Sharp slab obstructions and minimum repairs in C2 combinatorial metamaterials

This working paper develops an exact mathematical theory of prescribed defect patterns in C2 combinatorial mechanical metamaterials. It studies finite rectangular assemblies of cubic building blocks with three allowed orientations and free boundary orientations. The central questions are which defect patterns can be realized, when geometric obstructions first occur, how those obstructions can be certified, and how many prescribed defect-edge statuses must change to restore realizability. A slice-signature reduction converts three-dimensional realizability into a finite covering problem while preserving the local orientation constraints. For slices with two rows and arbitrary width, a symbolic four-pattern criterion characterizes the forbidden signatures. This proves that every parity-admissible target on a 2 x 2 x c or 2 x 3 x c box is realizable at every height. For the 2 x 4 cross-section, six forbidden slice signatures yield a complete arbitrary-height classification through vertex coverage in 32 disjoint complete graphs on four vertices. The first obstruction occurs at exactly 64 layers. Every nonrealizable target admits a compressed obstruction certificate using at most 96 selected layers, and this bound is sharp. Compression retains selected layers and composes the intervening interface constraints by exclusive-or; it does not require a contiguous physical subbox. For three-row slices, the paper establishes a symbolic cube-cover criterion and a sharp four-column compressed witness bound. For the 3 x 3 cross-section, it proves that the first obstruction occurs at exactly 24 layers: every parity-admissible target through 23 layers is realizable, while an explicit 24-layer target is not. The lower bound combines finite geometry, an excess-budget reduction, binary separation certificates and exhaustive searches that enforce shared tile identities across coupled fibers. Explicit fractional and local-quotient witnesses demonstrate why the particular relaxations studied cannot resolve this global consistency requirement. The paper also gives an exact minimum-repair algorithm for arbitrary fixed cross-sections, with runtime linear in height after cross-sectional precomputation. Explicit 2 x 4 x 64 and 3 x 3 x 24 obstructions both have optimal edge-Hamming repair distance two, supported by complete repaired block orientations. The accompanying materials include the manuscript source, obstruction and repair certificates, finite classifications, verification programs, separately implemented computational checks and SHA-256 integrity manifests. The results concern discrete compatibility with free boundaries. Repair distance counts changed target defect-edge bits, rather than block rotations or fabrication operations. The paper makes no claim of improved elastic strength, fatigue resistance or manufacturing tolerance, and does not determine the global minimum obstruction volume over arbitrary shapes.