Biorthogonal projection and a reciprocal identity in the dual boundary element method

The local expansion at a singular point carries two families of eigenfunctions. The admissible r + λ describes the field; the second, r − λ , is excluded from any physical field for unbounded energy yet satisfies the same homogeneous conditions on the surfaces meeting at the point. This paper identifies that discarded family as the natural dual basis of the first, and draws two exact consequences. Extraction of a singular amplitude becomes a projection rather than a fit, the pairing in Betti’s bilinear form annihilating every other mode at every radius, with no window of radius to choose and, for the traction-free crack, a normalisation in closed form. And the failure of that projection to be independent of its contour equals the violation of the boundary condition weighted by the same eigenfunction, exactly and with no asymptotic argument—a consistency check, not an estimator of the error in the amplitude. The proof needs only Betti’s theorem and the confinement of the discretisation error to the boundary, which no domain discretisation possesses. Neither result is a crack technique: biorthogonality follows from homogeneity in r alone, so the pairing rule is μ = − λ at any singular point, for complex exponents as for real. Verification spans exponents from 0.500 to 0.616 and a 270° re-entrant corner.