A Rooted Cyclic Gain Table of Order 56 with No Balanced Triangles

An explicit 56 × 56 matrix over Z/56Z is given with zero diagonal, nonzero off-diagonal entries satisfying g(j,i) = −g(i,j) modulo 56, a permutation root row, and all 55 rooted permutation equations. None of its 27,720 unordered triangles is balanced. Of the 3,080 complete ordered-pair equations, exactly the 110 involving the root hold. The matrix therefore demonstrates that the rooted conditions and the absence of balanced triangles are jointly insufficient to imply the complete ordered-pair system. The full matrix is printed, and a machine-readable copy and a self-contained Python verifier using exact arithmetic are supplied for independent validation. This certificate is not a complete Moore permutation system and does not settle the existence of a degree-57 Moore graph.