Proposition: Unchanged free analytical tests

The free Hessian, propagators, causal brackets, and the specific free numerical distributions entering the X-G-ST fish and null-wavefront diagnostics are independent of $\lambda$. Every conditional statement about those unchanged objects retains its original hypotheses and conclusion.

Proof. The added vertex has no quadratic part at the zero configuration. The definitions of the free operators in Section sec:free are therefore unchanged. Applying the same operations to the same free distributions gives the same diagnostic objects.