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, ²QX (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 =_ (2), we prove the spectral realization G p_* and an exact sequence 0 p_*_ G p_*0, =^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 (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 e3, exclude the entire quadratic-apolar stratum, or prove the weak Lefschetz property in full codimension three.