The Quintic Splitter and Universal (2,3)-Block Factor-Selection Covers

We develop a two-part factor-selection theory for polynomial equations in the Abel-Ruffini range. Part I gives the quintic prototype: a coefficient-defined scheme selecting a quadratic factor of a quintic, together with the associated cubic factor and decic first-coordinate resolvent. Part II extends the construction to every degree d >= 5. For each profile 2m + 3n = d, a finite coefficient-defined cover has branches given by decompositions into m quadratic and n cubic monic blocks. Once a branch is chosen over a field L, all roots lie in a radical extension of L, since only quadratic and cubic equations remain. The construction is not a radical formula from the original coefficients. For the generic polynomial with Galois group S_d, the labelled and unlabelled branch fields are the fixed fields of the corresponding block stabilizers, and their degrees are exactly d!/(2^m 6^n) and d!/(2^m 6^n m!n!). These bounds are sharp among intermediate fields determining the same block decomposition. For d >= 5, the normal closure is the full generic splitting field, so the generic branch-selection step is non-radical.