Antisymmetric Entropy: An Information Measure for Non-Commutative Dynamical Systems

We propose antisymmetric entropy (HAS), the first information-theoretic decom-position of a univariate time series into symmetric and antisymmetric components. Embedding the series as a grade-1 multivector in the Clifford algebra Cl(3, 0), thelagged geometric product decomposes by grade into a scalar (grade-0, symmetric correlation) and a bivector (grade-2, antisymmetric rotation). Unlike transfer en-tropy, which requires a bivariate system, the non-commutativity here arises from the geometry of the embedding, not from multivariate dependence. The papermakes three contributions. First, we define HAS = H0 + H2 and establish gradeadditivity from algebraic orthogonality plus statistical independence. Second, weprove that the maximum-entropy distribution in each grade, under symmetry and alog-variance constraint natural to ratio/product random variables, is Student-t—notGaussian—and confirm this empirically; the bivector’s ratio-of-normals structureprovides a generative mechanism for heavy tails. Third, we show that entropy(variability of rotation) and Frobenius energy (magnitude of rotation) carry distinctinformation and can diverge substantially. Classical scalar entropies—approximate,sample, permutation, multiscale—are blind to the antisymmetric grade.