Theorem: Proved finite coverage, not a finite-test assumption

For the complete residual family of Definition def:residual, set

Every admissible residual belongs to this known two-dimensional space. The evaluation

is injective on $\mathcal{V}$.

Consequently, within Definition def:stressclass,

The conclusion covers the whole declared residual family and test class, not only the two probe vectors.

Proof. The classification theorem proves, rather than assumes, that

Affine linearity and $\mathfrak{W}(t)=0$ give

This proves the spanning assertion for every component of the complete family, including its infinitely many field and test labels.

Equations (eq:probevalues) give

The determinant is $6$, so the evaluation is injective and the two displayed residual families are independent.

If both probes vanish, then $a=b=0$, hence $S=t$. All its local residuals vanish by the contact and implementation calculations, not by an unsupported extrapolation from sampled tests. The converse is immediate because the two probes are themselves components of the complete residual family.