Sets whose differences avoid a bracket quadratic

What the math gives to ML

The paper develops a concrete Heisenberg-nilmanifold representation of bracket-quadratic phases such as $e(-\theta n\lfloor \beta n\rfloor)$ and proves that quantitative equidistribution of the corresponding nilmanifold orbit forces exponential-sum cancellation. This gives a mathematically structured alternative to ordinary sinusoidal positional encodings: positions are mapped through a discontinuous, noncommutative phase whose long-range correlations can be bounded rather than learned from scratch. The most direct transfer is a deterministic bracket-polynomial positional encoding for long-context transformers, optionally combined with several irrational frequencies and a learned projection. The expected benefit is reduced positional aliasing and more uniform position-feature correlations at negligible inference cost.