Theorem: Natural first-order scalar non-descent

The differential (eq:nonlinear) does not preserve the unchanged free equation ideal, even locally on the plateau. In particular its natural first-order action cannot induce an interacting derivation on ${\mathfrak{A}}_0$ merely by using the unchanged free equations.

This failure occurs at order $\kappa$, so a second-order correction leaving $s_1\varphi$ unchanged cannot remove it.

Proof. Work in coordinates centred at a point $p\in U$, with $\tau =t-t_p$ and $y^1=x^1-{x_p}^1$. Take a nonzero odd generator $\theta$, a number $\lambda >0$, and

Set $r=b=\bar{c}=0$. These are smooth local free configurations: $\square c=0$, $P_m\varphi =0$, and (eq:freeequations) holds.

Since $s$ commutes with spacetime differentiation,

On free inputs the right side is

At $p$ it equals

Choose a compact real scalar test supported in a small neighbourhood of $p$ on which the coefficient has one nonzero sign. The corresponding free equation functional vanishes on all free configurations, but its $s$-image does not.

Every element of the free dynamical equation ideal has zero normal symbol on smooth free configurations: equation factors vanish there, and their contractions vanish by the free bisolution identities. Hence the displayed image cannot be an equation-ideal element. No term beginning at order ${\kappa }^2$ changes its nonzero order-$\kappa$ coefficient.