Normalization, Conductor Lines, and Exceptional Root Bridges in Codimension-Three Lefschetz Failure

A controlled codimension-three weak Lefschetz failure in the pure quadratic Pfaffian branch produces, on a smooth rational surface X resolving a plane root map, a pulled-back rank-two quotient bundle Q, a nowhere-zero binary quadratic \ q H⁰ (X, ²QX (M) ), \ and a finite flat root double cover p: X. The associated rank-two spectral quotient G satisfies 4c₂ (G) -c₁ (G) ²=3. The normal degree-two case was shown previously to make G exceptional. Here we analyze the nonnormal degree-two branch. Write the discriminant as (q) =2E+D₀ B₀=B-E, where E is the maximal divisorial square factor and D₀2B₀ is reduced in codimension one. We prove that normalization is the canonical elementary modification 0 G G i_*E (H+E) 0 and that 4c₂ (G) -c₁ (G) ²=-B₀². For comparison, a more general rank-one torsion-free conductor quotient would change the right side by a zero-dimensional correction divisible by four. In the present spectral setting the module is invertible; the relative Veronese factorization identifies the actual conductor quotient as the line bundle E (H+E), so no such correction occurs. At correspondence degree e=H M=2, every square-divisor component of positive plane degree is the strict transform of a line. Each repeated line has an exact finite branch-contact budget. Distinct conductor lines are disjoint on X and lie in distinct connected components of E; coexistence therefore requires exceptional curves outside the conductor. On any such exceptional bridge disjoint from the residual branch, the restricted quadratic factors into two moving roots, and their collision divisor is exactly the conductor attachment divisor. We obtain the unique minimal two-line normal form s²u²+2 st\, uv+t²v² ²1, together with explicit blowup and three-direction bridge models. Finally, if the normalized root cover is connected, the total positive plane degree d=H E of the square conductor satisfies d2. The maximal case d=3 would give a finite normal cover of ² unramified in codimension one; purity makes it etale, and Kummer theory forces it to split. Thus the plane discriminant has only the residual-sextic, squared-line--quartic, or squared-line-pair--conic forms described in the paper. We do not prove global existence or nonexistence of the remaining quartic and conic branches, treat e3, or solve the full codimension-three weak Lefschetz problem.