Spectral Endomorphism Factorization and Exceptional Degree-Two Root Correspondences in Codimension-Three Lefschetz Failure

Let X be a smooth rational surface obtained by resolving the generically reduced quadratic-root correspondence attached to a controlled codimension-three weak Lefschetz failure. The preceding Pfaffian analysis produces a pulled-back rank-two quotient bundle Q, a nowhere-zero section q∈ H⁰(X,²Q⊗X(M)), a finite flat root double cover p:→ X, and a rank-two quotient 0X(-M)²Q G0 with 4c₂(G)-c₁(G)²=3. We study the first remaining correspondence degree e=H· M=2. Writing Λ=_(2ξ), we prove the spectral realization G p_*Λ and an exact sequence 0 p_*_ G p_*Θ0, Θ=Λ⊗2⊗ p^*( G)⁻¹.$ Every non-scalar endomorphism is then represented by an explicit spectral factorization. A nilpotent endomorphism yields a square factorization indexed by a divisor class $T$ and forces the half-branch class to be effective. A semisimple endomorphism splits $G$ and yields two orthogonal quotient binary quadratics whose dual discriminants factor the branch section. For $e=2$, the nef kernel class is orthogonal to the half-branch class. On an integral root cover this excludes the nilpotent branch. In the split branch, the quotient of smaller plane degree descends to a fixed ternary symmetric tensor. Rank three contradicts the exceptional twisting, rank two forces a double-line component of the branch divisor, and rank one has identically zero discriminant. Consequently, if the connected root double cover is normal, then G is simple. Riemann--Roch gives χ( G)=1, while a degree-window argument gives ²(G,G)=0 for $e=2$; hence G is exceptional. Normality is retained as a hypothesis. We do not classify nonnormal degree-two correspondences, treat degrees e≥3, exclude the entire quadratic-apolar stratum, or prove the weak Lefschetz property in full codimension three.