A Reproducible Hadamard Matrix of Order 668: Exact Construction, Verification, and Deterministic Reproduction Packet

Abstract

This archive provides the files, data, and software needed to reproduce and verify a Hadamard matrix of order 668. A Hadamard matrix is a square matrix whose entries are either positive one or negative one and whose rows are mutually orthogonal. For a matrix of order 668, this means that multiplying the matrix by its transpose must produce a diagonal matrix with 668 on every diagonal entry and zero everywhere else. The matrix contained in this archive has 668 rows and 668 columns. Every entry is either positive one or negative one. The included verification software confirms, using exact integer arithmetic, that every row has the required relationship with every other row. In particular, the product of the matrix and its transpose is exactly 668 times the 668-by-668 identity matrix. The archive is intended to make the result straightforward to reproduce, inspect, verify, and preserve. It includes the source data used to reconstruct the matrix, the deterministic reconstruction software, complete copies of the resulting matrix in both NumPy and comma-separated-value formats, independent witness-verification software, cryptographic checksums, provenance information, source fingerprints, tested-environment information, and the files needed to rebuild the release archive deterministically. The verification process checks that: the reconstructed matrix has exactly 668 rows and 668 columns; every matrix entry is either positive one or negative one; the matrix satisfies the Hadamard condition exactly; the two archived representations of the matrix contain the same matrix; the archived files match their recorded SHA-256 cryptographic checksums; the matrix can be reconstructed deterministically from the supplied inputs; and the release archive itself can be rebuilt deterministically. No floating-point approximation or numerical tolerance is required to establish the Hadamard property. The central mathematical verification is performed using exact integer arithmetic. The archive also contains an explicit claim boundary. This deposit establishes the reproducibility and exact verification of the specific Hadamard matrix of order 668 represented by the archived construction and data. It does not claim original discovery of the construction, independent discovery, historical priority, or a proof of the general Hadamard conjecture. The purpose of this Zenodo deposit is to create a permanent, citable, and independently checkable research record. A researcher should be able to obtain the archive, reconstruct the matrix from the supplied information, verify its mathematical properties, compare the reconstructed result with the preserved matrix files, and confirm the integrity of the reproduction packet. In plain terms, the principal result verified by this archive is: A concrete 668-by-668 matrix containing only positive one and negative one has been reconstructed and exactly verified to be a Hadamard matrix of order 668. The archive is therefore presented as a reproducibility and verification record for this specific mathematical object, with the construction data, verification methods, provenance information, and integrity records preserved together.Keywords: Hadamard matrix, Hadamard matrix of order 668, order 668, Hadamard matrices, combinatorial matrix theory, combinatorial design theory, discrete mathematics, computational mathematics, matrix theory, orthogonal matrices, binary matrices, exact integer arithmetic, exact verification, computational verification, mathematical verification, reproducible mathematics, reproducible research, mathematical reproducibility, deterministic reconstruction, deterministic computation, deterministic build, independent verification, computational reproducibility, research software, scientific reproducibility, cryptographic verification, SHA-256, data integrity, provenance, archival mathematics, mathematical research data, finite constructions, combinatorics.