resulting limit is nonzero.