Clean matching and excess

For smooth maps $f_i:L_i\to W$, a matching fiber product is clean when it is a smooth manifold and its tangent space is exactly the space of tangent tuples with equal images under the differentials $\mathrm{d}{f}_i$.

For canonical composition, clean matching and the appropriate projection hypotheses precede the symbol integral. The clean composition excess is the dimension of the local fibers of that projection, not an arbitrary difference between two rank counts GS (Section 4.1)Wunsch (Section 9).

The test below is precisely this tangent-space test. No clean-composition theorem is applied unless it passes.