Non-vanishing of multiple correlation sequences

What the math gives to ML

The paper provides a constructive counterexample showing that multiple correlation sequences can remain nonzero in a 3-step nilsystem even when every observable is orthogonal to the natural 2-step Conze–Lesigne factor. The transferable mechanism is hierarchical noncommutative memory: information stored in third-order commutator coordinates can disappear under low-order projections while still producing persistent three-time correlations. A neural implementation is a bounded nilpotent recurrent state whose update follows a truncated Baker–Campbell–Hausdorff law and whose coordinates are reduced layer-by-layer to a fundamental domain. This yields a falsifiable test: a 3-step model should retain nonzero lagged triple correlations after its first- and second-layer states are projected away, whereas a 2-step or ordinary linear recurrent baseline should show decay.