Quadratic Apolar Residues and Pfaffian Root Correspondences in Codimension-Three Lefschetz Failure

Let $\kk$ be an algebraically closed field of characteristic zero, let $S=\Sym(V)$ with $\dim V=3$, and let $A=S/\Ann(F)$ be a compressed standard-graded Artinian Gorenstein algebra of socle degree $2m+1$, where $m\ge2$. We study a corank-one failure of multiplication by a general linear form in the middle degree, under the additional assumptions that multiplication is injective below the middle and that the induced binary inverse system belongs to the quadratic apolar stratum. We introduce a canonical quadratic residue with target $N_F=I_{m+2}/V I_{m+1},\qquad I=\Ann(F),$ and identify $\dim N_F$ with the defect of linear syzygies among the degree $m+1$ equations. In the pure branch $N_F=0$, we prove that $m$ is odd and that $I$ has a pure Buchsbaum--Eisenbud resolution defined by an odd alternating matrix of ternary quadrics. For a general hyperplane, the quotient has the exact binary presentation $A/(\ell)\cong \kk[u,v]\big/\big((h)+q\,(u,v)^{m-1}\big),\qquad \gcd(h,q)=1.$ The moving rank-one Pfaffian relation defines a rational kernel map. We exclude constant and curve images by classifying quadratic subspaces whose binary projection drops to rank at most one along a curve, and by a double-kernel argument for odd alternating matrices. The submaximal Pfaffians also define a morphism from the dual projective plane. The two roots of every general relative quadratic are identified by this Pfaffian morphism. We exclude the generically double-root, or tangent, branch. For the remaining squarefree root correspondence, a simultaneous resolution produces divisor classes $H,L,M$ satisfying $2L\equiv d(M+2H),\qquad L^2=\frac{d^2}{2}(H\cdot M),\qquad d=m+1,$ and an associated rank-two quotient bundle $G$ satisfying $4c_2(G)-c_1(G)^2=3.$ Thus the pure quadratic branch is reduced to the classification of a squarefree conic correspondence of bidegree $(2,e)$ and discriminant three. We do not classify that final correspondence and do not claim the weak Lefschetz property in full codimension three.