The current answer to the candidate is open. Cati and Pasechnik's construction database, revised on August 30, 2025, states that all known Hadamard orders through 1208 are implemented in SageMath and lists 668, 716, 892, and 1132 as the unknown orders in that range. Their Section 3 gives the direct query ``` hadamard_matrix(668, existence=True) Unknown ``` The result `Unknown` means that SageMath has no construction and makes no nonexistence assertion.

The appendix gives a second check. Table 2 lists a construction for each known order \(4n\) with odd \(n<300\); the entry at \(n=167\) is blank. Table 4 records \(167(3)\), meaning that the least known exponent \(m\) for an order \(2^m167\) construction is \(m=3\). Thus order \(1336=8\cdot167\) is known, while the desired order \(668=4\cdot167\) is the missing exponent-two case.

Eliahou's 2025 primary paper calls 668 the smallest open case and constructs the 64-modular near miss checked in the companion artifact. Suksmono's 2025 search paper independently lists 668, 716, and 892 as unresolved below 1000. Epoch AI still identifies 668 as the smallest unknown order on the access date. These sources report the state of knowledge. They cannot prove that a matrix does not exist.

Eliahou gives two run-length encoded sequences \(q,s\in\{\pm1\}^{167}\), forms the quadruple \[ (A,B,C,D)=(s,s',(sq),(sq)'), \] and inserts its four circulant matrices into the Goethals-Seidel array. Here the prime operation keeps the first 84 signs and reverses the signs of the last 83.

The independent replay reconstructs all 446,224 signs of the resulting \(668\)-square matrix \(H\). Every off-diagonal entry of \(HH^{\mathsf T}\) is divisible by 64, so \[ HH^{\mathsf T}\equiv668I_{668}\pmod {64}. \] Exact integer products expose the gap. Each row is orthogonal to 641 of the other rows and has 26 nonzero products. Per row, their values and multiplicities are \[ -512:2,\ -320:2,\ -256:2,\ -192:4,\ -64:4,\ 128:6,\ 256:4,\ 384:2. \] In particular, \(HH^{\mathsf T}\ne668I_{668}\). The construction is a verified 64-modular Hadamard matrix and supplies no real Hadamard matrix of order 668.

The audit followed the candidate's Epoch citation into the current construction tables and primary papers. Kharaghani and Tayfeh-Rezaie's `A Hadamard matrix of order 428`, Journal of Combinatorial Designs 13 (2005), 435-440, DOI 10.1002/jcd.20043, constructs order 428. It thereby moved the smallest unknown order to 668 and contains no order-668 construction.

Cati and Pasechnik checked the published construction record by implementing every known order through 1208. Their revised table leaves \(4\cdot167\) empty and reports `Unknown` for SageMath's order-668 existence query. Eliahou's later paper, received September 24, 2025, provides explicit data at order 668. Its equation is the congruence \[ HH^{\mathsf T}\equiv668I\pmod {64}, \] which allows nonzero off-diagonal multiples of 64. The exact replay finds those multiples and confirms the paper's own description of the true order-668 matrix as elusive.

No supplied citation claims a verified real matrix of order 668. The search also found no downloadable real witness to test. A future resolution should include either a complete \(668\)-square sign matrix with an exact Gram check or a construction theorem whose hypotheses are verified at the stated parameters.

The script expands the two run-length encodings in Fact 3.1, applies the stated half-sign switch, builds four circulant matrices, and assembles the Goethals-Seidel blocks. Each row is packed into a Python integer. XOR population counts then evaluate every row product exactly.

The replay reproduces all thirteen nonzero summed aperiodic autocorrelations in the paper. It verifies the congruence modulo 64 and finds the published 641 zero and 26 nonzero off-diagonal products in every row. The packed matrix uses 84 little-endian bytes per row, with bit \(j\) equal to 1 exactly when column \(j\) contains \(+1\). The concatenated bytes have SHA-256 digest `b9316f8fb407552f6c1301b027e8cddab64796d543801d0005d043cc61a668a1`.

The six-line stdout has SHA-256 digest `915e94a3afae5b1b4f29f7c59e5fe47f55a13c9ed2d1532eeb13ecc43bbccb55`.