Rank bounds and restoration in two-block cube covers

How large can a cover of the Boolean cube be if none of its members can be removed? For positive integers c and d, I study covers by coordinate subcubes with separate restrictions in two blocks: at most one fixed coordinate in a block of size c, and at most d in a block of size 2d. After refinement, chosen private points give an exact profile of the coordinate traces. An intersection matrix gives joint rank restrictions on repeated complementary supports and pairs of traces whose supports are not complementary. One consequence is that such covers of Q₃ × Q₈ have at most 488 members, compared with the elementary bound of 512. Attainment of 488 is not asserted. I also prove quantitative restoration bounds for these profiles. A support assignment on the whole second-block cube can be chosen while retaining every complementary trace pair already present. The bounds control the number of mask parts required, with refinements using vertex covers and cube parity. When the points with no selected private point contain no cube edge, the paper determines the exact number of masks required by the inherited complementary labels within each part. This 14-page author preprint includes the complete proofs and both appendices, together with the funding, competing-interest and AI-assistance statements. No external dataset or computer-generated certificate is required for the proofs. This deposit is a preprint, not a journal version of record. The earlier framework is credited in the article, including the preprints at https://doi.org/10.5281/zenodo.21471327 and https://doi.org/10.5281/zenodo.22174995.