Order 668 in the Hadamard Conjecture: Structural Bottlenecks and a Certified Local-Search Obstruction

The order 668 = 4 × 167 is the smallest order for which no Hadamard matrix is presently known. This paper gives a structural and computational map of one sharply delimited search frontier at this order, namely Goethals-Seidel difference families over the integers modulo 167, with particular emphasis on the quadratic-residue anchored skew supplementary-difference-set regime. It does not claim a non-existence result. The Paley-Todd quadratic-residue anchor, and the reduction from a four-block Goethals-Seidel family at a prime v congruent to 3 modulo 4 to a three-block cyclic supplementary difference set, are due to Đoković and Kotsireas (Special Matrices 4, 2016, 317 to 325), who also identified v = 167 as an admissible unresolved case beyond their published tables. The contribution here is deliberately narrower and explicit: a computational treatment of the still-untabulated v = 167 instance, a 668-specific enumeration of the ten row-sum chambers forced by the four-circulant zero-frequency constraint, a record of why the usual compression and multiplier-orbit reductions are weak at the prime length 167, and a reproducible exact-integer obstruction for a stated local-search regime. The best state reached in the reported computation has energy E = 2880, residual support 92, and maximum residual magnitude 12. Exact enumeration certifies that this state admits no improving size-preserving one-row pair flip and no improving two-row pair-flip compound move, so it is a local-minimum certificate relative to an explicitly stated neighborhood, in the standard sense of local search theory. It is a documented obstruction to a precise computational strategy, not a proof that a Hadamard matrix of order 668 does not exist, and no such matrix is supplied. All success checks use exact integer residuals. The accompanying open-source package provides the certified state files with SHA-256 hashes, an exact verifier, the one-row and two-row audit scripts, and one-command reproduction of the reported result.