Proposition: Reduction of the coefficient estimates to the traces

Assume

1. $v_{\alpha }\in C^{\infty }(S_0)$ for all $|\alpha |\le 3$;

2. $c_{D,\beta }\in {\mathcal{D}}^{\prime }(S_0\setminus Z)$ for all $|\beta |\le 3$.

Then:

1. If every $c_{D,\beta }$, $|\beta |\le 3$, extends to a distribution ${\tilde{c}}_{D,\beta }\in {\mathcal{D}}^{\prime }(S_0)$, then every ${b^{\circ }}_{D,\gamma }$ satisfies (eq:jetbound), the unrestricted supported realization of normal order at most three exists, and one realization is

   $$t_D=\sum _{|\gamma |\le 3}{\mathsf{E}}_{\gamma }{b}_{D,\gamma },b_{D,\gamma }=\sum _{\begin{array}{c}\beta \ge 0\\ |\gamma +\beta |\le 3\end{array}}(-1{)}^{|\beta |}(\genfrac{}{}{0ex}{}{\gamma +\beta }{\gamma })v_{\gamma +\beta }{\tilde{c}}_{D,\beta }.$$

2. Order by order, (eq:jetbound) for $|\gamma |=3$ involves only $c_{D,0}$; for $|\gamma |=2$ only $c_{D,0}$ and $c_{D,e_j}$; for $|\gamma |=1$ only $c_{D,\beta }$ with $|\beta |\le 2$; for $\gamma =0$ all $c_{D,\beta }$ with $|\beta |\le 3$.

3. If for some $|\gamma |=3$ and some compact $K$ the estimate (eq:jetbound) fails for ${b^{\circ }}_{D,\gamma }=v_{\gamma }{c}_{D,0}$, then $c_{D,0}$ has no extension to ${\mathcal{D}}^{\prime }(S_0)$; the failure sequence of XV-C for ${b^{\circ }}_{D,\gamma }$ is then a failure witness for the trace $c_{D,0}$ itself.

4. Changing the extensions ${\tilde{c}}_{D,\beta }$ by $Z$-supported distributions $s_{\beta }$ changes $b_{D,\gamma }$ by $\sum (-1{)}^{|\beta |}(\genfrac{}{}{0ex}{}{\gamma +\beta }{\gamma })v_{\gamma +\beta }{s}_{\beta }$, which is $Z$-supported. This family lies in XV-C's unrestricted supported ambiguity but is not asserted to exhaust it.

Proof. (i) Multiplication of a distribution on $S_0$ by a smooth function and finite sums are defined and restrict compatibly to $S_0\setminus Z$, so $b_{D,\gamma }{|}_{S_0\setminus Z}={b^{\circ }}_{D,\gamma }$. Distributional continuity of $b_{D,\gamma }$ on tests supported in $K$ gives (eq:jetbound); XV-C's supported realization theorem then gives existence, and its realization formula gives the display. (ii) is read off from (eq:collected), since $|\gamma +\beta |\le 3$ forces $|\beta |\le 3-|\gamma |$. (iii) If $c_{D,0}$ extended, then $v_{\gamma }{\tilde{c}}_{D,0}$ would extend $v_{\gamma }{c}_{D,0}$ and (eq:jetbound) would hold by (i). (iv) Smooth multiples and finite sums of $Z$-supported distributions are $Z$-supported; XV-C's ambiguity theorem identifies the unrestricted ambiguity as all $Z$-supported coefficient changes, of which these are some.