Counterterm-blind tests

For real distributions and real tests, the reality condition is automatic. Over complex tests it must not be omitted: tests annihilating $r$ alone cannot distinguish real from nonreal multiples of $r$.

Suppose a complete regulator difference has already been shown to converge,

Then a single compact test ${\Phi }^{\circ }$ with

obstructs removal of $L_{ab}$ by an allowed finite counterterm in $\mathbb{R}r$. These tests are diagnostics. Existence of a finite part still means convergence against all compact tests, not only against $\ker r$.

For an incidence representation, suppose a pushforward is defined and

For example, ${\gamma }_{ab}$ may be smooth and $\psi$ proper on the relevant supports. The correct condition is

Nonconstancy of ${\gamma }_{ab}$ by itself is not an obstruction unless the needed injectivity or separation property of the pushforward has also been proved.