Theorem: Ordering-independent exclusion

Let $f\in {\mathfrak{O}}_{\mathrm{G}}(g)$ and $C={\mathcal{C}}_3(f,g){|}_{{\mathcal{C}}_{\mathrm{sol}}}$. Then $C\notin {{\mathcal{F}}^{\mathrm{sol}}}_{\mathrm{mc},\mathrm{pol}}$. No classical-symbol coefficient of an element of ${\mathcal{E}}_H$ equals $C$, and the same holds for every admissible $H^{\prime }$. If $Y\in {{\mathcal{F}}^{\mathrm{sol}}}_{\mathrm{mc},\mathrm{pol}}$, then $Y+C\notin {{\mathcal{F}}^{\mathrm{sol}}}_{\mathrm{mc},\mathrm{pol}}$. Universal admission for all observations is excluded.

Proof. The first assertion is Input theorem input:XIG(b). Every classical-symbol coefficient belongs to ${{\mathcal{F}}^{\mathrm{sol}}}_{\mathrm{mc},\mathrm{pol}}$, proving the second. The underlying quotient vector space is the same for $H^{\prime }$; also ${\bar{\alpha }}_d$ is the identity modulo $\hslash$. Either fact proves ordering independence. If $Y,Y+C$ belonged to the vector space ${{\mathcal{F}}^{\mathrm{sol}}}_{\mathrm{mc},\mathrm{pol}}$, their difference $C$ would belong to it. Input theorem input:XIG(a) supplies excluded observations for every frozen admissible data set, ruling out universal admission into this carrier.