Explicit predecessor input

The cubic response ${\mathcal{C}}_3(f,g)$ is the frozen XI-F/XI-G functional, not an element already admitted to ${\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$. Write its polarized kernel as $u(f,g)$. For the detector, the frozen source normalization can be written

The label $g$ records the frozen predecessor data. The source is smooth and compact; no cubic source is multiplied by a new ramp in this revision.

For a free scalar ${\varphi }_0$, unit $\mathbf{n}$, and $\mathbf{p}\in {\mathbb{R}}^3$, set

Define

For the frozen admissible predecessor data:

1. If $f\ge 0$, $f\not\equiv 0$, $\mathrm{supp}f\subset (0,t_g)\times {\mathbb{R}}^3$, where $t_g=\min \{t:(t,\mathbf{x})\in \mathrm{supp}g\}>0$, and $\cos (mt)>0$ on $\mathrm{supp}f$, then $f\in {\mathfrak{O}}_{\mathrm{G}}(g)$. XI-G's scalar witness gives $\mathsf{b}(\mathbf{n},0)=-2{\Lambda }_f$ with ${\Lambda }_f\ne 0$. Such observations exist for every frozen admissible $g$ and $m\ge 0$.

2. If $f\in {\mathfrak{O}}_{\mathrm{G}}(g)$, no $F\in {\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$ satisfies

   $$F{|}_{{\mathcal{C}}_{\mathrm{sol}}}={\mathcal{C}}_3(f,g){|}_{{\mathcal{C}}_{\mathrm{sol}}}.$$

   Equivalently the shell class of the polarized cubic kernel has no microcausal polynomial representative.

This is the observation-qualified obstruction and reduction theorem of XI-G XIG; its proof is an external input. No sign of ${\Lambda }_f$ is needed here.

The wave-propagator criterion is compatible with the cone convention: its primed bicharacteristic relation transports covectors with their causal character and time orientation preserved. In Minkowski coordinates they are the same transported covectors. Thus a forbidden causal tuple in the wavefront set of the trisolution must pull back to a forbidden source tuple. This establishes the necessary cone-preservation implication; it does not evaluate the criterion for undetected observations.