The quadratic normalization

For every nonzero compact switch of the frozen class and every fixed $m\ge 0$,

The vacuum provides a nonempty fiber, so universal emptiness is false.

Proof. The inherited proof is XI-A :empty CheungXIA. For (eq:input),

The time component of (eq:compatibility) would require ${\partial }_{t}g/2=0$. A nonzero switch compactly supported in the slab interior cannot be independent of time. Hence compatibility fails.

For ${\varphi }_0=0$, the choice

solves the coefficient equations. A higher-order correction cannot cancel a nonzero order-one coefficient.

quantitative bump and its actual support

There is a relatively compact open region $B$ and a constant $c_{\mathrm{b}}>0$ with

Choose $f_{\mathrm{b}}\ge 0$ with

The retained XI-A detector is

If $\mathrm{supp}{f}_{\mathrm{b}}\subset D_{\ast }$, the nonzero pairing (eq:bump) transfers to the retained laboratory with the same value.

The detector is a switching-region detector: its support is not contained in the plateau $U$. If it does not fit the retained chart or shadow, the completion obstruction remains valid, but a report of its supplied value is not a locally acquired detector in that laboratory. Enlarging the shadow is not authorized.

Proof. Lemma lem:transfer preserves the compact pairing. On $U$, $g=1$ on a neighborhood and ${\partial }_{t}g=0$; thus a test supported there cannot have the strictly positive pairing (eq:bump). The remaining statements distinguish the domain of the analytical calculation from the location and access contract of its report.

Definition def:jet requires the equations on the completion slab. It is not the space obtained by requiring equations only on a plateau neighborhood. The obstruction proves no universal nonexistence theorem for plateau-local solutions.

Where the detecting switching region is in the chart, the same local compatibility identity excludes a jet satisfying the invariant equations on a neighborhood of that detector. This is the precise local consequence.