Uniform Quaternionic-Unimodular Matroids

This preprint classifies the uniform matroids admitting quaternionic-unimodular representations in the determinantal sense of Scott and Sokal. A uniform matroid of rank r on n elements admits such a representation exactly when it has at most six elements or its rank or corank is at most one. A Gram-matrix argument gives exactly six projective-equivalence classes of representations of the rank-three uniform matroid on six elements with fixed column order. Column permutations identify all six classes, and one representative has nonzero entries among the Hurwitz units. The same Gram constraints exclude the seven-element rank-three case; deletion, contraction, and duality complete the classification. The paper also proves that the determinantal and chain-group definitions agree for uniform matroids. As an application, inverse powers of the cubic elementary symmetric polynomial are completely monotone for exponent zero, exponent two, and every exponent at least three in five variables; in six variables the corresponding sufficient range is exponent zero, exponent two, and every exponent at least seven halves. These are sufficient exponent ranges. The deposit contains the manuscript PDF, LaTeX source, and an exact-arithmetic calculation supplement. Four Python programs verify the conference-matrix construction, all fourteen Rodrigues obstruction branches, the six normalized forms, independent complex-determinant checks, permutation-conjugacy certificates, Gram identities, and the full cubic basis-polynomial identity. The finite calculations support the algebraic proofs in the manuscript.