Proposition: Exact switch defect and the retained obstruction

With $g,\eta$ fixed,

A coupled invariant two-jet with free input $P_m{\varphi }_0=0$ must satisfy

For nonconstant $g$, an open dense set of smooth free inputs violates a pointwise necessary condition, so its specified-input coupled-jet fibres are empty.

Proof. Under $\delta q={\mathcal{L}}_{\xi }q$, $\delta \varphi ={\xi }^{\rho }{\partial }_{\rho }\varphi$, the curved densities vary by divergences. At fixed $\eta$,

Integrate against the switch factors and compare with the Euler variation. Using ${\partial }_{\mu }{T_0}^{\mu \nu }=(P_m\varphi ){\partial }^{\nu }\varphi$ gives (eq:Noether).

Alternatively expanding the exact Bianchi identity on

immediately gives (eq:Bianchi).

Choose $x_0$ with $\mathrm{d}g(x_0)\ne 0$. The violation at $x_0$ is a continuous quadratic polynomial in the free scalar's first jet. A free solution $v$ with $v(x_0)=0$ and $\partial v(x_0)=\mathrm{d}t$ has mixed stress $\mathrm{diag}(1/2,-1/2,-1/2,-1/2)$; such a first jet is realized by smooth Cauchy data. Its contraction with $\mathrm{d}g(x_0)$ is nonzero. For every $u$, at least one component at $u+\lambda v$ has a nonzero quadratic coefficient in $\lambda$. Arbitrarily small $\lambda$ therefore give a violation. This proves density; continuity proves openness.