A Logarithmic Fluctuation Hierarchy for Sequential Interacting Diffusions

What the math gives to ML

The paper shows that systems with the same deterministic mean-field limit can retain different order-sensitive information at the Gaussian fluctuation scale. Its transferable construction is a finite hierarchy of log-rank weighted prefix summaries, where successive channels preserve increasingly high moments of sequential position. These summaries can be computed online in linear time and used as a low-cost complement to causal attention or as a memory-efficient state in a sequence model. The most direct test is to add them to a small causal Transformer and evaluate long-range prediction at matched compute and memory.