Problem and contribution

A nonzero response of one specified tree need not survive the addition of every allowed local contact. Survival modulo contacts is a membership question in a space of response functions. Evaluating that question on a finite family gives a finite linear-algebra problem, but its interpretation depends on admissibility, normalization, and complete coverage of the allowed response changes.

We solve that finite problem for

The first label is blind by symmetry. The other two have reference response $16/3$, but the complete contact image on this family contains both responses.

Theorem thm:certificate gives the central result. Its proof derives four principal quantities:

They are respectively a reference-tree response, a raw contact contraction, an original matrix minor, and fitting coefficients.