Stage E: the separate obstruction instrument

Choose a separate finite dictionary of the reciprocity tasks (eq:W), with its own clock $C_{\mathrm{rec}}$. A task can contain the fixed analytic expression for $W$, a supplied rational enclosure $[L,U]\subset (0,\infty )$, and the tag $RECIPROCITY\_INCOMPATIBLE$.

Such an enclosure exists for each fixed witness: the proof gives $W>0$, so rational numbers $0<L\le W\le U$ can be chosen and supplied. This existence is not a general evaluation algorithm. For example, if $|{\sigma }_{\epsilon }^{\prime \prime }|\ge a>0$ on $\mathrm{supp}\zeta$, then

is an explicit lower-bound interface.

Define ${\mathcal{M}}_{\mathrm{rec}}$ by the same Kraus formula as (eq:instrument), using these separate report words and clock names.