Attachment Criteria and Cubic Reduction for SI-Orderings of Spanning Trees

We study inherited attachment of complementary spanning trees by subdividing one colored edge and adding an opposite-color edge to an old vertex. Exact interval criteria and constructive sufficient conditions yield coverage for both colors and every old attachment vertex. A computer-assisted theorem covers arbitrary FIRST-tree subdivisions of every six-vertex core when construction spokes end at original core vertices; it implies full attachment coverage for simple atomic 3-connected pairs with at most five wide source-tree cuts. Two arbitrary-rank constructions use whole-DAG two-root certificates and interval-blocked cyclic families. Physical decomposition and smoothing lemmas reduce a minimum graphic-SIBO counterexample to a specified simple atomic cubic residual class. A complete-parent-bank entry assertion for that class is sufficient for graphic SIBO but remains open. The supplement includes the complete finite qualification, proofs, physical constructors and portable verifiers.