The Quintic Splitter: Universal Quintic Solvability in a Quadratic-Factor-Selection Function System

The general quintic is not solvable by ordinary radicals. We prove the complementary statement that a monic quintic over a characteristic-zero field is uniformly solvable after adjoining one branch of a natural quadratic-factor-selection cover. For P=x⁵+ax⁴+bx³+cx²+dx+e, this cover is XP=Spec K[U,V]/(F₁,F₂); its points are monic quadratic factors x²+Ux+V. A branch $(u,v)$ gives P=(x²+ux+v)(x³+px²+qx+r), with $p=a-u$, q=b-au+u²-v, and r=c-bu+au²-u³-av+2uv, so the remaining roots come from the quadratic and cubic formulae. The first coordinate satisfies the decic resolvent RP(U)=∏i<j(U+αᵢ+αⱼ), but the intrinsic object is the two-coordinate cover XP. For S₅-quintics, its branches form S₅/(S₂× S₃); degree $10$ is the sharp lower bound for acquiring a quadratic factor, and each branch field has full splitting-field normal closure. Finally, P is ordinary-radical-solvable exactly when XP has a radical branch. Thus the Abel--Ruffini obstruction is exactly the non-radicality of the splitter branch.