A Third Non-Equivalent Proper Euler Magic Matrix of Order 7 via Hexagonal Bireflections

An Euler magic matrix is an integer matrix M with MMᵀ = γI whose squared entries sum to γ along both main diagonals; it is proper when all entry squares are distinct. The existence of a proper example of order 7, asked by Mehat (arXiv:2609.05530, Problem 9.3), was settled in an earlier preprint (doi:10.5281/zenodo.23041131) by a witness with constant 789² obtained from a Cayley transform, and a second non-equivalent witness with constant 935² was constructed in a subsequent preprint (doi:10.5281/zenodo.23083645) from quaternionic rotation blocks. This preprint exhibits the results of a research that was conducted with the assistance of Google’s Gemini Argon model, presenting a third primitive, non-equivalent proper Euler magic matrix of order 7, with constant 360² = 129,600 and no zero entry, obtained by a third method: a product of three hexagonal A₂ bireflections with parameters 5, 8 and 9, followed by a compatible diagonal matching. This lowers the upper bound on the smallest constant in order 7 from 789² to 360²; together with the known lower bound 74², the smallest constant in order 7 lies in [74², 360²].