Domination, component spectra, and sharp augmentation losses in matroid base-intersection graphs

For all finite matroids of rank at least two, proves exact prescribed-family augmentation and cocircuit optimization formulas, a complete minimum-family component spectrum, and a sharp augmentation-strategy loss bound. Gives an explicit laminar-tree algorithm, paving/Fano-free zero-loss classes, and graphic/simple-binary sharp examples. The graph has all bases as vertices, with adjacency by nonempty intersection; it is not a basis-exchange or disjointness graph. Unweighted; not a classification of all zero-loss matroids or a general oracle-input polynomial algorithm. The value complexity for explicit laminar trees is separated from output complexity. Classic partition/union tools, ordinary rank-two endpoints, density bounds, tree-DP techniques and the hidden-instance method are not claimed as new. This is not a proof of a general hypergraph domination conjecture. Status: independent project-level manuscript audit, fixed finite verification and full-page artifact acceptance are recorded separately; external mathematical review and formal peer review are pending. Three important prior-work gaps remain: the final identity/full text of a Zhang-Liu work cited as forthcoming in a 2012 chapter; Akkari's 1995 packing paper; and the full later versions of adjacent Weninger-Fukasawa work. These are not represented as excluded coverage, and no global-priority guarantee is asserted. The archive contains the frozen manuscript, accepted TeX/PDF, bounded verification code and fixed data, reproduction instructions, artifact reports and checksums. The separately downloadable PDF is identical to the accepted PDF inside the archive.