Combinatorics of the Fourier transform: Stokes data, Gale duality and frieze patterns

What the math gives to ML

The paper gives an explicit, constructive form of Gale duality: a configuration of b projective points in a projective space, represented by an a by b matrix X, is transformed into a complementary matrix Y satisfying Y D X-transpose equals zero for an invertible diagonal gauge D. This suggests neural modules that replace a redundant set of token or feature directions by a complementary annihilator representation, with exact orthogonality constraints rather than a learned approximate projection. A second transferable construction is the superperiodic order-a recurrence, which represents a cyclic sequence using only a-1 local coefficients and a fixed boundary sign; this can serve as a linear-time structured alternative to dense sequence mixing. The main caveat is that these structures impose strong algebraic constraints, so experiments should compare expressivity and optimization stability against standard attention and state-space baselines.