Theorem: Finite-plane extension with an explicit dual family

There is $\rho \ge 0$, compactly supported in the open response slab, for which

is positive definite. Define

Then $⟨J_a,v_c⟩={\delta }_{ac}$, and

represents $B_f$ on ${\mathcal{V}}_2$.

Proof. The two curvature fields are independent on the full open slab. Otherwise a nonzero combination $v$ would have $\Delta v=0$ there. Each spatial slice is Schwartz, hence zero, and wave uniqueness would contradict independence.

Consequently finitely many component evaluations separate the two-dimensional coefficient space. Small nonnegative bumps around those points produce a positive definite Gram matrix: its limiting quadratic form is a positive sum of squares of separating evaluations. Fix a sufficiently small choice.

Equation (eq:curvaturecurrent) and matrix inversion give the duality. Evaluating (eq:Jplane) on ${\lambda }_1{v}_1+{\lambda }_2{v}_2$ proves the claim.